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Go Math Grade 8 Answer Key Chapter 2 Exponents and Scientific Notation

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Go Math Grade 8 Chapter 2 Exponents and Scientific Notation Answer Key

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Lesson 1: Integer Exponents

Lesson 2: Scientific Notation with Positive Powers of 10

Lesson 3: Scientific Notation with Negative Powers of 10

Lesson 4: Operations with Scientific Notation

Model Quiz

Mixed Review

Guided Practice – Integer Exponents – Page No. 36

Find the value of each power.

Question 1.
8−1 =
\(\frac{□}{□}\)

Answer:
\(\frac{1}{8}\)

Explanation:
Base = 8
Exponent = 1
8−1 = (1/8)1 = 1/8

Question 2.
6−2 =
\(\frac{□}{□}\)

Answer:
\(\frac{1}{36}\)

Explanation:
Base = 6
Exponent = 2
6−2 = (1/6)2 = 1/36

Exponents Grade 8 Worksheet Question 3.
2560 =
______

Answer:
1

Explanation:
2560
Base = 256
Exponent = 0
Anything raised to the zeroth power is 1.
2560 = 1

Question 4.
102 =
______

Answer:
100

Explanation:
Base = 10
Exponent = 2
102 = 10 × 10 = 100

Question 5.
54 =
______

Answer:
625

Explanation:
Base = 5
Exponent = 4
54 = 5 × 5 × 5 × 5 = 625

Question 6.
2−5 =
\(\frac{□}{□}\)

Answer:
\(\frac{1}{32}\)

Explanation:
Base = 2
Exponent = 5
2−5 = (1/2)5 = (1/2) × (1/2) × (1/2) × (1/2) × (1/2) = 1/32

Question 7.
4−5 =
\(\frac{□}{□}\)

Answer:
\(\frac{1}{1,024}\)

Explanation:
Base = 4
Exponent = 5
4−5 = (1/4)5 = (1/4) × (1/4) × (1/4) × (1/4) × (1/4) = 1/1,024

Question 8.
890 =
______

Answer:
1

Explanation:
890
Base = 89
Exponent = 0
Anything raised to the zeroth power is 1.
890 = 1

Exponents and Scientific Notation Unit Test Answer Key 8th Grade Question 9.
11−3 =
\(\frac{□}{□}\)

Answer:
\(\frac{1}{1,331}\)

Explanation:
Base = 11
Exponent = 3
11−3 = (1/11)3 = (1/11) × (1/11) × (1/11) = 1/1,331

Use properties of exponents to write an equivalent expression.

Question 10.
4 ⋅ 4 ⋅ 4 = 4?
Type below:
_____________

Answer:
43

Explanation:
The same number 4 is multiplying 3 times.
The number of times a term is multiplied is called the exponent.
So the base is 4 and the exponent is 3
4 ⋅ 4 ⋅ 4 = 43

Question 11.
(2 ⋅ 2) ⋅ (2 ⋅ 2 ⋅ 2) = 2? ⋅ 2? = 2?
Type below:
_____________

Answer:
25

Explanation:
The same number 2 is multiplying 5 times.
The number of times a term is multiplied is called the exponent.
So the base is 2 and the exponent is 5
(2 ⋅ 2) ⋅ (2 ⋅ 2 ⋅ 2) = 22 ⋅ 23 = 25

Question 12.
\(\frac { { 6 }^{ 7 } }{ { 6 }^{ 5 } } \) = \(\frac{6⋅6⋅6⋅6⋅6⋅6⋅6}{6⋅6⋅6⋅6⋅6}\) = 6?
Type below:
_____________

Answer:
62

Explanation:
\(\frac { { 6 }^{ 7 } }{ { 6 }^{ 5 } } \) = \(\frac{6⋅6⋅6⋅6⋅6⋅6⋅6}{6⋅6⋅6⋅6⋅6}\)
Cancel the common factors
6.6
Base = 6
Exponent = 2
62

Question 13.
\(\frac { { 8 }^{ 12 } }{ { 8 }^{ 9 } } \) = 8?-? = 8?
Type below:
_____________

Answer:
83

Explanation:
\(\frac { { 8 }^{ 12 } }{ { 8 }^{ 9 } } \)
Bases are common. So, the exponents are subtracted
812-9 = 83

Add and Subtract Scientific Notation Calculator Question 14.
510 ⋅ 5 ⋅ 5 = 5?
Type below:
_____________

Answer:
512

Explanation:
Bases are common and multiplied. So, the exponents are added
Base = 5
Exponents = 10 + 1 + 1 = 12
512

Question 15.
78 ⋅ 75 = 7?
Type below:
_____________

Answer:
713

Explanation:
Bases are common and multiplied. So, the exponents are added
Base = 7
Exponents = 8 + 5 = 13
713

Question 16.
(62)4 = (6 ⋅ 6)? = (6 ⋅ 6) ⋅ (6 ⋅ 6) ⋅ (? ⋅ ?) ⋅ ? = 6?
Type below:
_____________

Answer:
68

Explanation:
(62)4 = (6 ⋅ 6)4 = (6 ⋅ 6) ⋅ (6 ⋅ 6) ⋅ (6 ⋅ 6) ⋅ (6 ⋅ 6) = 62 ⋅ 62 . 62 ⋅ 62
Bases are common and multiplied. So, the exponents are added
= 62+2+2+2
68

Question 17.
(33)3 = (3 ⋅ 3 ⋅ 3)3 = (3 ⋅ 3 ⋅ 3) ⋅ (? ⋅ ? ⋅ ?) ⋅ ? = 3?
Type below:
______________

Answer:
39

Explanation:
(3 ⋅ 3 ⋅ 3) ⋅ (3 ⋅ 3 ⋅ 3) ⋅ (3 ⋅ 3 ⋅ 3) = 33 ⋅ 33 ⋅ 33
Bases are common and multiplied. So, the exponents are added
33 + 3 + 3
39

Simplify each expression.

Question 18.
(10 − 6)3⋅42 + (10 + 2)2
______

Answer:
1,168

Explanation:
4³. 4² + (12)² = 45 + (12)² = 45 + (12 . 12)²
45 + (144) = 1,024 + 144 = 1,168

Question 19.
\(\frac { { (12-5) }^{ 7 } }{ { [(3+4)^{ 2 }] }^{ 2 } } \)
________

Answer:
343

Explanation:
77 ÷ (7²)² = 77 ÷ 74
77-4

7 . 7 . 7 = 343

ESSENTIAL QUESTION CHECK-IN

Question 20.
Summarize the rules for multiplying powers with the same base, dividing powers with the same base, and raising a power to a power.
Type below:
______________

Answer:
The exponent “product rule” tells us that, when multiplying two powers that have the same base, you can add the exponents.
The quotient rule tells us that we can divide two powers with the same base by subtracting the exponents.
The “power rule” tells us that to raise a power to a power, just multiply the exponents.

Independent Practice – Integer Exponents – Page No. 37

Question 21.
Explain why the exponents cannot be added in the product 123 ⋅ 113.
Type below:
______________

Answer:
The exponent “product rule” tells us that, when multiplying two powers that have the same base, you can add the exponents.
The bases are not the same in the given problem.
=> (12)³ x (11)³
If we solve this equation following the rule of exponent will get the correct answer:
=> (12 x 12 x 12) x (11 x 11 x 11)
=> 1728 X 1331
=> The answer is 2 299 968
But if we add the exponent, the answer would be wrong
=> (12)³ x (11)³
=> 132^6
=> 5289852801024 which is wrong.

Question 22.
List three ways to express 35 as a product of powers.
Type below:
______________

Answer:
3¹ . 34
3² . 33
3³ . 32

Question 23.
Astronomy
The distance from Earth to the moon is about 224 miles. The distance from Earth to Neptune is about 227 miles. Which distance is the greater distance and about how many times greater is it?
_______ times

Answer:
(22)³ or 10,648 times

Explanation:
The distance from Earth to the moon is about 224 miles. The distance from Earth to Neptune is about 227 miles.
227 – 224  = (22)³
The greatest distance is from Earth to Neptune
The distance from Earth to Neptune is greater by (22)³ or 10,648 miles

Question 24.
Critique Reasoning
A student claims that 83 ⋅ 8-5 is greater than 1. Explain whether the student is correct or not.
______________

Answer:
83 ⋅ 8-5 is = 8-2
(1/8)²
(1/8) . (1/8) = 1/64 = 0.015
The student is not correct.

Find the missing exponent.

Question 25.
(b2)? = b-6
_______

Answer:
(b2)-8

Explanation:
(b2)? = b-6
(b-6) = b2-8
(b2-8) = b2 . b-8
(b2)-8 = b-6

Grade 8 Lesson 1 Properties of Integer Exponents Quiz Answer Key Question 26.
x? ⋅ x6 = x9
_______

Answer:

Explanation:
x? ⋅ x6 = x9
x9 = x3 + 6
x³ x6

Question 27.
\(\frac { { y }^{ 25 } }{ { y }^{ ? } } \) = y6
_______

Answer:
y25 ÷ y16

Explanation:
\(\frac { { y }^{ 25 } }{ { y }^{ ? } } \) = y
y6 = y25 – 16
y25 ÷ y16

Question 28.
Communicate Mathematical Ideas
Why do you subtract exponents when dividing powers with the same base?
Type below:
______________

Answer:
To divide exponents (or powers) with the same base, subtract the exponents. The division is the opposite of multiplication, so it makes sense that because you add exponents when multiplying numbers with the same base, you subtract the exponents when dividing numbers with the same base.

Question 29.
Astronomy
The mass of the Sun is about 2 × 1027 metric tons, or 2 × 1030 kilograms. How many kilograms are in one metric ton?
________ kgs in one metric ton

Answer:
1,000 kgs in one metric ton

Explanation:
The mass of the Sun is about 2 × 1027 metric tons, or 2 × 1030 kilograms.
2 × 1027 metric tons = 2 × 1030 ki
1 metric ton = 2 × 1030 ki ÷ 2 × 1027 = (10)³ = 1,000 kgs in one metric ton

Question 30.
Represent Real-World Problems
In computer technology, a kilobyte is 210 bytes in size. A gigabyte is 230 bytes in size. The size of a terabyte is the product of the size of a kilobyte and the size of a gigabyte. What is the size of a terabyte?
Type below:
______________

Answer:
240 bytes

Explanation:
In computer technology, a kilobyte is 210 bytes in size. A gigabyte is 230 bytes in size. The size of a terabyte is the product of the size of a kilobyte and the size of a gigabyte.
terabyte = 210 bytes × 230 bytes = 210+30 bytes = 240 bytes

Integer Exponents – Page No. 38

Question 31.
Write equivalent expressions for x7 ⋅ x-2 and \(\frac { { x }^{ 7 } }{ { x }^{ 2 } } \). What do you notice? Explain how your results relate to the properties of integer exponents.
Type below:
______________

Answer:
x^a * x^b = x^(a+b)
and
x^-a = 1/x^a
Therefore, x^7 * x^-2 = x^7/x^2 = x^5
or
x^7 * x^-2 = x^(7-2) = x^5
x^7 / x^2 = x^7 * x^-2

A toy store is creating a large window display of different colored cubes stacked in a triangle shape. The table shows the number of cubes in each row of the triangle, starting with the top row.
Go Math Grade 8 Answer Key Chapter 2 Exponents and Scientific Notation Lesson 1: Integer Exponents img 1

Question 32.
Look for a Pattern
Describe any pattern you see in the table.
Type below:
______________

Answer:
As the number of rows increased, the number of cubes in each row by a multiple of 3.

Exponents Questions and Answers Grade 8 Question 33.
Using exponents, how many cubes will be in Row 6? How many times as many cubes will be in Row 6 than in Row 3?
_______ times more cubes

Answer:
(33) times more cubes

Explanation:
For row 6, the number of cubes in each row = (36)
(36) ÷ (33) = (36-3) = (33)
(33) times more cubes

Question 34.
Justify Reasoning
If there are 6 rows in the triangle, what is the total number of cubes in the triangle? Explain how you found your answer.
______ cubes

Answer:
1,092 cubes

Explanation:
(31) + (32) + (33) + (34) + (35) + (36)
3 + 9 + 27 + 81 + 243 + 729 = 1,092

H.O.T.

Focus on Higher Order Thinking

Question 35.
Critique Reasoning
A student simplified the expression \(\frac { { 6 }^{ 2 } }{ { 36 }^{ 2 } } \) as \(\frac{1}{3}\). Do you agree with this student? Explain why or why not.
______________

Answer:
\(\frac { { 6 }^{ 2 } }{ { 36 }^{ 2 } } \)
(62) ÷ (62
(62) ÷ (64)
(62 – 4)
(6-2) = 1/36
I don’t agree with the student

Question 36.
Draw Conclusions
Evaluate –an when a = 3 and n = 2, 3, 4, and 5. Now evaluate (–a)n when a = 3 and n = 2, 3, 4, and 5. Based on this sample, does it appear that –an = (–a)n? If not, state the relationships, if any, between –an and (–a)n.
Type below:
______________

Answer:
–an when a = 3 and n = 2, 3, 4, and 5.
-3n
-(32 )= -9
(–a)n = -3 . -3 = 9
–an = (–a)n are not equal.

Properties of Integer Exponents Worksheet Question 37.
Persevere in Problem-Solving
A number to the 12th power divided by the same number to the 9th power equals 125. What is the number?
_______

Answer:
Let’s call our number a.
(a12 ) ÷ (a9 )
(a12-9 ) = (a3 )
(a3 ) = 125
a = (125)1/3
a = 5

Guided Practice – Scientific Notation with Positive Powers of 10 – Page No. 42

Write each number in scientific notation.

Question 1.
58,927
(Hint: Move the decimal left 4 places)
Type below:
______________

Answer:
5.8927 × (10)4

Explanation:
58,927
Move the decimal left 4 places
5.8927 × (10)4

Scientific Notation Exercises Worksheet Question 2.
1,304,000,000
(Hint: Move the decimal left 9 places.)
Type below:
______________

Answer:
1.304 × (10)9

Explanation:
1,304,000,000
Move the decimal left 9 places
1.304 × (10)9

Question 3.
6,730,000
Type below:
______________

Answer:

Explanation:
6,730,000
Move the decimal left 6 places
6.73 × (10)6

Question 4.
13,300
Type below:
______________

Answer:

Explanation:
13,300
Move the decimal left 4 places
1.33 × (10)4

Question 5.
An ordinary quarter contains about 97,700,000,000,000,000,000,000 atoms.
Type below:
______________

Answer:

Explanation:
97,700,000,000,000,000,000,000
Move the decimal left 22 places
9.77 × (10)22

Question 6.
The distance from Earth to the Moon is about 384,000 kilometers.
Type below:
______________

Answer:
3.84 × (10)6

Explanation:
384,000
Move the decimal left 6 places
3.84 × (10)6

Write each number in standard notation.

Question 7.
4 × 105
(Hint: Move the decimal right 5 places.)
Type below:
______________

Answer:
400,000

Explanation:
4 × 105
Move the decimal right 5 places
400,000

Lesson 2 Problem Set 2.1 Answer Key Question 8.
1.8499 × 109
(Hint: Move the decimal right 9 places.)
Type below:
______________

Answer:
1849900000

Explanation:
1.8499 × 109
Move the decimal right 9 places
1849900000

Question 9.
6.41 × 103
Type below:
______________

Answer:
6410

Explanation:
6.41 × 103
Move the decimal right 3 places
6410

Question 10.
8.456 × 107
Type below:
______________

Answer:
84560000

Explanation:
8.456 × 107
Move the decimal right 7 places
84560000

Question 11.
8 × 105
Type below:
______________

Answer:
800,000

Explanation:
8 × 105
Move the decimal right 5 places
800,000

Question 12.
9 × 1010
Type below:
______________

Answer:
90000000000

Explanation:
9 × 1010
Move the decimal right 10 places
90000000000

Scientific Notation Worksheet 8th Grade Pdf Question 13.
Diana calculated that she spent about 5.4 × 104 seconds doing her math homework during October. Write this time in standard notation.
Type below:
______________

Answer:
5400

Explanation:
Diana calculated that she spent about 5.4 × 104 seconds doing her math homework during October.
5.4 × 104
Move the decimal right 4 places

5400

Question 14.
The town recycled 7.6 × 106 cans this year. Write the number of cans in standard notation
Type below:
______________

Answer:
7600000

Explanation:
The town recycled 7.6 × 106 cans this year.
7.6 × 106
Move the decimal right 10 places
7600000

ESSENTIAL QUESTION CHECK-IN

Question 15.
Describe how to write 3,482,000,000 in scientific notation.
Type below:
______________

Answer:
3.482 × (10)9

Explanation:
3,482,000,000
Move the decimal left 9 places
3.482 × (10)9

Independent Practice – Scientific Notation with Positive Powers of 10 – Page No. 43

Paleontology

Use the table for problems 16–21. Write the estimated weight of each dinosaur in scientific notation.
Go Math Grade 8 Answer Key Chapter 2 Exponents and Scientific Notation Lesson 2: Scientific Notation with Positive Powers of 10 img 2

Question 16.
Apatosaurus ______________
Type below:
______________

Answer:
6.6 × (10)4

Explanation:
66,000
Move the decimal left 4 places
6.6 × (10)4

Question 17.
Argentinosaurus ___________
Type below:
______________

Answer:
2.2 × (10)5

Explanation:
220,000
Move the decimal left 5 places
2.2 × (10)5

Question 18.
Brachiosaurus ______________
Type below:
______________

Answer:
1 × (10)5

Explanation:
100,000
Move the decimal left 5 places
1 × (10)5

Lesson 2 Extra Practice Powers and Exponents Answer Key Question 19.
Camarasaurus ______________
Type below:
______________

Answer:
4 × (10)4

Explanation:
40,000
Move the decimal left 4 places
4 × (10)4

Question 20.
Cetiosauriscus ____________
Type below:
______________

Answer:
1.985 × (10)4

Explanation:
19,850
Move the decimal left 4 places
1.985 × (10)4

Question 21.
Diplodocus _____________
Type below:
______________

Answer:
5 × (10)4

Explanation:
50,000
Move the decimal left 4 places
5 × (10)4

Question 22.
A single little brown bat can eat up to 1,000 mosquitoes in a single hour. Express in scientific notation how many mosquitoes a little brown bat might eat in 10.5 hours.
Type below:
______________

Answer:
1.05 × (10)4

Explanation:
(1000 x 10.5) = 10500.
The little brown bat can eat 10500 mosquitoes in 10.5 hours.
1.05 × (10)4

Question 23.
Multistep
Samuel can type nearly 40 words per minute. Use this information to find the number of hours it would take him to type 2.6 × 105 words.
Type below:
______________

Answer:
Samuel can type 40 words per minute.
Then how many hours will it take for him to type 2.6 words times 10 to the power of five words
2.6 words time 10 to the power of 5
2.6 × (10)4
2.6 x 100 000 = 260 000 words in all.
Now, we need to find the number of words Samuel can type in an hour
40 words/minutes, in 1 hour there are 60 minutes
40 x 60
2,400 words /hour
Now, let’s divide the total of words he needs to type to the number of words he can type in an hour
260 000 / 2 400
108.33 hours.

Question 24.
Entomology
A tropical species of mite named Archegozetes longisetosus is the record holder for the strongest insect in the world. It can lift up to 1.182 × 103 times its own weight.
a. If you were as strong as this insect, explain how you could find how many pounds you could lift.
Type below:
______________

Answer:
Number of pounds you can lift by 1.182 × 103 by your weight

Question 24.
b. Complete the calculation to find how much you could lift, in pounds, if you were as strong as an Archegozetes longisetosus mite. Express your answer in both scientific notation and standard notation.
Type below:
______________

Answer:
scientific notation: 1.182 × 105
standard notation: 118200

Explanation:
1.182 × 103 × 102
1.182 × 105
118200

Question 25.
During a discussion in science class, Sharon learns that at birth an elephant weighs around 230 pounds. In four herds of elephants tracked by conservationists, about 20 calves were born during the summer. In scientific notation, express approximately how much the calves weighed all together.
Type below:
______________

Answer:
4.6 × 103

Explanation:
During a discussion in science class, Sharon learns that at birth an elephant weighs around 230 pounds. In four herds of elephants tracked by conservationists, about 20 calves were born during the summer.
Total weight of the claves = 230 × 20 = 4600
Move the decimal left 3 places
4.6 × 103

Question 26.
Classifying Numbers
Which of the following numbers are written in scientific notation?
0.641 × 103          9.999 × 104
2 × 101                 4.38 × 510
Type below:
______________

Answer:
0.641 × 103
4.38 × 510

Scientific Notation with Positive Powers of 10 – Page No. 44

Question 27.
Explain the Error
Polly’s parents’ car weighs about 3500 pounds. Samantha, Esther, and Polly each wrote the weight of the car in scientific notation. Polly wrote 35.0 × 102, Samantha wrote 0.35 × 104, and Esther wrote 3.5 × 104.
a. Which of these girls, if any, is correct?
______________

Answer:
None of the girls is correct

Question 27.
b. Explain the mistakes of those who got the question wrong.
Type below:
______________

Answer:
Polly did not express the number such first part is greater than or equal to 1 and less than 10
Samantha did not express the number such first part is greater than or equal to 1 and less than 10
Esther did not express the exponent of 10 correctly

Question 28.
Justify Reasoning
If you were a biologist counting very large numbers of cells as part of your research, give several reasons why you might prefer to record your cell counts in scientific notation instead of standard notation.
Type below:
______________

Answer:
It is easier to comprehend the magnitude of large numbers when in scientific notation as multiple zeros in the number are removed and express as an exponent of 10.
It is easier to compare large numbers when in scientific notation as numbers are be expressed as a product of a number greater than or equal to 1 and less than 10
It is easier to multiply the numbers in scientific notation.

H.O.T.

Focus on Higher Order Thinking

Question 29.
Draw Conclusions
Which measurement would be least likely to be written in scientific notation: number of stars in a galaxy, number of grains of sand on a beach, speed of a car, or population of a country? Explain your reasoning.
Type below:
______________

Answer:
speed of a car

Explanation:
As we know scientific notation is used to express measurements that are extremely large or extremely small.
The first two are extremely large, then, they could be expressed in scientific notation.
If we compare the speed of a car and the population of a country, it is clear that the larger will be the population of a country.
Therefore, it is more likely to express that in scientific notation, so the answer is the speed of a car.

Question 30.
Analyze Relationships
Compare the two numbers to find which is greater. Explain how you can compare them without writing them in standard notation first.
4.5 × 106              2.1 × 108
Type below:
______________

Answer:
2.1 × 108

Explanation:
2.1 × 108 is greater because the power of 10 is greater in  2.1 × 108

Question 31.
Communicate Mathematical Ideas
To determine whether a number is written in scientific notation, what test can you apply to the first factor, and what test can you apply to the second factor?
Type below:
______________

Answer:
The first term must have one number before the decimal point
The second term (factor) must be 10 having some power.

Guided Practice – Scientific Notation with Negative Powers of 10 – Page No. 48

Write each number in scientific notation.

Question 1.
0.000487
Hint: Move the decimal right 4 places.
Type below:
______________

Answer:
4.87 × 10-4

Explanation:
0.000487
Move the decimal right 4 places
4.87 × 10-4

How to Multiply Scientific Notation with Negative Exponents Question 2.
0.000028
Hint: Move the decimal right 5 places
Type below:
______________

Answer:
2.8 × 10-5

Explanation:
0.000028
Move the decimal right 5 places
2.8 × 10-5

Question 3.
0.000059
Type below:
______________

Answer:
5.9 × 10-5

Explanation:
0.000059
Move the decimal right 5 places
5.9 × 10-5

Question 4.
0.0417
Type below:
______________

Answer:
4.17 × 10-2

Explanation:
0.0417
Move the decimal right 2 places
4.17 × 10-2

Question 5.
Picoplankton can be as small as 0.00002 centimeters.
Type below:
______________

Answer:
2 × 10-5

Explanation:
0.00002
Move the decimal right 5 places
2 × 10-5

Question 6.
The average mass of a grain of sand on a beach is about 0.000015 grams.
Type below:
______________

Answer:
1.5 × 10-5

Explanation:
0.000015
Move the decimal right 5 places
1.5 × 10-5

Write each number in standard notation.

Question 7.
2 × 10-5
Hint: Move the decimal left 5 places.
Type below:
______________

Answer:
0.00002

Explanation:
2 × 10-5
Move the decimal left 5 places
0.00002

Question 8.
3.582 × 10-6
Hint: Move the decimal left 6 places.
Type below:
______________

Answer:
0.000003582

Explanation:
3.582 × 10-6
Move the decimal left 6 places
0.000003582

Question 9.
8.3 × 10-4
Type below:
______________

Answer:
0.00083

Explanation:
8.3 × 10-4
Move the decimal left 4 places
0.00083

Question 10.
2.97 × 10-2
Type below:
______________

Answer:
0.0297

Explanation:
2.97 × 10-2
Move the decimal left 2 places
0.0297

Question 11.
9.06 × 10-5
Type below:
______________

Answer:
0.0000906

Explanation:
9.06 × 10-5
Move the decimal left 5 places
0.0000906

Question 12.
4 × 10-5
Type below:
______________

Answer:
0.00004

Explanation:
4 × 10-5
Move the decimal left 5 places
0.00004

Question 13.
The average length of a dust mite is approximately 0.0001 meters. Write this number in scientific notation.
Type below:
______________

Answer:
1 × 10-4

Explanation:
The average length of a dust mite is approximately 0.0001 meters.
0.0001
Move the decimal right 4 places
1 × 10-4

Question 14.
The mass of a proton is about 1.7 × 10-24 grams. Write this number in standard notation.
Type below:
______________

Answer:
0.000000000000000000000017

Explanation:
The mass of a proton is about 1.7 × 10-24 grams.
1.7 × 10-24
Move the decimal left 24 places
0.000000000000000000000017

ESSENTIAL QUESTION CHECK-IN

Question 15.
Describe how to write 0.0000672 in scientific notation.
Type below:
______________

Answer:
6.72 × 10-5

Explanation:
0.0000672
Move the decimal right 5 places
6.72 × 10-5

Independent Practice – Scientific Notation with Negative Powers of 10 – Page No. 49

Use the table for problems 16–21. Write the diameter of the fibers in scientific notation.
Go Math Grade 8 Answer Key Chapter 2 Exponents and Scientific Notation Lesson 3: Scientific Notation with Negative Powers of 10 img 3

Question 16.
Alpaca _______
Type below:
______________

Answer:
2.77 × 10-3

Explanation:
0.00277
Move the decimal right 3 places
2.77 × 10-3

Question 17.
Angora rabbit _____________
Type below:
______________

Answer:
1.3 × 10-3

Explanation:
0.0013
Move the decimal right 3 places
1.3 × 10-3

Question 18.
Llama ____________
Type below:
______________

Answer:
3.5 × 10-3

Explanation:
0.0035
Move the decimal right 3 places
3.5 × 10-3

Question 19.
Angora goat ____________
Type below:
______________

Answer:
4.5 × 10-3

Explanation:
0.0045
Move the decimal right 3 places
4.5 × 10-3

Question 20.
Orb web spider ___________
Type below:
______________

Answer:
1.5 × 10-2

Explanation:
0.015
Move the decimal right 2 places
1.5 × 10-2

Question 21.
Vicuña __________
Type below:
______________

Answer:
8 × 10-4

Explanation:
0.0008
Move the decimal right 4 places
8 × 10-4

Question 22.
Make a Conjecture
Which measurement would be least likely to be written in scientific notation: the thickness of a dog hair, the radius of a period on this page, the ounces in a cup of milk? Explain your reasoning.
Type below:
______________

Answer:
The ounces in a cup of milk would be least likely to be written in scientific notation. The ounces in a cup of milk is correct.
Scientific notation is used for either very large or extremely small numbers.
The thickness of dog hair is very small as the hair is thin. Hence can be converted to scientific notation.
The radius of a period on this page is also pretty small. Hence can be converted to scientific notation.
The ounces in a cup of milk. There are 8 ounces in a cup, so this is least likely to be written in scientific notation.

Question 23.
Multiple Representations
Convert the length 7 centimeters to meters. Compare the numerical values when both numbers are written in scientific notation
Type below:
______________

Answer:
7 centimeters convert to meters
In every 1 meter, there are 100 centimeters = 7/100 = 0.07
Therefore, in 7 centimeters there are 0.07 meters.
7 cm is a whole number while 0.07 m is a decimal number
Scientific Notation of each number
7 cm = 7 x 10°
7 m = 1 x 10¯²
Scientific notation, by the way, is an expression used by the scientist to make a large number of very small number easy to handle.

Question 24.
Draw Conclusions
A graphing calculator displays 1.89 × 1012 as 1.89E12. How do you think it would display 1.89 × 10-12? What does the E stand for?
Type below:
______________

Answer:
1.89E-12. E= Exponent

Explanation:

Question 25.
Communicate Mathematical Ideas
When a number is written in scientific notation, how can you tell right away whether or not it is greater than or equal to 1?
Type below:
______________

Answer:
A number written in scientific notation is of the form
a × 10-n where 1 ≤ a < 10 and n is an integer
The number is greater than or equal to one if n ≥ 0.

Question 26.
The volume of a drop of a certain liquid is 0.000047 liter. Write the volume of the drop of liquid in scientific notation.
Type below:
______________

Answer:
4.7 × 10-5

Explanation:
The volume of a drop of a certain liquid is 0.000047 liter.
Move the decimal right 5 places
4.7 × 10-5

Question 27.
Justify Reasoning
If you were asked to express the weight in ounces of a ladybug in scientific notation, would the exponent of the 10 be positive or negative? Justify your response.
______________

Answer:
Negative

Explanation:
Scientific notation is used to express very small or very large numbers.
Very small numbers are written in scientific notation using negative exponents.
Very large numbers are written in scientific notation using positive exponents.
Since a ladybug is very small, we would use the very small scientific notation, which uses negative exponents.

Physical Science – Scientific Notation with Negative Powers of 10 – Page No. 50

The table shows the length of the radii of several very small or very large items. Complete the table.
Go Math Grade 8 Answer Key Chapter 2 Exponents and Scientific Notation Lesson 3: Scientific Notation with Negative Powers of 10 img 4

Question 28.
Type below:
______________

Answer:
1.74 × (10)6

Explanation:
The moon = 1,740,000
Move the decimal left 6 places
1.74 × (10)6

Question 29.
Type below:
______________

Answer:
1.25e-10

Explanation:
1.25 × (10)-10
Move the decimal left 10 places
1.25e-10

Question 30.
Type below:
______________

Answer:
2.8 × (10)3

Explanation:
0.0028
Move the decimal left 3 places
2.8 × (10)3

Question 31.
Type below:
______________

Answer:
71490000

Explanation:
7.149 × (10)7
Move the decimal left 7 places
71490000
Question 32.
Type below:
______________

Answer:
1.82 × (10)-10

Explanation:
0.000000000182
Move the decimal right 10 places
1.82 × (10)-10

Question 33.
Type below:
______________

Answer:
3397000

Explanation:
3.397 × (10)6
Move the decimal left 6 places
3397000

Question 34.
List the items in the table in order from the smallest to the largest.
Type below:
______________

Answer:
1.82 × (10)-10
1.25 × (10)-10
2.8 × (10)3
1.74 × (10)6
3.397 × (10)6
7.149 × (10)7

H.O.T.

Focus on Higher Order Thinking

Question 35.
Analyze Relationships
Write the following diameters from least to greatest. 1.5 × 10-2m ; 1.2 × 102 m ; 5.85 × 10-3 m ; 2.3 × 10-2 m ; 9.6 × 10-1 m.
Type below:
______________

Answer:
5.85 × 10-3 m, 1.5 × 10-2m, 2.3 × 10-2 m, 9.6 × 10-1 m, 1.2 × 102 m

Explanation:
1.5 × 10-2m = 0.015
1.2 × 102 m = 120
5.85 × 10-3 m = 0.00585
2.3 × 10-2 m = 0.023
9.6 × 10-1 m = 0.96
0.00585, 0.015, 0.023, 0.96, 120

Question 36.
Critique Reasoning
Jerod’s friend Al had the following homework problem:
Express 5.6 × 10-7 in standard form.
Al wrote 56,000,000. How can Jerod explain Al’s error and how to correct it?
Type below:
______________

Answer:

Explanation:
5.6 × 10-7 in
0.000000056
Al wrote 56,000,000. AI wrote the zeroes to the right side of the 56 which is not correct. As the exponent of 10 is negative zero’s need to add to the left of the number.

Question 37.
Make a Conjecture
Two numbers are written in scientific notation. The number with a positive exponent is divided by the number with a negative exponent. Describe the result. Explain your answer.
Type below:
______________

Answer:
When the division is performed, the denominator exponent is subtracted from the numerator exponent. Subtracting a negative value from the numerator exponent will increase its value.

Guided Practice – Operations with Scientific Notation – Page No. 54

Add or subtract. Write your answer in scientific notation.

Question 1.
4.2 × 106 + 2.25 × 105 + 2.8 × 106
4.2 × 106 + ? × 10 ? + 2.8 × 106
4.2 + ? + ?
? × 10?
Type below:
______________

Answer:
4.2 × 106 + 0.225 × 10 × 105 + 2.8 × 106
Rewrite 2.25 = 0.225 × 10
(4.2 + 0.225 + 2.8) × 106
7.225 × 106

Question 2.
8.5 × 103 − 5.3 × 103 − 1.0 × 102
8.5 × 103 − 5.3 × 103 − ? × 10?
? − ? − ?
? × 10?
Type below:
______________

Answer:
8.5 × 103 − 5.3 × 103 − 0.1 × 103
(8.5 − 5.3 − 0.1) × 103
(3.1) × 103

Lesson 2 Multiplication of Numbers in Exponential Form Answer Key Question 3.
1.25 × 102 + 0.50 × 102 + 3.25 × 102
Type below:
______________

Answer:
1.25 × 102 + 0.50 × 102 + 3.25 × 102
(1.25 + 0.50 + 3.25) × 102
5 × 102

Question 4.
6.2 × 105 − 2.6 × 104 − 1.9 × 102
Type below:
______________

Answer:
6.2 × 105 − 2.6 × 104 − 1.9 × 102
6.2 × 105 − 0.26 × 105 − 0.0019 × 105
(6.2 – 0.26 – 0.0019) × 105
5.9381 × 105

Multiply or divide. Write your answer in scientific notation.

Question 5.
(1.8 × 109)(6.7 × 1012)
Type below:
______________

Answer:
12.06 × 1021

Explanation:
(1.8 × 109)(6.7 × 1012)
1.8 × 6.7 = 12.06
109+12 = 1021
12.06 × 1021

Question 6.
\(\frac { { 3.46×10 }^{ 17 } }{ { 2×10 }^{ 9 } } \)
Type below:
______________

Answer:
1.73 × 108

Explanation:
3.46/2 = 1.73
1017/109 = 1017-9 = 108
1.73 × 108

Question 7.
(5 × 1012)(3.38 × 106)
Type below:
______________

Answer:
16.9 × 1018

Explanation:
(5 × 1012)(3.38 × 106)
5 × 3.38 = 16.9
106+12 = 1018
16.9 × 1018

Question 8.
\(\frac { { 8.4×10 }^{ 21 } }{ { 4.2×10 }^{ 14 } } \)
Type below:
______________

Answer:
2 × 107

Explanation:
8.4/4.2 = 2
1021/1014 = 1021-14 = 107
2 × 107

Write each number using calculator notation.

Question 9.
3.6 × 1011
Type below:
______________

Answer:
3.6e11

Question 10.
7.25 × 10-5
Type below:
______________

Answer:
7.25e-5

Question 11.
8 × 10-1
Type below:
______________

Answer:
8e-1

Write each number using scientific notation.

Question 12.
7.6E − 4
Type below:
______________

Answer:
7.6 × 10-4

Question 13.
1.2E16
Type below:
______________

Answer:
1.2 × 1016

Question 14.
9E1
Type below:
______________

Answer:
9 × 101

ESSENTIAL QUESTION CHECK-IN

Question 15.
How do you add, subtract, multiply, and divide numbers written in scientific notation?
Type below:
______________

Answer:
Numbers with exponents can be added and subtracted only when they have the same base and exponent.
To multiply two numbers in scientific notation, multiply their coefficients and add their exponents.
To divide two numbers in scientific notation, divide their coefficients, and subtract their exponents.

Independent Practice – Operations with Scientific Notation – Page No. 55

Question 16.
An adult blue whale can eat 4.0 × 107 krill in a day. At that rate, how many krill can an adult blue whale eat in 3.65 × 102 days?
Type below:
______________

Answer:
14.6 × 109

Explanation:
(4.0 × 107 )(3.65 × 102 )
4.0 × 3.65 = 14.6
107+2  =  109
14.6 × 109

How to Multiply and Divide in Scientific Notation Question 17.
A newborn baby has about 26,000,000,000 cells. An adult has about 4.94 × 1013 cells. How many times as many cells does an adult have as a newborn? Write your answer in scientific notation.
Type below:
______________

Answer:
1.9 × 103

Explanation:
26,000,000,000 = 2.6 × 1010
4.94 × 1013
(4.94 × 1013 )/(2.6 × 1010 )
1.9 × 103

Represent Real-World Problems

The table shows the number of tons of waste generated and recovered (recycled) in 2010.
Go Math Grade 8 Answer Key Chapter 2 Exponents and Scientific Notation Lesson 4: Operations with Scientific Notation img 5

Question 18.
What is the total amount of paper, glass, and plastic waste generated?
Type below:
______________

Answer:
11.388 × 107

Explanation:
7.131 × 107 + 1.153 × 107 + 3.104 × 107
11.388 × 107

Question 19.
What is the total amount of paper, glass, and plastic waste recovered?
Type below:
______________

Answer:
5.025 × 107

Explanation:
4.457 × 107  + 0.313 × 107  + 0.255 × 107
5.025 × 107

How to Multiply Scientific Notation Question 20.
What is the total amount of paper, glass, and plastic waste not recovered?
Type below:
______________

Answer:
6.363 × 107

Explanation:
(11.388 × 107 )  – (5.025 × 107)
6.363 × 107

Question 21.
Which type of waste has the lowest recovery ratio?
Type below:
______________

Answer:
Plastics

Explanation:
7.131 × 107  – 4.457 × 107 = 2.674 × 107
1.153 × 107  – 0.313 × 107 = 0.84 × 107
3.104 × 107  – 0.255 × 107 = 2.849 × 107
Plastics have the lowest recovery ratio

Social Studies

The table shows the approximate populations of three countries.
Go Math Grade 8 Answer Key Chapter 2 Exponents and Scientific Notation Lesson 4: Operations with Scientific Notation img 6

Question 22.
How many more people live in France than in Australia?
Type below:
______________

Answer:
4.33 × 107

Explanation:
(6.48 × 107 )  – (2.15× 107)
4.33 × 107

Question 23.
The area of Australia is 2.95 × 106 square miles. What is the approximate average number of people per square mile in Australia?
Type below:
______________

Answer:
About 7 people per square mile

Explanation:
2.95 × 106 square miles = (2.15× 107)
1 square mile = (2.15× 107)/(2.95 × 106) = 7.288

Question 24.
How many times greater is the population of China than the population of France? Write your answer in standard notation.
Type below:
______________

Answer:
20.52; there are about 20 people in China for every 1 person in France.

Multiplication of Numbers in Exponential Form Answer Key Question 25.
Mia is 7.01568 × 106 minutes old. Convert her age to more appropriate units using years, months, and days. Assume each month to have 30.5 days.
Type below:
______________

Answer:
13 years 3 months 22.5 days

Explanation:
7.01568 × 106 minutes
(7.01568 × 106 minutes) ÷ (6 × 101)(2.4 × 101)(1.2 × 101)(3.05 × 101)
= (1.331 × 101)
= 13 years 3 months 22.5 days

Operations with Scientific Notation – Page No. 56

Question 26.
Courtney takes 2.4 × 104 steps during her long-distance run. Each step covers an average of 810 mm. What total distance (in mm) did Courtney cover during her run? Write your answer in scientific notation. Then convert the distance to the more appropriate unit kilometers. Write that answer in standard form.
______ km

Answer:
19.4 km

Explanation:
Courtney takes 2.4 × 104 steps during her long-distance run. Each step covers an average of 810 mm.
(2.4 × 104 steps) × 810mm
(2.4 × 104 ) × (8.1 × 102 )
The total distance covered = (19.44 × 106 )
Convert to unit kilometers:
(19.44 × 106 ) × (1 × 10-6 )
(1.94 × 101 )
19.4 km

Question 27.
Social Studies
The U.S. public debt as of October 2010 was $9.06 × 1012. What was the average U.S. public debt per American if the population in 2010 was 3.08 × 108 people?
$ _______

Answer:
$29,400 per American

Explanation:
($9.06 × 1012.)/(3.08 × 108 )
($2.94 × 104.) = $29,400 per American

H.O.T.

Focus on Higher Order Thinking

Question 28.
Communicate Mathematical Ideas
How is multiplying and dividing numbers in scientific notation different from adding and subtracting numbers in scientific notation?
Type below:
______________

Answer:
When you multiply or divide in scientific notation, you just add or subtract the exponents. When you add or subtract in scientific notation, you have to make the exponents the same before you can do anything else.

Question 29.
Explain the Error
A student found the product of 8 × 106 and 5 × 109 to be 4 × 1015. What is the error? What is the correct product?
Type below:
______________

Answer:
The error the student makes is he multiplies the terms instead of the addition.

Explanation:
product of 8 × 106 and 5 × 109
40 × 1015
4 × 1016
The student missed the 10 while multiplying the product of 8 × 106 and 5 × 109

Question 30.
Communicate Mathematical Ideas
Describe a procedure that can be used to simplify \(\frac { { (4.87×10 }^{ 12 }) – { (7×10 }^{ 10 }) }{ { (3×10 }^{ 7 })-{ (6.1×10 }^{ 8 }) } \). Write the expression in scientific notation in simplified form.
Type below:
______________

Answer:
\(\frac { { (4.87×10 }^{ 12 }) – { (7×10 }^{ 10 }) }{ { (3×10 }^{ 7 })-{ (6.1×10 }^{ 8 }) } \)
\(\frac { { (487×10 }^{ 10 }) – { (7×10 }^{ 10 }) }{ { (3×10 }^{ 7 })-{ (61×10 }^{ 7 }) } \)
(480 × 1010 )/(64 × 107 )
7.50 × 10³

2.1 Integer Exponents – Model Quiz – Page No. 57

Find the value of each power.

Question 1.
3-4
\(\frac{□}{□}\)

Answer:
\(\frac{1}{81}\)

Explanation:
Base = 3
Exponent = 4
3-4 = (1/3)4 = 1/81

Question 2.
350
______

Answer:
1

Explanation:
350
Base = 35
Exponent = 0
Anything raised to the zeroth power is 1.
350 = 1

Question 3.
44
______

Answer:
256

Explanation:
Base = 4
Exponent = 4
44 = 4 . 4 . 4 . 4 = 2561

Use the properties of exponents to write an equivalent expression.

Question 4.
83 ⋅ 87
Type below:
____________

Answer:
810

Explanation:
83 ⋅ 87
83+7
810

Question 5.
\(\frac { 12^{ 6 } }{ 12^{ 2 } } \)
Type below:
____________

Answer:
124

Explanation:
126 ÷ 122
126-2
124

Question 6.
(103)5
Type below:
____________

Answer:
108

Explanation:
(103)5
(103+5)
(108)

2.2 Scientific Notation with Positive Powers of 10

Convert each number to scientific notation or standard notation.

Question 7.
2,000
Type below:
____________

Answer:
2 × (103)

Explanation:
2 × 1,000
Move the decimal left 3 places
2 × (103)

Question 8.
91,007,500
Type below:
____________

Answer:
9.10075 × (107)

Explanation:
91,007,500
Move the decimal left 7 places
9.10075 × (107)

Question 9.
1.0395 × 109
Type below:
____________

Answer:
1039500000

Explanation:
1.0395 × 109
Move the decimal right 9 places
1039500000

Question 10.
4 × 102
Type below:
____________

Answer:
400

Explanation:
4 × 102
Move the decimal right 2 places
400

2.3 Scientific Notation with Negative Powers of 10

Convert each number to scientific notation or standard notation.

Question 11.
0.02
Type below:
____________

Answer:
2 × 10-2

Explanation:
0.02
Move the decimal right 2 places
2 × 10-2

Practice and Homework Lesson 2.3 Answer Key Question 12.
0.000701
Type below:
____________

Answer:
7.01 × 10-4

Explanation:
0.000701
Move the decimal right 4 places
7.01 × 10-4

Question 13.
8.9 × 10-5
Type below:
____________

Answer:
0.000089

Explanation:
8.9 × 10-5
Move the decimal left 5 places
0.000089

Question 14.
4.41 × 10-2
Type below:
____________

Answer:
0.0441

Explanation:
4.41 × 10-2
Move the decimal left 2 places
0.0441

2.4 Operations with Scientific Notation

Perform the operation. Write your answer in scientific notation.

Question 15.
7 × 106 − 5.3 × 106
Type below:
____________

Answer:
1.7 × 106

Explanation:
7 × 106 − 5.3 × 106
(7 – 5.3) × 106
1.7 × 106

Question 16.
3.4 × 104 + 7.1 × 105
Type below:
____________

Answer:
7.44 × 104

Explanation:
3.4 × 104 + 7.1 × 105
0.34 × 105 + 7.1 × 105
(0.34 + 7.1) × 105
7.44 × 105

Question 17.
(2 × 104)(5.4 × 106)
Type below:
____________

Answer:
10.8 × 1010

Explanation:
(2 × 104)(5.4 × 106)
(2 × 5.4)(104 × 106)
10.8 × 1010

Question 18.
\(\frac { 7.86×10^{ 9 } }{ 3×10^{ 4 } } \)
Type below:
____________

Answer:
2.62 × 105

Explanation:
7.86/3 = 2.62
109/104 = 105
2.62 × 105

Question 19.
Neptune’s average distance from the Sun is 4.503×109 km. Mercury’s average distance from the Sun is 5.791 × 107 km. About how many times farther from the Sun is Neptune than Mercury? Write your answer in scientific notation.
Type below:
____________

Answer:
(0.7776 × 102 km) = 77.76 times

Explanation:
As Neptune’s average distance from the sun is 4.503×109 km and Mercury’s is 5.791 × 107 km
(4.503×109 km)/(5.791 × 107 km)
(0.7776 × 109-7 km)
(0.7776 × 102 km)
77.76 times

Essential Question

Question 20.
How is scientific notation used in the real world?
Type below:
____________

Answer:
Scientific notation is used to write very large or very small numbers using fewer digits.

Selected Response – Mixed Review – Page No. 58

Question 1.
Which of the following is equivalent to 6-3?
Options:
a. 216
b. \(\frac{1}{216}\)
c. −\(\frac{1}{216}\)
d. -216

Answer:
b. \(\frac{1}{216}\)

Explanation:
Base = 6
Exponent = 3
63 = (1/6)3 = 1/216

Question 2.
About 786,700,000 passengers traveled by plane in the United States in 2010. What is this number written in scientific notation?
Options:
a. 7,867 × 105 passengers
b. 7.867 × 102 passengers
c. 7.867 × 108 passengers
d. 7.867 × 109 passengers

Answer:
c. 7.867 × 108 passengers

Explanation:
786,700,000
Move the decimal left 8 places
7.867 × 108 passengers

Question 3.
In 2011, the population of Mali was about 1.584 × 107 people. What is this number written in standard notation?
Options:
a. 1.584 people
b. 1,584 people
c. 15,840,000 people
d. 158,400,000 people

Answer:
c. 15,840,000 people

Explanation:
1.584 × 107
Move the decimal right 7 places
15,840,000 people

Question 4.
The square root of a number is between 7 and 8. Which could be the number?
Options:
a. 72
b. 83
c. 51
d. 66

Answer:
c. 51

Explanation:
7²= 49
8²=64
(49+64)/2
56.5

Question 5.
Each entry-level account executive in a large company makes an annual salary of $3.48 × 104. If there are 5.2 × 102 account executives in the company, how much do they make in all?
Options:
a. $6.69 × 101
b. $3.428 × 104
c. $3.532 × 104
d. $1.8096 × 107

Answer:
d. $1.8096 × 107

Explanation:
Each entry-level account executive in a large company makes an annual salary of $3.48 × 104. If there are 5.2 × 102 account executives in the company,
($3.48 × 104)( 5.2 × 102)
$1.8096 × 107

Question 6.
Place the numbers in order from least to greatest.
0.24,4 × 10-2, 0.042, 2 × 10-4, 0.004
Options:
a. 2 × 10-4, 4 × 10-2, 0.004, 0.042, 0.24
b. 0.004, 2 × 10-4, 0.042, 4 × 10-2, 0.24
c. 0.004, 2 × 10-4, 4 × 10-2, 0.042, 0.24
d. 2 × 10-4, 0.004, 4 × 10-2, 0.042, 0.24

Answer:
d. 2 × 10-4, 0.004, 4 × 10-2, 0.042, 0.24

Explanation:
2 × 10-4 = 0.0002
4 × 10-2 = 0.04

Question 7.
Guillermo is 5 \(\frac{5}{6}\) feet tall. What is this number of feet written as a decimal?
Options:
a. 5.7 feet
b. 5.\(\bar{7}\) feet
c. 5.83 feet
d. 5.8\(\bar{3}\) feet

Answer:
c. 5.83 feet

Question 8.
A human hair has a width of about 6.5 × 10-5 meters. What is this width written in standard notation?
Options:
a. 0.00000065 meter
b. 0.0000065 meter
c. 0.000065 meter
d. 0.00065 meter

Answer:
c. 0.000065 meter

Explanation:
6.5 × 10-5 meter = 0.000065

Mini-Task

Question 9.
Consider the following numbers: 7000, 700, 70, 0.7, 0.07, 0.007
a. Write the numbers in scientific notation.
Type below:
_____________

Answer:
7000 = 7 × 10³
700 = 7 × 10²
70 = 7 × 10¹
0.7 = 7 × 10¯¹
0.07 = 7 × 10¯²
0.007 = 7 × 10¯³

Question 9.
b. Look for a pattern in the given list and the list in scientific notation. Which numbers are missing from the lists?
Type below:
_____________

Answer:
In the given list the decimal is moving to the left by one place. From the scientific notation, numbers are decreasing by 10. The number missing is 7

Question 9.
c. Make a conjecture about the missing numbers.
Type below:
_____________

Answer:
The numbers will continue to decrease by 10 in the given list.

Conclusion:

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Go Math Grade 8 Answer Key Chapter 1 Real Numbers

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Go Math Grade 8 Chapter 1 Real Numbers Answer Key

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Lesson 1: Rational and Irrational Numbers

Lesson 2: Sets of real Numbers

Lesson 3: Ordering Real Numbers

Model Quiz

Mixed Review

Guided Practice – Rational and Irrational Numbers – Page No. 12

Write each fraction or mixed number as a decimal.

Question 1.
\(\frac{2}{5}\) =

Answer:
0.4

Explanation:
\(\frac{2}{5}\) = \(\frac{2 × 2}{5 × 2}\) = \(\frac{4}{10}\) = 0.4

Lesson 1-1 Rational and Irrational Numbers Reteach Answer Key Question 2.
\(\frac{8}{9}\) =

Answer:
0.88

Explanation:
\(\frac{8}{9}\) = \(\frac{8 × 10}{9 × 10}\) = \(\frac{80}{9 × 10}\) = \(\frac{8.88}{10}\) = 0.88

Question 3.
3 \(\frac{3}{4}\) =

Answer:
3.75

Explanation:
3 \(\frac{3}{4}\) =\(\frac{15}{4}\) = 3.75

Question 4.
\(\frac{7}{10}\) =

Answer:
0.7

Explanation:
\(\frac{7}{10}\) = 0.7

Question 5.
2 \(\frac{3}{8}\) =

Answer:
2.375

Explanation:
2 \(\frac{3}{8}\) = \(\frac{19}{8}\) = 2.375

Grade 8 Mathematics Unit 1 Answer Key Question 6.
\(\frac{5}{6}\) =

Answer:
0.833

Explanation:
\(\frac{5}{6}\) = \(\frac{5 × 10}{6 × 10}\) = \(\frac{50}{6 × 10}\) = \(\frac{8.33}{10}\) = 0.833

Write each decimal as a fraction or mixed number in simplest form

Question 7.
0.675
\(\frac{□}{□}\)

Answer:
\(\frac{27}{40}\)

Explanation:
\(\frac{0.675 × 1000}{1 × 1000}\) = \(\frac{675}{1000}\) = \(\frac{675/25}{1000/25}\) = \(\frac{27}{40}\)

Topic 1 Solve Problems Involving Real Numbers Question 8.
5.6
______ \(\frac{□}{□}\)

Answer:
5 \(\frac{3}{5}\)

Explanation:
\(\frac{5.6 × 10}{10}\) = \(\frac{56}{10}\) = 5 \(\frac{6}{10}\) = 5 \(\frac{6/2}{10/2}\) = 5 \(\frac{3}{5}\)

Question 9.
0.44
\(\frac{□}{□}\)

Answer:
\(\frac{11}{25}\)

Explanation:
\(\frac{0.44 × 100}{1 × 100}\) = \(\frac{44}{100}\) = \(\frac{44/4}{100/4}\) = \(\frac{11}{25}\)

Question 10.
0.\(\bar{4}\)
\(\frac{□}{□}\)

Answer:
\(\frac{4}{9}\)

Explanation:
Let x = 0.\(\bar{4}\)
Now, 10x = 4.\(\bar{4}\)
10x – x = 4.\(\bar{4}\) – 0.\(\bar{4}\)
9x = 4
x = \(\frac{4}{9}\)

Question 11.
0.\(\overline { 26 } \)
\(\frac{□}{□}\)

Answer:
\(\frac{26}{99}\)

Explanation:
Let x = 0.\(\overline {26}\)
Now, 100x = 26.\(\overline{26}\)
100x – x = 26.\(\overline{26}\) – 0.\(\overline {26}\)
99x = 26
x = \(\frac{26}{99}\)

Question 12.
0.\(\overline { 325 } \)
\(\frac{□}{□}\)

Answer:
\(\frac{325}{999}\)

Explanation:
Let x = 0.\(\overline {325}\)
Now, 1000x = 325.\(\overline{325}\)
1000x – x = 325.\(\overline{325}\) – 0.\(\overline {325}\)
999x = 325
x = \(\frac{325}{999}\)

Solve each equation for x

Question 13.
x2 = 144
± ______

Answer:
x=±12

Explanation:
x2 = 144
Taking square roots on both sides
x2=±144
x = ±12

Real Number System Study Guide Answer Key 8th Grade Question 14.
x2 = \(\frac{25}{289}\)
± \(\frac{□}{□}\)

Answer:
x = ±\(\frac{5}{17}\)

Explanation:
x2 = \(\frac{25}{289}\)
Taking square roots on both the sides
x2=±√\(\frac{25}{289}\)
x = ±\(\frac{5}{17}\)

Question 15.
x3 = 216
______

Answer:
x = 6

Explanation:
x3 = 216
Taking cube roots on both the sides
3x3= 3√216
x = 6

Approximate each irrational number to two decimal places without a calculator.

Question 16.
\(\sqrt { 5 } \) ≈ ______

Answer:
2.236

Explanation:
x = \(\sqrt { 5 } \)
The 5 is in between 4 and 6
Take square root of each year
√4 < √5 < √6
2 < √5 < 3
√5 = 2.2
(2.2)² = 4.84
(2.25)² = 5.06
(2.5)³ = 5.29
A good estimate for √5 is 2.25

Question 17.
\(\sqrt { 3 } \) ≈ ______

Answer:
1.75

Explanation:
\(\sqrt { 3 } \)
1 < 3 < 4
√1 < √3 < √4
1 < √3 < 2
√3 = 1.6
(1.65)² = 2.72
(1.7)² = 2.89
(1.75)² = 3.06
A good estimate for √3 is 1.75

Question 18.
\(\sqrt { 10 } \) ≈ ______

Answer:
3.15

Explanation:
\(\sqrt { 10 } \)
9 < 10 < 16
√9 < √10 < √16
3 < √10 < 4
√10 = 3.1
(3.1)² = 9.61
(3.15)² = 9.92
(3.2)² = 10.24
A good estimate for √10 is 3.15

Real Numbers 8th Grade Math Question 19.
What is the difference between rational and irrational numbers?
Type below:
_____________

Answer:

Rational number can be expressed as a ration of two integers such as 5/2
Irrational number cannot be expressed as a ratio of two integers such as √13

Explanation:
A rational number is a number that can be express as the ratio of two integers. A number that cannot be expressed that way is irrational.

1.1 Independent Practice – Rational and Irrational Numbers – Page No. 13

Question 20.
A \(\frac{7}{16}\)-inch-long bolt is used in a machine. What is the length of the bolt written as a decimal?
______ -inch-long

Answer:
0.4375 inch

Explanation:
The length of the bolt is \(\frac{7}{16}\)-inch
Let, x = \(\frac{7}{16}\)
Multiplying by 125 on both nominator and denominator
x = \(\frac{7×125}{16×125}\) = \(\frac{875}{2000}\) =\(\frac{437.5}{1000}\) = 0.4375

Question 21.
The weight of an object on the moon is \(\frac{1}{6}\) its weight on Earth. Write \(\frac{1}{6}\) as a decimal.
______

Answer:
0.1666

Explanation:
The weight of the object on the moon is \(\frac{1}{6}\)
Let, x = \(\frac{1}{6}\)
Multiplying by 100 on both nominator and denominator
x = \(\frac{1×100}{6×100}\) = \(\frac{16.6}{100}\) =0.166

Lesson 1-1 Operations on Real Numbers Answer Key Question 22.
The distance to the nearest gas station is 2 \(\frac{4}{5}\) kilometers. What is this distance written as a decimal?
______

Answer:
2.8

Explanation:
The distance of the nearest gas station is 2 \(\frac{4}{5}\)
Let, x = 2 \(\frac{4}{5}\)
Multiplying by 100 on both nominator and denominator
x = 2 \(\frac{4×100}{5×100}\) = \(\frac{80}{100}\) =0.8

Question 23.
A baseball pitcher has pitched 98 \(\frac{2}{3}\) innings. What is the number of innings written as a decimal?
______

Answer:
98.6

Explanation:
A baseball pitcher has pitched 98 \(\frac{2}{3}\) innings.
98 \(\frac{2}{3}\) = 98 + 2/3
= (294/3) + (2/3)
296/3
98.6

Question 24.
A heartbeat takes 0.8 second. How many seconds is this written as a fraction?
\(\frac{□}{□}\)

Answer:
\(\frac{4}{5}\)

Explanation:
A heartbeat takes 0.8 seconds.
0.8
There are 8 tenths.
8/10 = 4/5

Question 25.
There are 26.2 miles in a marathon. Write the number of miles using a fraction.
\(\frac{□}{□}\)

Answer:
26\(\frac{1}{5}\)

Explanation:
There are 26.2 miles in a marathon.
26.2 miles
262/10
131/5
26 1/5 miles

Question 26.
The average score on a biology test was 72.\(\bar{1}\). Write the average score using a fraction.
\(\frac{□}{□}\)

Answer:
80 \(\frac{1}{9}\)

Explanation:
The average score on a biology test was 72.\(\bar{1}\).
72.\(\bar{1}\)
Let x = 72.\(\bar{1}\)
10x = 10(72.\(\bar{1}\))
10x = 721.1
-x = -0.1
9x = 721
x = 721/9
x = 80 1/9

Question 27.
The metal in a penny is worth about 0.505 cent. How many cents is this written as a fraction?
\(\frac{□}{□}\)

Answer:
\(\frac{101}{200}\)

Explanation:
The metal in a penny is worth about 0.505 cent.
0.505 cent
505 thousandths
505/1000
101/200 cents

Question 28.
Multistep An artist wants to frame a square painting with an area of 400 square inches. She wants to know the length of the wood trim that is needed to go around the painting.
Go Math Grade 8 Answer Key Chapter 1 Real Numbers Lesson 1: Rational and Irrational Numbers img 1
a. If x is the length of one side of the painting, what equation can you set up to find the length of a side?
x2 = ______

Answer:
x² = 400

Explanation:
The area of a square is the square of its equal side, x
x² = 400

Question 28.
b. Solve the equation you wrote in part a. How many solutions does the equation have?
x = ± ______

Answer:
x = ± 20

Explanation:
Take the square root on both sides. Solve
x = ± 20

Question 28.
c. Do all of the solutions that you found in part b make sense in the context of the problem? Explain.
Type below:
_____________

Answer:
No. Both values of x do not make sense.

Explanation:
The length cannot be negative, hence negative value does not make sense.
No. Both values of x do not make sense.

Question 28.
d. What is the length of the wood trim needed to go around the painting?
P = ______ inches

Answer:
Length P = 20 + 2y

Rational and Irrational Numbers – Page No. 14

Question 29.
Analyze Relationships To find \(\sqrt { 15 } \), Beau found 32 = 9 and 42 = 16. He said that since 15 is between 9 and 16, \(\sqrt { 15 } \) must be between 3 and 4. He thinks a good estimate for \(\sqrt { 15 } \) is \(\frac { 3+4 }{ 2 } \) = 3.5. Is Beau’s estimate high, low, or correct? Explain.
_____________

Answer:
3.85

Explanation:
15 is closer to 16
√15 is closer to √16
Beau’s estimate is low.
(3.8)² = 14.44
(3.85)² = 14.82
(3.9)² = 15.21
√15 is 3.85

Simple Solutions Math Grade 8 Answer Key Pdf Lesson 1 Question 30.
Justify Reasoning What is a good estimate for the solution to the equation x3 = 95? How did you come up with your estimate?
x ≈ ______

Answer:
x ≈  4.55

Explanation:
3√x = 95
x = 3√95
64 < 95 < 125
Take the cube root of each number
3√64 < 3√95  < 3√125
4 < 3√95 < 5
3√95 = 4.6
(4.5)³ = 91.125
(4.55)³ = 94.20
(4.6)³ = 97.336
3√95 = 4.55

Question 31.
The volume of a sphere is 36π ft3. What is the radius of the sphere? Use the formula V = \(\frac { 4 }{ 3 } \)πr3 to find your answer.
Go Math Grade 8 Answer Key Chapter 1 Real Numbers Lesson 1: Rational and Irrational Numbers img 2
r = ______

Answer:
r = 3

Explanation:
V = 4/3 πr³
36π = 4/3 πr³
r³ = 36π/π . 3/4
r³ = 27
r = 3√27
r = 3

FOCUS ON HIGHER ORDER THINKING

Question 32.
Draw Conclusions Can you find the cube root of a negative number? If so, is it positive or negative? Explain your reasoning.
_____________

Answer:
Yes

Explanation:
Yes. The cube root of a negative number would be negative. Because the product of three negative signs is always negative.

Question 33.
Make a Conjecture Evaluate and compare the following expressions.
\(\sqrt { \frac { 4 }{ 25 } } \) and \(\frac { \sqrt { 4 } }{ \sqrt { 25 } } \) \(\sqrt { \frac { 16 }{ 81 } } \) and \(\frac { \sqrt { 16 } }{ \sqrt { 81 } } \) \(\sqrt { \frac { 36 }{ 49 } } \) and\(\frac { \sqrt { 36 } }{ \sqrt { 49 } } \)
Use your results to make a conjecture about a division rule for square roots. Since division is multiplication by the reciprocal, make a conjecture about a multiplication rule for square roots.
Expressions are: _____________

Answer:
Evaluating and comparing
√4/25 = 2/5
√16/81 = 4/9
√36/49 = 6/7
Conjecture about a division rule for square roots
√a/√b = √(a/b)
Conjecture about a multiplication rule for square roots
√a × √b

Question 34.
Persevere in Problem Solving
The difference between the solutions to the equation x2 = a is 30. What is a? Show that your answer is correct.
_____

Answer:
30

Explanation:
x2 = a
x = ±√a
√a – (-√a) = 30
√a + √a = 30
2√a = 30
√a = 15
a = 225
x2 = 225
x = ±225
x = ±15
15 – (-15) = 15 + 15 = 30

Guided Practice – Sets of real Numbers – Page No. 18

Write all names that apply to each number.

Question 1.
\(\frac{7}{8}\)
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers

Mid Topic Performance Task Topic 1 Answer Key Question 2.
\(\sqrt { 36 } \)
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers

Explanation:
\(\sqrt { 36 } \) = 6

Question 3.
\(\sqrt { 24 } \)
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
e. Irrational Numbers

Question 4.
0.75
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers

Question 5.
0
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers

Question 6.
−\(\sqrt { 100 } \)
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integers

Explanation:
−\(\sqrt { 100 } \) = – 10

Grade 8 Math Unit 1 Performance Assessment Task 1 Answer Key Question 7.
5.\(\overline { 45 } \)
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers

Question 8.
−\(\frac{18}{6}\)
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integers

Explanation:
−\(\frac{18}{6}\) = -3

Tell whether the given statement is true or false. Explain your choice.

Question 9.
All whole numbers are rational numbers.
i. True
ii. False

Answer:
i. True

Explanation:
All whole numbers are rational numbers.
Whole numbers are a subset of the set of rational numbers and can be written as ratio of the whole number to 1.

Question 10.
No irrational numbers are whole numbers.
i. True
ii. False

Answer:
i. True

Explanation:
True. Whole numbers are ration numbers.

Identify the set of numbers that best describes each situation. Explain your choice.

Question 11.
the change in the value of an account when given to the nearest dollar
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
c. Integer Numbers

Explanation:
The change can be a whole dollar amount and can be positive, negative or zero.

Lesson 1-1 Additional Practice Operations on Real Numbers Question 12.
The markings on a standard ruler
Go Math Grade 8 Answer Key Chapter 1 Real Numbers Lesson 2: Sets of real Numbers img 3
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
b. Rational Numbers

Explanation:
The ruler is marked every 1/16t inch.

ESSENTIAL QUESTION CHECK-IN

Question 13.
What are some ways to describe the relationships between sets of numbers?

Answer:
There are two ways that we have been using until now to describe the relationships between sets of numbers

  • Using a scheme or a diagram as the one on page 15.
  • Verbal description, for example, “All irrational numbers are real numbers.”

1.2 Independent Practice – Sets of real Numbers – Page No. 19

Write all names that apply to each number. Then place the numbers in the correct location on the Venn diagram.
Go Math Grade 8 Answer Key Chapter 1 Real Numbers Lesson 2: Sets of real Numbers img 4

Question 14.
\(\sqrt { 9 } \)
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers

Explanation:
\(\sqrt { 9 } \) = 3

Question 15.
257
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers

Question 16.
\(\sqrt { 50 } \)
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
e. Irrational Numbers

Question 17.
8 \(\frac{1}{2}\)
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers

Question 18.
16.6
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers

Question 19.
\(\sqrt { 16 } \)
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers

Explanation:
\(\sqrt { 16 } \) = 4

Identify the set of numbers that best describes each situation. Explain your choice.

Question 20.
the height of an airplane as it descends to an airport runway
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
d. Whole Numbers

Explanation:
Whole. The height of an airplane as it descents to an airport runway is a whole number greater than 0

Question 21.
the score with respect to par of several golfers: 2, – 3, 5, 0, – 1
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
c. Integer Numbers

Explanation:
Integers. The scores are counting numbers, their opposites, and zero.

Question 22.
Critique Reasoning Ronald states that the number \(\frac{1}{11}\) is not rational because, when converted into a decimal, it does not terminate. Nathaniel says it is rational because it is a fraction. Which boy is correct? Explain.
i. Ronald
ii. Nathaniel

Answer:
ii. Nathaniel

Explanation:
Nathaniel is correct.
A fraction is a rational real number, even if it is not a terminating decimal.

Sets of real Numbers – Page No. 20

Question 23.
Critique Reasoning The circumference of a circular region is shown. What type of number best describes the diameter of the circle? Explain your answer.
Go Math Grade 8 Answer Key Chapter 1 Real Numbers Lesson 2: Sets of real Numbers img 5
Options:
a. Real Numbers
b. Rational Numbers
c. Irrational Numbers
d. Integers
e. Whole Numbers

Answer:
e. Whole Numbers

Explanation:
Circumference of the circle
A = 2πr
π = 2πr
Diameter is twice the radius
2r = 1
Whole

Question 24.
Critical Thinking A number is not an integer. What type of number can it be?
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
b. Rational Numbers
e. Irrational Numbers

Question 25.
A grocery store has a shelf with half-gallon containers of milk. What type of number best represents the total number of gallons?
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
b. Rational Numbers

FOCUS ON HIGHER ORDER THINKING

Question 26.
Explain the Error Katie said, “Negative numbers are integers.” What was her error?
Type below:
_______________

Answer:
Her error is that she stated that all negative numbers are integers. Some negative numbers are integers such as -4 but some are not such an -0.8

Question 27.
Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational? Explain.
Type below:
_______________

Answer:
Not always.

Explanation:
Not always.
If the calculator shows a terminating decimal, the number is rational but otherwise, it is not possible as you can only see a few digits.

Question 28.
Draw Conclusions The decimal 0.\(\bar{3}\) represents \(\frac{1}{3}\). What type of number best describes 0.\(\bar{9}\) , which is 3 × 0.\(\bar{3}\)? Explain.
Type below:
_______________

Answer:
1

Explanation:
let x = 0.9999999
10x = 9.99999999
10x = 9 + 0.999999999
10x = 9 + x
9x = 9
x=1.

Question 29.
Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form. Why is this?

Answer:
Because irrational numbers are nonrepeating, otherwise they could be represented as a fraction. Although a potential counter-example to this claim is that some irrational numbers can only be represented in decimal form, for example, 0.1234567891011121314151617…, 0.24681012141618202224…, 0.101101110111101111101111110… are all irrational numbers.

Guided Practice – Ordering Real Numbers – Page No. 24

Compare. Write <, >, or =.

Question 1.
\(\sqrt { 3 } \) + 2 ________ \(\sqrt { 3 } \) + 3

Answer:
\(\sqrt { 3 } \) + 2 < \(\sqrt { 3 } \) + 3

Explanation:
\(\sqrt { 3 } \) is between 1 and 2
\(\sqrt { 3 } \) + 2 is between 3 and 4
\(\sqrt { 3 } \) + 3 is between 4 and 5
\(\sqrt { 3 } \) + 2 < \(\sqrt { 3 } \) + 3

Lesson 1.3 Ordering Real Numbers Question 2.
\(\sqrt { 11 } \) + 15 _______ \(\sqrt { 8 } \) + 15

Answer:
\(\sqrt { 11 } \) + 15 > \(\sqrt { 8 } \) + 15

Explanation:
\(\sqrt { 11 } \) is between 3 and 4
\(\sqrt { 8 } \) is between 2 and 3
\(\sqrt { 11 } \) + 15 is between 18 and 19
\(\sqrt { 8 } \) + 15 is between 17 and 18
\(\sqrt { 11 } \) + 15 > \(\sqrt { 8 } \) + 15

Question 3.
\(\sqrt { 6 } \) + 5 _______ 6 + \(\sqrt { 5 } \)

Answer:
\(\sqrt { 6 } \) + 5 < 6 + \(\sqrt { 5 } \)

Explanation:
\(\sqrt { 6 } \) is between 2 and 3
\(\sqrt { 5 } \) is between 2 and 3
\(\sqrt { 6 } \) is between 7 and 8
\(\sqrt { 5 } \) is between 8 and 9
\(\sqrt { 6 } \) + 5 < 6 + \(\sqrt { 5 } \)

Question 4.
\(\sqrt { 9 } \) + 3 _______ 9 + \(\sqrt { 3 } \)

Answer:
\(\sqrt { 9 } \) + 3 < 9 + \(\sqrt { 3 } \)

Explanation:
\(\sqrt { 9 } \) + 3
9 + \(\sqrt { 3 } \)
\(\sqrt { 3 } \) is between 1 and 2
\(\sqrt { 9 } \) + 3 = 3 + 3 = 6
9 + \(\sqrt { 3 } \) is between 10 and 11
\(\sqrt { 9 } \) + 3 < 9 + \(\sqrt { 3 } \)

Question 5.
\(\sqrt { 17 } \) – 3 _______ -2 + \(\sqrt { 5 } \)

Answer:
\(\sqrt { 17 } \) – 3 > -2 + \(\sqrt { 5 } \)

Explanation:
\(\sqrt { 17 } \) is between 4 and 5
\(\sqrt { 5 } \) is between 2 and 3
\(\sqrt { 17 } \) – 3 is between 1 and 2
-2 + \(\sqrt { 5 } \) is between 0 and 1
\(\sqrt { 17 } \) – 3 > -2 + \(\sqrt { 5 } \)

Question 6.
10 – \(\sqrt { 8 } \) _______ 12 – \(\sqrt { 2 } \)

Answer:
10 – \(\sqrt { 8 } \) < 12 – \(\sqrt { 2 } \)

Explanation:
\(\sqrt { 8 } \) is between 2 and 3
\(\sqrt { 2 } \) is between 1 and 2
10 – \(\sqrt { 8 } \) is between 8 and 7
12 – \(\sqrt { 2 } \) is between 11 and 10
10 – \(\sqrt { 8 } \) < 12 – \(\sqrt { 2 } \)

Question 7.
\(\sqrt { 7 } \) + 2 _______ \(\sqrt { 10 } \) – 1

Answer:
\(\sqrt { 7 } \) + 2 > \(\sqrt { 10 } \) – 1

Explanation:
\(\sqrt { 7 } \) is between 2 and 3
\(\sqrt { 10 } \) is between 3 and 4
\(\sqrt { 7 } \) + 2 is between 4 and 5
\(\sqrt { 10 } \) – 1 is between 2 and 3
\(\sqrt { 7 } \) + 2 > \(\sqrt { 10 } \) – 1

Question 8.
\(\sqrt { 17 } \) + 3 _______ 3 + \(\sqrt { 11 } \)

Answer:
\(\sqrt { 17 } \) + 3 > 3 + \(\sqrt { 11 } \)

Explanation:
\(\sqrt { 17 } \) is between 4 and 5
\(\sqrt { 11 } \) is between 3 and 4
\(\sqrt { 17 } \) + 3 is between 7 and 8
3 + \(\sqrt { 11 } \) is between 6 and 7
\(\sqrt { 17 } \) + 3 > 3 + \(\sqrt { 11 } \)

Comparing and Ordering Real Numbers Worksheet 8th Grade Answer Key Question 9.
Order \(\sqrt { 3 } \), 2 π, and 1.5 from least to greatest. Then graph them on the number line.
\(\sqrt { 3 } \) is between _________ and _____________ , so \(\sqrt { 3 } \) ≈ ____________.
π ≈ 3.14, so 2 π ≈ _______________.
Go Math Grade 8 Answer Key Chapter 1 Real Numbers Lesson 3: Ordering Real Numbers img 6
From least to greatest, the numbers are ______________, _____________________ ,_________________.
Type below:
___________

Answer:
1.5, \(\sqrt { 3 } \), 2 π

Explanation:
\(\sqrt { 3 } \) is between 1.7 and 1.75
π = 3.14; 2 π = 6.28
grade 8 chapter 1 image 1
1.5, \(\sqrt { 3 } \), 2 π

Question 10.
Four people have found the perimeter of a forest using different methods. Their results are given in the table. Order their calculations from greatest to least.
Go Math Grade 8 Answer Key Chapter 1 Real Numbers Lesson 3: Ordering Real Numbers img 7
Type below:
___________

Answer:
\(\sqrt { 17 } \) – 2, 1+ π/2, 2.5, 12/5

Explanation:
\(\sqrt { 17 } \) – 2
\(\sqrt { 17 } \) is between 4 and 5
Since, 17 is closer to 16, the estimated value is 4.1
1+ π/2
1 + (3.14/2) = 2.57
12/5 = 2.4
2.5
\(\sqrt { 17 } \) – 2, 1+ π/2, 2.5, 12/5

ESSENTIAL QUESTION CHECK-IN

Question 11.
Explain how to order a set of real numbers.
Type below:
___________

Answer:
Evaluate the given numbers and write in decimal form. Plot on number line and arrange the numbers accordingly.

Independent Practice – Ordering Real Numbers – Page No. 25

Order the numbers from least to greatest.

Question 12.
\(\sqrt { 7 } \), 2, \(\frac { \sqrt { 8 } }{ 2 } \)
Type below:
____________

Answer:
\(\frac { \sqrt { 8 } }{ 2 } \), 2, \(\sqrt { 7 } \)

Explanation:
\(\sqrt { 7 } \), 2, \(\frac { \sqrt { 8 } }{ 2 } \)
\(\sqrt { 7 } \) is between 2 and 3
Since 7 is closer to 9, (2.65)² = 7.02, hence the estimated value is 2.65
\(\frac { \sqrt { 8 } }{ 2 } \)
\(\sqrt { 8 } \) is between 2 and 3
Since 8 is closer to 9, (2.85)² = 8.12, hence the estimated value is 2.85
2.85/2 = 1.43
grade 8 chapter 1 image 3
\(\frac { \sqrt { 8 } }{ 2 } \), 2, \(\sqrt { 7 } \)

Question 13.
\(\sqrt { 10 } \), π, 3.5
Type below:
____________

Answer:
π, \(\sqrt { 10 } \), 3.5

Explanation:
\(\sqrt { 10 } \), π, 3.5
\(\sqrt { 10 } \) is between 3 and 4
Since, 10 is closer to 9, (3.15)² = 9.92, hence the estimated value is 3.15
π = 3.14
3.5
grade 8 chapter 1 image 4
π, \(\sqrt { 10 } \), 3.5

Question 14.
\(\sqrt { 220 } \), −10, \(\sqrt { 100 } \), 11.5
Type below:
____________

Answer:
-10, √100, 11.5, √220

Explanation:
\(\sqrt { 220 } \), −10, \(\sqrt { 100 } \), 11.5
196 < 220 < 225
√196 < √220 < √225
14 < √220 < 15
√220 = 14.5
√100 = 10
grade 8 chapter 1 image 5
-10, √100, 11.5, √220

Question 15.
\(\sqrt { 8 } \), −3.75, 3, \(\frac{9}{4}\)
Type below:
____________

Answer:
−3.75, \(\frac{9}{4}\), \(\sqrt { 8 } \)

Explanation:
\(\sqrt { 8 } \), −3.75, 3, \(\frac{9}{4}\)
\(\sqrt { 8 } \) is between 2 and 3
Since, 8 is closer to 9, (2.85)² = 8.12, hence the estimated value is 2.85
-3.75 = 3
9/4 = 2.25
grade 8 chapter 1 image 6
−3.75, \(\frac{9}{4}\), \(\sqrt { 8 } \)

Ordering Real Numbers Worksheet 8th Grade Pdf Question 16.
Your sister is considering two different shapes for her garden. One is a square with side lengths of 3.5 meters, and the other is a circle with a diameter of 4 meters.
a. Find the area of the square.
_______ m2

Answer:
(3.5)² = 12.25

Explanation:
Area of the square = x²
Area = (3.5)² = 12.25

Question 16.
b. Find the area of the circle.
_______ m2

Answer:
π(2)² = 12.56

Explanation:
Area of the circle = πr² where r = d/2 = 4/2 = 2
Area = π(2)² = 12.56

Question 16.
c. Compare your answers from parts a and b. Which garden would give your sister the most space to plant?
___________

Answer:
12.25 < 12.56
The circle will give more space

Question 17.
Winnie measured the length of her father’s ranch four times and got four different distances. Her measurements are shown in the table.
a. To estimate the actual length, Winnie first approximated each distance to the nearest hundredth. Then she averaged the four numbers. Using a calculator, find Winnie’s estimate.
Go Math Grade 8 Answer Key Chapter 1 Real Numbers Lesson 3: Ordering Real Numbers img 8
______

Answer:
7.4815

Explanation:
\(\sqrt { 60 } \) = 7.75
58/8 = 7.25
7.3333
7 3/5 = 7.60
Average = (7.75 + 7.25 + 7.33 + 7.60)/4 = 7.4815

Question 17.
b. Winnie’s father estimated the distance across his ranch to be \(\sqrt { 56 } \) km. How does this distance compare to Winnie’s estimate?
____________

Answer:
They are nearly identical

Explanation:
\(\sqrt { 56 } \) = 7.4833
They are nearly identical

Give an example of each type of number.

Question 18.
a real number between \(\sqrt { 13 } \) and \(\sqrt { 14 } \)
Type below:
____________

Answer:
A real number between \(\sqrt { 13 } \) and \(\sqrt { 14 } \)
Example: 3.7

Explanation:
\(\sqrt { 13 } \) = 3.61
\(\sqrt { 13 } \) = 3.74
A real number between \(\sqrt { 13 } \) and \(\sqrt { 14 } \)
Example: 3.7

Question 19.
an irrational number between 5 and 7
Type below:
____________

Answer:
An irrational number between 5 and 7
Example: \(\sqrt { 29 } \)

Explanation:
5² = 25 and 7² = 49
An irrational number between 5 and 7
Example: \(\sqrt { 29 } \)

Ordering Real Numbers – Page No. 26

Question 20.
A teacher asks his students to write the numbers shown in order from least to greatest. Paul thinks the numbers are already in order. Sandra thinks the order should be reversed. Who is right?
Go Math Grade 8 Answer Key Chapter 1 Real Numbers Lesson 3: Ordering Real Numbers img 9
_____________

Answer:
Neither are correct

Explanation:
\(\sqrt { 115 } \), 115/11, 10.5624
\(\sqrt { 115 } \) is between 10 and 11
Since, 115 is closer to 121, (10.7)² = 114.5, hence the estimated value is 10.7
115/11 = 10.4545
10.5624
Neither are correct

Compare and Order Numbers Lesson 1.3 Answer Key Question 21.
Math History
There is a famous irrational number called Euler’s number, symbolized with an e. Like π, its decimal form never ends or repeats. The first few digits of e are 2.7182818284.
a. Between which two square roots of integers could you find this number?
Type below:
_____________

Answer:
The square of e lies between 7 and 8
2.718281828
(2.72)² = 7.3984
Hence, it lies between \(\sqrt { 7 } \) = 2.65 and \(\sqrt { 8 } \) = 2.82

Question 21.
b. Between which two square roots of integers can you find π?
Type below:
_____________

Answer:
3.142
(3.14)² = 9.8596
Hence. it lies between \(\sqrt { 9 } \) = 3 and \(\sqrt { 10 } \) = 3.16

H.O.T.

FOCUS ON HIGHER ORDER THINKING

Question 22.
Analyze Relationships
There are several approximations used for π, including 3.14 and \(\frac{22}{7}\). π is approximately 3.14159265358979 . . .
a. Label π and the two approximations on the number line.
Go Math Grade 8 Answer Key Chapter 1 Real Numbers Lesson 3: Ordering Real Numbers img 10
Type below:
_____________

Answer:
grade 8 chapter 1 image 7

Question 22.
b. Which of the two approximations is a better estimate for π? Explain.
Type below:
_____________

Answer:
As we can see from the number line, 22/7 is closer to π, so we can conclude that 22/7 is a better estimation for π.

Question 22.
c. Find a whole number x so that the ratio \(\frac{x}{113}\) is a better estimate for π than the two given approximations.
Type below:
_____________

Answer:
355/113 is a better estimation for π, because 355/113 = 3.14159292035 = 3.14159265358979 = π

Lesson 1-3 Compare and Order Real Numbers Question 23.
Communicate Mathematical Ideas
What is the fewest number of distinct points that must be graphed on a number line, in order to represent natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers? Explain.
_______ points

Answer:
2 points

Explanation:
There need to be plotting of at least 2 points because a rational number can never be equal to an irrational number. So let’s say 5 points are the same among six but the 6th will be different as there both rational numbers and irrational numbers included.

Question 24.
Critique Reasoning
Jill says that 12.\(\bar{6}\) is less than 12.63. Explain her error.
Type below:
_____________

Answer:
12.\(\bar{6}\) = 12.666
12.\(\bar{6}\) > 12.63

1.1 Rational and Irrational Numbers – Model Quiz – Page No. 27

Write each fraction as a decimal or each decimal as a fraction.

Question 1.
\(\frac{7}{20}\)
_______

Answer:
0.35

Explanation:
\(\frac{7}{20}\) = 0.35

Question 2.
1.\(\overline { 27} \)
______ \(\frac{□}{□}\)

Answer:
1\(\frac{28}{99}\)

Explanation:
1.\(\overline { 27} \)
x = 1.\(\overline { 27} \)
100x = 100(1.\(\overline { 27} \))
100x = 127(\(\overline { 27} \))
x = .\(\overline { 27} \)
99x = 127
x = 127/99
x = 1 28/99

Question 3.
1 \(\frac{7}{8}\)
______

Answer:
1.875

Explanation:
1 \(\frac{7}{8}\)
1 + 7/8
8/8 + 7/8
15/8 = 1.875

Solve each equation for x.

Question 4.
x2 = 81
± ______

Answer:
± 9

Explanation:
x2 = 81
x = ± 81
x = ± 9

Question 5.
x3 = 343
______

Answer:
x = 7

Explanation:
x3 = 343
x = 7

Question 6.
x2 = \(\frac{1}{100}\)
± \(\frac{□}{□}\)

Answer:
± \(\frac{1}{10}\)

Explanation:
x2 = \(\frac{1}{100}\)
x = ± \(\frac{1}{10}\)

Question 7.
A square patio has an area of 200 square feet. How long is each side of the patio to the nearest 0.05?
______ feet

Answer:
14.15 feet

Explanation:
The area of a square is found by multiplying the side of the square by itself. Therefore, to find the side of the square, we have to take the square root of the area.
Let’s denote with A the area of the patio and with s each side of the square.
We have:
A = 200
A = s.s
s = \(\sqrt { A } \) = \(\sqrt { 200 } \)
Following the steps as in “Explore Activity” on page 9, we can make an estimation for the irrational number:
196 < 200 < 225
\(\sqrt { 196 } \) < \(\sqrt { 200 } \) < \(\sqrt { 225 } \)
14 < \(\sqrt { 200 } \) < 15
We see that 200 is much closer to 196 than to 225, therefore the square root of it should be between 14 and 14.5. To make a better estimation, we pick some numbers between 14 and 14.5 and calculate their squares:
(14.1)² = 198.81
(14.2)² = 201.64
14.1 < \(\sqrt { 200 } \) < 14.2
\(\sqrt { 200 } \) = 14.15
We see that 200 is much closer to 14.1 than to 14.2, therefore the square root of it should be between 14.1 and 14.15. If we round to the nearest 0.05, we have:
s = 14.15

1.2 Sets of Real Numbers

Write all names that apply to each number.

Question 8.
\(\frac { 121 }{ \sqrt { 121 } }\)
Type below:
___________

Answer:
Rational, whole, integer, real numbers

Explanation:
\(\frac { 121 }{ \sqrt { 121 } }\)
121/11 = 11

Question 9.
\(\frac{π}{2}\)
Type below:
___________

Answer:
Irrational, real numbers

Question 10.
Tell whether the statement “All integers are rational numbers” is true or false. Explain your choice.
___________

Answer:
True

Explanation:
“All integers are rational numbers” is true, because every integer can be expressed as a fraction with a denominator equal to 1. The set of integer A a subset of rational numbers.

1.3 Ordering Real Numbers

Compare. Write <, >, or =.

Question 11.
\(\sqrt { 8 }\) + 3 _______ 8 + \(\sqrt { 3 }\)

Answer:
\(\sqrt { 8 }\) + 3 < 8 + \(\sqrt { 3 }\)

Explanation:
4 < 8 < 9
\(\sqrt { 4 }\) < \(\sqrt { 8 }\) < \(\sqrt { 9 }\)
2 < \(\sqrt { 8 }\) < 3
1 < 3 < 4
\(\sqrt { 1 }\) < \(\sqrt { 3 }\) < \(\sqrt { 4 }\)
1 < \(\sqrt { 3 }\) < 2
\(\sqrt { 8 }\) + 3 is between 5 and 6
8 + \(\sqrt { 3 }\) is between 9 and 10
\(\sqrt { 8 }\) + 3 < 8 + \(\sqrt { 3 }\)

Real Number System Study Guide Answer Key Question 12.
\(\sqrt { 5 }\) + 11 _______ 5 + \(\sqrt { 11 }\)

Answer:
\(\sqrt { 5 }\) + 11 > 5 + \(\sqrt { 11 }\)

Explanation:
\(\sqrt { 5 }\) lies in between 2 and 3
\(\sqrt { 11 }\) lies in between 3 and 4
\(\sqrt { 5 }\) + 11 lies in between 13 and 14
5 + \(\sqrt { 11 }\) lies in between 8 and 9
\(\sqrt { 5 }\) + 11 > 5 + \(\sqrt { 11 }\)

Order the numbers from least to greatest.

Question 13.
\(\sqrt { 99 }\), π2, 9.\(\bar { 8 }\)
Type below:
_______________

Answer:
π2, 9.\(\bar { 8 }\), \(\sqrt { 99 }\)

Explanation:
\(\sqrt { 99 }\), π2, 9.\(\bar { 8 }\)
99 lies between 9² and 10²
99 is closer to 100, hence \(\sqrt { 99 }\) is closer to 10
(9.9)² = 98.01
(9.95)² = 99.0025
(10)² = 100
\(\sqrt { 99 }\) = 9.95
π² = 9.86
9.88888 = 9.89
grade 8 chapter 1 image 8
π2, 9.\(\bar { 8 }\), \(\sqrt { 99 }\)

Question 14.
\(\sqrt { \frac { 1 }{ 25 } } \), \(\frac{1}{4}\), 0.\(\bar { 2 }\)
Type below:
____________

Answer:
\(\sqrt { \frac { 1 }{ 25 } } \), 0.\(\bar { 2 }\), \(\frac{1}{4}\)

Explanation:
\(\sqrt { \frac { 1 }{ 25 } } \), \(\frac{1}{4}\), 0.\(\bar { 2 }\)
\(\sqrt { \frac { 1 }{ 25 } } \) = 1/5 = 0.2
1/4 = 0.25
0.\(\bar { 2 }\) = 0.222 = 0.22
grade 8 chapter 1 image 9
\(\sqrt { \frac { 1 }{ 25 } } \), 0.\(\bar { 2 }\), \(\frac{1}{4}\)

Essential Question

Question 15.
How are real numbers used to describe real-world situations?
Type below:
_______________

Answer:
In real-world situations, we use real numbers to count or make measurements. They can be seen as a convention for us to quantify things around, for example, the distance, the temperature, the height, etc.

Selected Response – Mixed Review – Page No. 28

Question 1.
The square root of a number is 9. What is the other square root?
Options:
a. -9
b. -3
c. 3
d. 81

Answer:
a. -9

Explanation:
We know that every positive number has two square roots, one positive and one negative. We are given the principal square root (9), so the other square root would be its negative (-9). To prove that, we square both numbers and we compare the results:
9 • 9 = 81
(-9). (-9)= 81

Question 2.
A square acre of land is 4,840 square yards. Between which two integers is the length of one side?
Options:
a. between 24 and 25 yards
b. between 69 and 70 yards
c. between 242 and 243 yards
d. between 695 and 696 yards

Answer:
b. between 69 and 70 yards

Explanation:
The area of a square is found by multiplying the side of the square by itself. Therefore, to Bud the side of the square, we have to take the square root of the area.
Let’s denote with A the area of the land and with each side of the square. We have:
A = 4840
A = s . s
A = s²
s = √A = √4840
Following the steps as in °Explore Activity on page 9, we can make an estimation for the irrational number:
4761 < 4840 < 4900
\(\sqrt { 4761 }\) < \(\sqrt { 4840 }\) < \(\sqrt { 4900 }\)
69 < \(\sqrt { 4840 }\) < 70
Each side of the land is between 69 and 70 yards.

Question 3.
Which of the following is an integer but not a whole number?
Options:
a. -9.6
b. -4
c. 0
d. 3.7

Answer:
b. -4

Explanation:
Whole numbers are not negative
-4 is an integer but not a whole number

Question 4.
Which statement is false?
Options:
a. No integers are irrational numbers.
b. All whole numbers are integers.
c. No real numbers are irrational numbers.
d. All integers greater than 0 are whole numbers.

Answer:
c. No real numbers are irrational numbers.

Explanation:
Rational and irrational numbers are real numbers.

Question 5.
Which set of numbers best describes the displayed weights on a digital scale that shows each weight to the nearest half pound?
Options:
a. whole numbers
b. rational numbers
c. real numbers
d. integers

Answer:
b. rational numbers

Explanation:
The scale weighs nearest to 1/2 pound.

Question 6.
Which of the following is not true?
Options:
a. π2 < 2π + 4
b. 3π > 9
c. \(\sqrt { 27 }\) + 3 > 172
d. 5 – \(\sqrt { 24 }\) < 1

Answer:
c. \(\sqrt { 27 }\) + 3 > 172

Explanation:
a. π2 < 2π + 4
(3.14)² < 2(3.14) + 4
9.86 < 10.28
True
b. 3π > 9
9.42 > 9
True
c. \(\sqrt { 27 }\) + 3 > 172
5.2 + 3 > 8.5
8.2 > 8.5
False
d. 5 – \(\sqrt { 24 }\) < 1
5 – 4.90 < 1
0.1 < 1
True

Question 7.
Which number is between \(\sqrt { 21 }\) and \(\frac{3π}{2}\) ?
Options:
a. \(\frac{14}{3}\)
b. 2 \(\sqrt { 6 }\)
c. 5
d. π + 1

Answer:

Explanation:
a. \(\sqrt { 21 }\) and \(\frac{3π}{2}\)
\(\sqrt { 21 }\) = 4.58
\(\frac{3π}{2}\) = 4.71
14/3 = 4.67
b. 2\(\sqrt { 6 }\) = 4.90
c. 5
d. π + 1 = 3.14 + 1 = 4.14

Question 8.
What number is shown on the graph?
Go Math Grade 8 Answer Key Chapter 1 Real Numbers Mixed Review img 11
Options:
a. π+3
b. \(\sqrt { 4 }\) + 2.5
c. \(\sqrt { 20 }\) + 2
d. 6.\(\overline { 14 } \)

Answer:
c. \(\sqrt { 20 }\) + 2

Explanation:
6.48
a. π+3 = 3.14 + 3 = 6.14
b. \(\sqrt { 4 }\) + 2.5 = 2 + 2.5 = 4.5
c. \(\sqrt { 20 }\) + 2 = 4.47 + 2 = 6.47
d. 6.\(\overline { 14 } \) = 6.1414

Question 9.
Which is in order from least to greatest?
Options:
a. 3.3, \(\frac{10}{3}\), π, \(\frac{11}{4}\)
b. \(\frac{10}{3}\), 3.3, \(\frac{11}{4}\), π
c. π, \(\frac{10}{3}\), \(\frac{11}{4}\), 3.3
d. \(\frac{11}{4}\), π, 3.3, \(\frac{10}{3}\)

Answer:
d. \(\frac{11}{4}\), π, 3.3, \(\frac{10}{3}\)

Explanation:
10/3 = 3.3333333
11/4 = 2.75
grade 8 chapter 1 image 10

Mini-Task

Question 10.
The volume of a cube is given by V = x3, where x is the length of an edge of the cube. The area of a square is given by A = x2, where x is the length of a side of the square. A given cube has a volume of 1728 cubic inches.
a. Find the length of an edge.
______ inches

Answer:
12 inches

Explanation:
V = x3
A = x2
1728 = x3
x = 12
The length of an edge = 12 in

Question 10.
b. Find the area of one side of the cube.
______ in2

Answer:
144 in2

Explanation:
A = (12)² = 144
Area of the side of the cube = 144 in2

Question 10.
c. Find the surface area of the cube.
______ in2

Answer:
864 in2

Explanation:
SA = 6 (12)² = 864
The surface area of the cube = 864 in2

Question 10.
d. What is the surface area in square feet?
______ ft2

Answer:
6 ft2

Explanation:
SA = 864/144 = 6
The surface area of the cube = 6 ft2

Conclusion:

If you are looking for the Grade 8 maths notes and textbook, then refer to Go Math Grade 8 Answer Key Chapter 1 Real Numbers. It is the best source for students to learn maths and get a good score in the exam.

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Lesson 1: Writing Linear Equations from Situations and Graphs 

Lesson 2: Writing Linear Equations from a Table

Lesson 3: Linear Relationships and Bivariate Data

Model Quiz

Mixed Review

Guided Practice – Writing Linear Equations from Situations and Graphs – Page No. 130

Question 1.
Li is making beaded necklaces. For each necklace, she uses 27 spacers, plus 5 beads per inch of necklace length. Write an equation to find how many beads Li needs for each necklace.
a. input variable:
____________

Answer:
The length of the necklace in inches

Question 1.
b. output variable:
____________

Answer:
The total number of beads in the necklace

Question 1.
c. equation:
Type below:
____________

Answer:
y = 5x

Writing Linear Equations Worksheet Answer Key Question 2.
Kate is planning a trip to the beach. She estimates her average speed to graph her expected progress on the trip. Write an equation in slope-intercept form that represents the situation.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 1: Writing Linear Equations from Situations and Graphs img 1
Type below:
____________

Answer:
y = -60x + 300

Explanation:
Choose two points on the graph to find the slope.
Find the slope
m = (y2 -y1)/(x2 – x1)
m = -300/5 = -60
Read the y-intercept from the graph: b = 300
Use your slope and y-intercept values to write an equation in slope-intercept
form.
y = -60x + 300

Question 3.
At 59 °F, crickets chirp at a rate of 76 times per minute, and at 65 °F, they chirp 100 times per minute. Write an equation in slope-intercept form that represents the situation.
Type below:
____________

Answer:
y = 4x – 160

Explanation:
Input variable: Temperature
Output variable: Number of chirps per minute
Slope:
m = (y2 -y1)/(x2 – x1) = (100 – 76)/(65 – 59) = 24/6 = 4
100 = 4(65) + b
y-intercept:
b = -160
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = 4 and b = -160.
y = 4x – 160

Essential Question Check-In

Question 4.
Explain what m and b in the equation y = mx + b tell you about the graph of the line with that equation.
Type below:
____________

Answer:
The equation of any straight line, called a linear equation, can be written as: y = mx + b, where m is the slope of the line and b is the y-intercept.
m = change in y-value/change in x-value
The y-intercept of this line is the value of y at the point where the line crosses the y-axis.

Independent Practice – Writing Linear Equations from Situations and Graphs – Page No. 131

Question 5.
A dragonfly can beat its wings 30 times per second. Write an equation in slope-intercept form that shows the relationship between flying time in seconds and the number of times the dragonfly beats its wings
Type below:
____________

Answer:
y = 30x

Explanation:
The linear equation is y = mx + b where m is the slope and b is the y-intercept.
y variable will be the number of times the dragonfly has beat its wings and the x variable will be the time.
A dragonfly can beat its wings 30 times per second.
To find b, let’s consider how many times the dragonfly has beat its wings at time 0s.
So, the equation of the line is y = 30x

5.1 Understanding Linear Functions Answer Key Question 6.
A balloon is released from the top of a platform that is 50 meters tall. The balloon rises at the rate of 4 meters per second. Write an equation in slope-intercept form that tells the height of the balloon above the ground after a given number of seconds.
Type below:
____________

Answer:
y = 4x + 50

Explanation:
Input variable: Number of seconds
Output variable: Height of the balloon
The balloon rises at a rate of 4 meters per second. m = 4;
A balloon is released from the top of a platform that is 50 meters tall. b = 50.
y = 4x + 50

The graph shows a scuba diver’s ascent over time.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 1: Writing Linear Equations from Situations and Graphs img 2

Question 7.
Use the graph to find the slope of the line. Tell what the slope means in this context.
Slope = _______ meters per second

Answer:
Slope = 1/8 or 0.125 meters per second

Explanation:
10m/80sec = 1/8
The scuba diver’s ascent gives up 1 meter per 16 seconds.

Question 8.
Identify the y-intercept. Tell what the y-intercept means in this context.
y-intercept = _______ meters

Answer:
y-intercept = -10 meters

Explanation:
The scuba divers start 10 meters below the water surface

Question 9.
Write an equation in slope-intercept form that represents the diver’s depth over time.
Type below:
____________

Answer:
y = 1/8 x – 10

Writing Linear Equations Quiz Answer Key Question 10.
The formula for converting Celsius temperatures to Fahrenheit temperatures is a linear equation. Water freezes at 0°C, or 32°F, and it boils at 100°C, or 212°F. Find the slope and y-intercept for a graph that gives degrees Celsius on the horizontal axis and degrees Fahrenheit on the vertical axis. Then write an equation in slope-intercept form that converts degrees Celsius into degrees Fahrenheit.
Type below:
____________

Answer:
Input variable: Celsius
Output variable: Fahrenheit
(0, 32) (100, 212)
m = (y2 -y1)/(x2 – x1) = (212 – 32)/(100 – 0) = 180/100 = 1.8
y intercepts = 32 when x = 0
F = 1.8C + 32

Question 11.
The cost of renting a sailboat at a lake is $20 per hour plus $12 for lifejackets. Write an equation in slope-intercept form that can be used to calculate the total amount you would pay for using this sailboat.
Type below:
____________

Answer:
y = 20x + 12

Explanation:
Input variable: Number of hours the sailboat is rented
Output variable: Total cost
The cost of renting a sailboat at a lake is $20 per hour plus $12 for lifejackets.
Slope m = 20; y-intercept b = 12
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = 20 and b = 12.
y = 20x + 12

Writing Linear Equations from Situations and Graphs – Page No. 132

The graph shows the activity in a savings account.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 1: Writing Linear Equations from Situations and Graphs img 3

Question 12.
What was the amount of the initial deposit that started this savings account?
$ _______

Answer:
$1000

Explanation:
The amount of the initial deposit that started this savings account is $1000

Question 13.
Find the slope and y-intercept of the graphed line.
Type below:
____________

Answer:
slope = 500
y-intercept = 1000

Explanation:
slope m = (y2 -y1)/(x2 – x1) = (2000 – 1500)/(2 – 1) = 500/1 = 500
y-intercept = 1000

Chapter 5 Analyzing Linear Equations Answer Key Question 14.
Write an equation in slope-intercept form for the activity in this savings account.
Type below:
____________

Answer:
y = 500x + 1000

Explanation:
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = 500 and b = 1000.
y = 500x + 1000

Question 15.
Explain the meaning of the slope in this graph.
Type below:
____________

Answer:
The slope represents the amount of money saved in dollars per month in the plan.

H.O.T.

Focus on Higher Order Thinking

Question 16.
Communicate Mathematical Ideas
Explain how you decide which part of a problem will be represented by the variable x, and which part will be represented by the variable y in a graph of the situation.
Type below:
____________

Answer:
y=mx+b
M-slope
B-Y intercept
and
(x,y)- would be any point on the graph and if you were to plug those points in for x and y the equation would solve if done properly

Question 17.
Represent Real-World Problems
Describe what would be true about the rate of change in a situation that could not be represented by a graphed line and an equation in the form y = mx + b.
Type below:
____________

Answer:
The rate of change would not be constant. Using different pairs of points in the slope formula would give different results.

Question 18.
Draw Conclusions
Must m, in the equation y = mx + b, always be a positive number? Explain.
Type below:
____________

Answer:
No; The slope m can be negative or positive. If the slope of the number is positive (the graph goes upward from left to right), then m will be positive, but if the slope is negative (the graph goes down from left to right), then m is negative.

Guided Practice – Writing Linear Equations from a Table – Page No. 136

Question 1.
Jaime purchased a $20 bus pass. Each time he rides the bus, a certain amount is deducted from the pass. The table shows the amount, y, left on his pass after x rides. Graph the data, and find the slope and y-intercept from the graph or from the table. Then write the equation for the graph in slope-intercept form.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 2: Writing Linear Equations from a Table img 4
Type below:
____________

Answer:
Grade 8 Chapter 5 image 1
y = -5/4 x + 20

Explanation:
Slope = -20/16 = -5/4 = -1.25
y-intercepts = 20
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = -1.25 and b = 20.
y = -5/4 x + 20

The table shows the temperature (y) at different altitudes (x). This is a linear relationship.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 2: Writing Linear Equations from a Table img 5

Question 2.
Find the slope for this relationship.
_______

Answer:
Slope m = (y2 -y1)/(x2 – x1) = (51 – 59)/(2000 – 0) = -8/2000 = -0.004

Explanation:
(x1, y1) = (0, 59), (x2, y2) = (2000, 51)
Slope m = (y2 -y1)/(x2 – x1) = (51 – 59)/(2000 – 0) = -8/2000 = -0.004

Writing Equations from a Table Worksheet Answer Key Pdf Question 3.
Find the y-intercept for this relationship.
y-intercept = _______

Answer:
b = 50

Explanation:
y-intercept = 59 when x = 0

Question 4.
Write an equation in slope-intercept form that represents this relationship.
Type below:
____________

Answer:
y = -0.004x + 59

Explanation:
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = -0.004 and b = 59.

Question 5.
Use your equation to determine the temperature at an altitude of 5000 feet.
_______ °F

Answer:
39°F

Explanation:
y = -0.004x + 59
y = -0.004(5000) + 59 = -20 + 59 = 39
The temperature is 39°F at the altitude of 5000 feet.

Essential Question Check-In

Question 6.
Describe how you can use the information in a table showing a linear relationship to find the slope and y-intercept for the equation.
Type below:
____________

Answer:
Use any two points from the table to fins teh slope.
Slope m = (y2 -y1)/(x2 – x1)
If the value of y-intercept, when x=0 is not given in the table, use the slope and any ordered pair from the table and substitute in slope-intercept from y=mx + b to find b.

Independent Practice – Writing Linear Equations from a Table – Page No. 137

Question 7.
The table shows the costs of a large cheese pizza with toppings at a local pizzeria. Graph the data, and find the slope and y-intercept from the graph. Then write the equation for the graph in slope-intercept form.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 2: Writing Linear Equations from a Table img 6
Type below:
____________

Answer:
Grade 8 Chapter 5 image 2

Slope m = (y2 -y1)/(x2 – x1) = (10 – 8)/(1 – 0) = 2/1 = 2
y-intercept b = 8
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = 2 and b = 8.
y = 2x + 8

Writing Linear Equations from a Table Worksheet Question 8.
The table shows how much an air-conditioning repair company charges for different numbers of hours of work. Graph the data, and find the slope and y-intercept from the graph. Then write the equation for the graph in slope-intercept form.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 2: Writing Linear Equations from a Table img 7
Type below:
____________

Answer:
Grade 8 Chapter 5 image 3

Slope m = (y2 -y1)/(x2 – x1) = (100 – 50)/(1 – 0) = 50/1 = 50
y-intercept b = 50
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = 50 and b = 50.
y = 50x + 50

Question 9.
A friend gave Ms. Morris a gift card for a local car wash. The table shows the linear relationship of how the value left on the card relates to the number of car washes.
a. Write an equation that shows the number of dollars left on the card.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 2: Writing Linear Equations from a Table img 8
Type below:
____________

Answer:
y = -1.50x + 30

Explanation:
Slope m = (y2 -y1)/(x2 – x1) = (18 – 30)/(8 – 0) = -12/8 = -1.5
y-intercept b = 30
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = -1.50 and b = 30.
y = -1.50x + 30

Question 9.
b. Explain the meaning of the negative slope in this situation.
Type below:
____________

Answer:
The negative slope means that the amount on the card decreased by $1.5 per car wash

Question 9.
c. What is the maximum value of x that makes sense in this context? Explain.
x = _______

Answer:
x = 20

Explanation:
0 = -1.50x + 30
1.5x = 30
x = 30/1.5 = 20
the maximum value of x = 20

The tables show linear relationships between x and y. Write an equation in slope-intercept form for each relationship.

Question 10.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 2: Writing Linear Equations from a Table img 9
Type below:
____________

Answer:
Slope m = (y2 -y1)/(x2 – x1) = (3 – 1)/(2 – 0) =  2/2 = 1
y-intercept b = 1
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = 1 and b = 1.
y =  x + 1

Writing Equations from a Table Worksheet y=mx+b Answer Key Question 11.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 2: Writing Linear Equations from a Table img 10
Type below:
____________

Answer:
Slope m = (y2 -y1)/(x2 – x1) = (6 – 4)/(0 – 1) =  -2/1 = -2
y-intercept b = 6
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = -2 and b = 6.
y = -2x + 6

Writing Linear Equations from a Table – Page No. 138

Question 12.
Finance
Desiree starts a savings account with $125.00. Every month, she deposits $53.50.
a. Complete the table to model the situation.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 2: Writing Linear Equations from a Table img 11
Type below:
____________

Answer:
Grade 8 Chapter 5 image 4

Question 12.
b. Write an equation in slope-intercept form that shows how much money Desiree has in her savings account after x months.
Type below:
____________

Answer:
y = 53.5x + 125

Explanation:
Desiree starts a savings account with $125.00. Every month, she deposits $53.50.
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = 53.5 and b = 125.
y = 53.5x + 125

Question 12.
c. Use the equation to find how much money Desiree will have in savings after 11 months.
$ ________

Answer:
$713.5

Explanation:
y = 53.5x + 125
The value of x is 11
y = 53.5 (11) + 125 = 588.5 + 125 = 713.5
Desiree will have $713.5 in savings after 11 months.

Graphing and Writing Linear Equations Answer Key Question 13.
Monty documented the amount of rain his farm received on a monthly basis, as shown in the table.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 2: Writing Linear Equations from a Table img 12
a. Is the relationship linear? Why or why not?
Type below:
____________

Answer:
No

Explanation:
The change in the months is constant but the change in rainfall is not constant.

Question 13.
b. Can an equation be written to describe the amount of rain? Explain.
Type below:
____________

Answer:
No

Explanation:
There is no apparent pattern in the given data.

H.O.T.

Focus on Higher Order Thinking

Question 14.
Analyze Relationships
If you have a table that shows a linear relationship, when can you read the value for b, in y = mx + b, directly from the table without drawing a graph or doing any calculations? Explain.
Type below:
____________

Answer:
You can read the value of b directly from the table whrn the table contains the input value of 0 and its corresponding output values (value of y when x = 0)

Question 15.
What If?
Jaíme graphed linear data given in the form (cost, number). The y-intercept was 0. Jayla graphed the same data given in the form (number, cost). What was the y-intercept of her graph? Explain.
Type below:
____________

Answer:
Jaíme graphed linear data given in the form (cost, number). The y-intercept was 0. Jayla graphed the same data given in the form (number, cost).
Jaíme’s graph contained (0, 0). Since Jayal’s data were the same y-intercept is 0 but x and y are switched.

Guided Practice – Linear Relationships and Bivariate Data – Page No. 144

Use the following graphs to find the equation of the linear relationship.

Question 1.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 3: Linear Relationships and Bivariate Data img 13
Type below:
____________

Answer:
y = 30x

Explanation:
Grade 8 Chapter 5 image 5
Find the slope using two points from the grapgh by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (1, 30) and (x2, y2) = (2, 60)
Slope m = (y2 -y1)/(x2 – x1) = (60 – 30)/(2 – 1) = 30/1 = 30
y-intercept b = 0
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = 30 and b = 0.
y = 30x

Linear Relationships and Bivariate Data Answer Key Question 2.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 3: Linear Relationships and Bivariate Data img 14
Type below:
____________

Answer:
y = 2.5x + 2

Explanation:
Grade 8 Chapter 5 image 6
Find the slope using two points from the grapgh by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (2, 7) and (x2, y2) = (4, 12)
Slope m = (y2 -y1)/(x2 – x1) = (12 – 7)/(4 – 2) = 5/2 = 2.5
y-intercept b = 2
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = 2.5 and b = 2.
y = 2.5x + 2

Question 3.
The graph shows the relationship between the number of hours a kayak is rented and the total cost of the rental. Write an equation of the relationship. Then use the equation to predict the cost of a rental that lasts 5.5 hours.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 3: Linear Relationships and Bivariate Data img 15
Type below:
____________

Answer:
y = 20x + 30
The cost of a rental that lasts 5.5 hours is $140

Explanation:
Find the slope using two points from the grapgh by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (2, 70) and (x2, y2) = (4, 110)
Slope m = (y2 -y1)/(x2 – x1) = (110 – 70)/(4 – 2) = 40/2 = 20
y-intercept b = 30
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = 20 and b = 30.
y = 20x + 30
Substitute the value of x is 5.5 hrs
y = 20 (5.5) + 30
y = 110 + 30 = 140
The cost of a rental that lasts 5.5 hours is $140

Does each of the following graphs represent a linear relationship? Why or why not?

Question 4.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 3: Linear Relationships and Bivariate Data img 16
____________

Answer:
Yes; The graph has a constant rate of change

Explanation:
Find the slope using two points from the graph by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (2, 6) and (x2, y2) = (5, 15)
Slope m = (y2 -y1)/(x2 – x1) = (15 – 6)/(5 – 2) = 9/3 = 3
y-intercept b = 6
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = 3 and b = 6.
y = 3x + 6
The values change according to the constant change in the x values.

Question 5.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 3: Linear Relationships and Bivariate Data img 17
____________

Answer:
No; The graph does not have a constant rate of change

Essential Question Check-In

Question 6.
How can you tell if a set of bivariate data shows a linear relationship?
Type below:
____________

Answer:
It is a linear relationship if the rate of change is constant or the graph is a straight line.

Independent Practice – Linear Relationships and Bivariate Data – Page No. 145

Does each of the following tables represent a linear relationship? Why or why not?

Question 7.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 3: Linear Relationships and Bivariate Data img 18
____________

Answer:
Linear Relationship

Explanation:
Find Rate of Change using Slope m = (y2 -y1)/(x2 – x1)
Rate of Change = (45 – 15)/(9 – 3) = 30/6 = 5
Rate of Change = (105 – 45)/(21 – 9) = 60/12 = 5
It is a Linear Relationship as the rate of the change is constant.

Question 8.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 3: Linear Relationships and Bivariate Data img 19
____________

Answer:
not a Linear Relationship

Explanation:
Find Rate of Change using Slope m = (y2 -y1)/(x2 – x1)
Rate of Change = (76.8 – 30)/(8 – 5) = 46./3 = 15.6
Rate of Change = (235.2 – 76.8)/(14 – 8) = 158.4/6 = 26.4
It is not a Linear Relationship as the rate of the change is varies.

Explain whether or not you think each relationship is linear.

Question 9.
the cost of equal-priced DVDs and the number purchased
____________

Answer:
y = cx

Explanation:
The relation between the cost of equal-priced DVDs and the number purchased is linear because the change rate is constant. If we denote with c the cost of a DVD, with x the number of purchased DVDs and with y the total cost of the purchased DVDs, we get an equation of the form:
y = cx

Question 10.
the height of a person and the person’s age
____________

Answer:
Non – Linear relationship

Explanation:
The height of a person does not increase at a constant rate with a person’s age

Question 11.
the area of a square quilt and its side length
____________

Answer:
Non – Linear relationship

Explanation:
The rate of change in the area of a square quilt increase as the side length increases.

Question 12.
the number of miles to the next service station and the number of kilometers
____________

Answer:
Linear relationship

Explanation:
The number of miles increases at a constant rate with the number of kilometers.

Question 13.
Multistep
The Mars Rover travels 0.75 feet in 6 seconds. Add the point to the graph. Then determine whether the relationship between distance and time is linear, and if so, predict the distance that the Mars Rover would travel in 1 minute.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 3: Linear Relationships and Bivariate Data img 20
Distance traveled in 1 min.: _______ ft.

Answer:
Relationship is linear
Distance traveled in 1 min.: 7.5 ft.

Explanation:
Grade 8 Chapter 5 image 7
The graph is a straight line. So, the Relationship is linear
The equation representing the relationship between distance and time.
y = 0.125x
The value of x is 1 min = 60 sec
y = 0.125 (60) = 7.5 ft
Distance traveled in 1 min.: 7.5 ft.

Linear Relationships and Bivariate Data – Page No. 146

Question 14.
Make a Conjecture
Zefram analyzed a linear relationship, found that the slope-intercept equation was y=3.5x+16, and made a prediction for the value of y for a given value of x. He realized that he made an error calculating the y-intercept and that it was actually 12. Can he just subtract 4 from his prediction if he knows that the slope is correct? Explain.
____________

Answer:
Yes

Explanation:
The value of y is calculated using y = 3.5x+ 16. Since the slope of the point remains the same, 4 can be subtracted from the predicted answer as the value of y would be: y = 3.5x+ 16 – 4

H.O.T.

Focus on Higher Order Thinking

Question 15.
Communicate Mathematical Ideas
The table shows a linear relationship. How can you predict the value of y when x = 6 without finding the equation of the relationship?
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Lesson 3: Linear Relationships and Bivariate Data img 21
Type below:
____________

Answer:
57

Explanation:
Find Rate of Change using Slope m = (y2 -y1)/(x2 – x1)
Rate of Change = (76 – 38)/(8 – 4) = 38/4 = 9.5
Since the difference between 8 and S is 2, subtract twice the value of the rate of change from 76
y = 76 – (9.5 × 2) =57

Question 16.
Critique Reasoning
Louis says that if the differences between the values of x are constant between all the points on a graph, then the relationship is linear. Do you agree? Explain.
____________

Answer:
No

Explanation:
The difference between y should be constant for a constant interval of x.

Question 17.
Make a Conjecture
Suppose you know the slope of a linear relationship and one of the points that its graph passes through. How could you predict another point that falls on the graph of the line?
Type below:
____________

Answer:
Find the equation of the linear relationship using the slope and given point. The insert any x-value to find a y value on the graph of the line.

Question 18.
Explain the Error
Thomas used (7, 17.5) and (18, 45) from a graph to find the equation of a linear relationship as shown. What was his mistake?
m = \(\frac{45-7}{18-17.5}=\frac{38}{0.5}\) = 79
y = 79x + b =
49 = 79 × 18 + b
45 = 1422 + b, so b = −1337
The equation is y = 79x − 1377
Type below:
____________

Answer:
He subtracted the x value of the first point from the y of the second point and the y value of the second point from the x value of the first point.
Hence, the slope is incorrect and the equation is incorrect as well. The correct slope is
(45 – 17.5)/(18 – 7) = 27.5/11 = 2.5

5.1 Writing Linear Equations from Situations and Graphs – Model Quiz – Page No. 147

Write the equation of each line in slope-intercept form.

Question 1.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Model Quiz img 22
Type below:
____________

Answer:
y = 30x + 20

Explanation:
Find the slope using two points from the graph by m = (y2 -y1)/(x2 – x1) where (x1, y1) = (0, 20), (x2, y2) = (2, 80)
Substitute the value of m and (x1, y1) = (0, 20), (x2, y2) = (2, 80)
Slope m = (y2 -y1)/(x2 – x1) = (80 – 20)/(2 – 0) = 60/2 = 30
Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b):
80 = 30(2) + b
y-intercept b = 20
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = 30 and b = 20.
y = 30x + 20

Writing Linear Equations from Situations and Graphs Lesson 5.1 Answer Key Question 2.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Model Quiz img 23
Type below:
____________

Answer:
y = -10x + 60

Explanation:
Find the slope using two points from the graph by m = (y2 -y1)/(x2 – x1) where (x1, y1) = (6, 0), (x2, y2) = (0, 60)
Substitute the value of m and (x1, y1) = (6, 0), (x2, y2) = (0, 60)
Slope m = (y2 -y1)/(x2 – x1) = (60 – 0)/(0 – 6) = -60/6 = -10
Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b):
60 = -10(0) + b
y-intercept b = 60
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = -10 and b = 60.
y = -10x + 60

5.2 Writing Linear Equations from a Table

Write the equation of each linear relationship in slope-intercept form.

Question 3.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Model Quiz img 24
Type below:
____________

Answer:
y = 0.35x + 1.5

Explanation:
Find the slope using two points from the graph by m = (y2 -y1)/(x2 – x1) where (x1, y1) = (0, 1.5), (x2, y2) = (100, 36.5)
Substitute the value of m and (x1, y1) = (0, 1.5), (x2, y2) = (100, 36.5)
Slope m = (y2 -y1)/(x2 – x1) = (36.5 – 1.5)/(100 – 0) = 35/100 = 0.35
Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b):
1.5 =0.35(0) + b
y-intercept b = 1.5
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = 0.35 and b = 1.5.
y = 0.35x + 1.5

Question 4.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Model Quiz img 25
Type below:
____________

Answer:
y = -0.6x + 109

Explanation:
Find the slope using two points from the graph by m = (y2 -y1)/(x2 – x1) where (x1, y1) = (25, 94), (x2, y2) = (35, 88)
Substitute the value of m and (x1, y1) = (25, 94), (x2, y2) = (35, 88)
Slope m = (y2 -y1)/(x2 – x1) = (88 – 94)/(35 – 25) = -6/10 = -0.6
Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b):
88 = -0.6(35) + b
y-intercept b = 109
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = -0.6 and b = 109.
y = -0.6x + 109

5.3 Linear Relationships and Bivariate Data

Write the equation of the line that connects each set of data points.

Question 5.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Model Quiz img 26
Type below:
____________

Answer:
y = 2/3x + 26 2/3

Explanation:
Grade 8 Chapter 5 image 8
Find the slope using two points from the graph by m = (y2 -y1)/(x2 – x1) where (x1, y1) = (20, 40), (x2, y2) = (50, 60)
Substitute the value of m and (x1, y1) = (20, 40), (x2, y2) = (50, 60)
Slope m = (y2 -y1)/(x2 – x1) = (60 – 40)/(50 – 20) = 20/30 = 2/3
Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b):
40 = 2/3(20) + b
y-intercept b = 26 2/3
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = 2/3 and b = 26 2/3.
y = 2/3x + 26 2/3

8th Grade Math Linear Equations Question 6.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Model Quiz img 27
Type below:
____________

Answer:
y = -3x + 140

Explanation:
Grade 8 Chapter 5 image 9
Find the slope using two points from the graph by m = (y2 -y1)/(x2 – x1) where (x1, y1) = (25, 65), (x2, y2) = (30, 50)
Substitute the value of m and (x1, y1) = (25, 65), (x2, y2) = (30, 50)
Slope m = (y2 -y1)/(x2 – x1) = (50 – 65)/(30 – 25) = -15/5 = -3
Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b):
50 = -3(30) + b
y-intercept b = 140
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = -3 and b = 140.
y = -3x + 140

Essential Question

Question 7.
Write a real-world situation that can be represented by a linear relationship.
Type below:
____________

Answer:
There is an hourly fee of $15 for renting the ski gear per hour and an entry fee of $10 for the ski lodge.

Selected Response – Mixed Review – Page No. 148

Question 1.
An hourglass is turned over with the top part filled with sand. After 3 minutes, there are 855 mL of sand in the top half. After 10 minutes, there are 750 mL of sand in the top half. Which equation represents this situation?
Options:
a. y = 285x
b. y = −10.5x + 900
c. y = −15x + 900
d. y = 75x

Answer:
c. y = −15x + 900

Explanation:
Identify the input and output variable
Input: Number of minutes
Output: Quantity of sand in the hourglass
Write the given information as ordered pair (3, 855), (10, 750)
Find the slope using two points from the graph by m = (y2 -y1)/(x2 – x1) where (x1, y1) = (10, 750), (x2, y2) =(3, 855)
Substitute the value of m and (x1, y1) = (10, 750), (x2, y2) =(3, 855)
Slope m = (y2 -y1)/(x2 – x1) = (855 – 750)/(3 – 10) = -105/7 = -15
Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b):
750 = -15(10) + b
y-intercept b = 900
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = -15 and b = 900.
y = -15x + 900

Linear Equations Project-Based Learning Answer Key Question 2.
Which graph shows a linear relationship?
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Mixed Review img 28
Options:
a. A
b. B
c. C
d. D

Answer:
b. B

Explanation:
Graph B represents the linear relationship
The data appears to lie on a straight line

Question 3.
What are the slope and y-intercept of the relationship shown in the table?
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Mixed Review img 29
Options:
a. slope = 0.05, y-intercept = 1,500
b. slope = 0.5, y-intercept = 1,500
c. slope = 0.05, y-intercept = 2,000
d. slope = 0.5, y-intercept = 2,000

Answer:
c. slope = 0.05, y-intercept = 2,000

Explanation:
Find the slope using two points from the graph by m = (y2 -y1)/(x2 – x1) where (x1, y1) = (10,000, 2,500), (x2, y2) =(20,000, 3,000)
Substitute the value of m and (x1, y1) = (10,000, 2,500), (x2, y2) =(20,000, 3,000)
Slope m = (y2 -y1)/(x2 – x1) = (3,000 – 2,500)/(20,000 – 10,000) = 500/10,000 = 0.05
Substituting the value of slope (m) and (x, y) in the slope intercept form to find y intercept (b):
3,000 = 0.05(20,000) + b
y-intercept b = 2,000

Question 4.
Which is the sum of 3.15 × 107 + 9.3 × 106? Write your answer in scientific notation.
Options:
a. 4.08 × 107
b. 4.08 × 106
c. 0.408 × 108
d. 40.8 × 106

Answer:
a. 4.08 × 107

Explanation:
Given 3.15 × 107 + 9.3 × 106?
(3.15 + 0.93) × 107
4.08 × 107

Mini-Task

Lesson 5 Skills Practice Graph A Line Using Intercepts Question 5.
Franklin’s faucet was leaking, so he put a bucket underneath to catch the water. After a while, Franklin started keeping track of how much water was in the bucket. His data is in the table below.
Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations Mixed Review img 30
a. Is the relationship linear or nonlinear?
_____________

Answer:
The relationship linear

Explanation:
Find the rate of change by Difference in Quarts/Hours
(6.5 – 5)/(3 – 2) = 1.5
(8 – 6.5)/(4 – 3) = 1.5
(9.5 – 8)/(5 – 4) = 1.5
Since the rate of change is constant, the relationship is linear.

Question 5.
b. Write the equation for the relationship.
Type below:
_____________

Answer:
Rate of change is the slope of the linear equation
m = 1.5

Question 5.
c. Predict how much water will be in the bucket after 14 hours if Franklin doesn’t stop the leak.
_______ quarts

Answer:
23 quarts

Explanation:
Substituting the value of slope (m) and (x, y) in the slope intercept form to find y-intercept (b)
y = mx + b
5 = 1.5(2) + b
y-intercept b = 2
Substituting the value of the slope m and y-intercept in the slope-intercept form. y = mx + b where, m = 1.5 and b = 2.
y = 1.5x + 2
The x value is 2
y = 1.5(2) + 2 = 23
There will be 23 quarts after 14 hrs.

Conclusion:

Go Math Grade 8 Answer Key Chapter 5 Writing Linear Equations PDF to Download. Learn the modern maths in a simple way with the help of Go Math Grade 8 Chapter 5 Answer Key. Quickly begin your learning by downloading a PDF of Go Math Grade 8 Chapter 5 Writing Linear Equations Solution Key.

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Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships

The best Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships PDF for more people who are seeking for math learning in an easy way. Find the top-suggested ways of math problem-solving methods and learn the best way to solve math. The list of all practice questions of Go Math Grade 8 Answer Key are given here in this article. The students can find and practice all the questions to score the good marks in the exam.

Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships

You can enjoy solving math problems with the help of Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships. Download Go Math Grade 8 Chapter 4 Nonproportional Relationships Solution Key. Many students refer to HMH Go Math Grade 8 Chapter 4 Answer Key for the best practice.

Lesson 1: Representing Linear Nonproportional Relationships

Lesson 2: Determining Slope and y-intercept

Lesson 3: Graphing Linear Nonproportional Relationships Using Slope and y-intercept

Lesson 4: Proportional and Nonproportional Situations 

Lesson 5: Representing Linear Nonproportional Relationships – Model Quiz

Mixed Review 

Guided Practice – Representing Linear Nonproportional Relationships – Page No. 98

Make a table of values for each equation.

Question 1.
y = 2x + 5
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 1: Representing Linear Nonproportional Relationships img 1
Type below:
____________

Answer:
grade 8 chapter 4 image 13

Explanation:
y = 2x + 5
Choose several values for x and substitute in the equation to find y.
x = 2(-2) + 5 = 1
x = 2(-1) + 5 = 3
x = 2(0) + 5 = 5
x = 2(1) + 5 = 7
x = 2(2) + 5 = 9

Graphing Linear Nonproportional Relationships Worksheet Answers Question 2.
y = \(\frac{3}{8}\)x − 5
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 1: Representing Linear Nonproportional Relationships img 2
Type below:
____________

Answer:
grade 8 chapter 4 image 14

Explanation:
y = \(\frac{3}{8}\)x − 5
Choose several values for x and substitute in the equation to find y.
x = 3/8(-8) – 5 = -8
x = 3/8(0) – 5 = -5
x = 3/8(8) – 5 = -2
x = 3/8(16) – 5 = 1
x = 3/8(24) – 5 = 4

Explain why each relationship is not proportional.

Question 3.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 1: Representing Linear Nonproportional Relationships img 3
First calculate \(\frac{y}{x}\) for the values in the table.
____________

Answer:
The relationship is not proportional

Explanation:
Find y/x
3/0 = undefined
7/2 = 3.5
11/4 = 2.75
15/6 = 2.5
19/8 = 2.375
The ratio is not constant, hence relationship is not proportional.

Question 4.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 1: Representing Linear Nonproportional Relationships img 4
__________________

Answer:
The graph is a straight line but does not pass through the origin. So, the relationship is not proportional.

Complete the table for the equation. Then use the table to graph the equation.

Question 5.
y = x − 1
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 1: Representing Linear Nonproportional Relationships img 5
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 1: Representing Linear Nonproportional Relationships img 6
Type below:
____________

Answer:
grade 8 chapter 4 image 15

grade 8 chapter 4 image 16

Explanation:
y = x – 1
Choose several values of x and substitute in the equation to find y to draw a table.
x = -2; y = -2 – 1 = -2
x = -1; y = -1 -1 = -2
x = 0; y = 0 -1 = -1
x = 1; y = 1 – 1 = 0
x = 2; y = 2 -1 = 1
Also, Plot the ordered pairs from the table. Then draw a line connecting the points to represent all the possible solutions

Essential Question Check-In

Question 6.
How can you choose values for x when making a table of values representing a real-world situation?
Type below:
____________

Answer:
When choosing values for x in a real-world situation, you choose positive values with an appropriate interval to represent the array of data.

Independent Practice – Representing Linear Nonproportional Relationships – Page No. 99

State whether the graph of each linear relationship is a solid line or a set of unconnected points. Explain your reasoning.

Question 7.
The relationship between the number of $4 lunches you buy with a $100 school lunch card and the money remaining on the card
____________

Answer:
Set of unconnected points.

Explanation:
You cannot buy a fractional part of a lunch.
Set of unconnected points.

Question 8.
The relationship between time and the distance remaining on a 3-mile walk for someone walking at a steady rate of 2 miles per hour.
____________

Answer:
A solid line

Explanation:
The relationship between time and the distance remaining on a 3-mile walk for someone walking at a steady rate of 2 miles per hour. The distance remaining can be a fraction. The time can be in a fraction as well.
A solid line

Nonproportional Relationship Graph Question 9.
Analyze Relationships
Simone paid $12 for an initial year’s subscription to a magazine. The renewal rate is $8 per year. This situation can be represented by the equation y = 8x + 12, where x represents the number of years the subscription is renewed and y represents the total cost.
a. Make a table of values for this situation.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 1: Representing Linear Nonproportional Relationships img 7
Type below:
____________

Answer:
grade 8 chapter 4 image 17

Explanation:
y = 8x + 12
Choose several values for x and substitute in the equation to find y.

Question 9.
b. Draw a graph to represent the situation. Include a title and axis labels.
Type below:
____________

Answer:
grade 8 chapter 4 image 18

Explanation:
Plot the ordered pairs from the table. Then draw a line connecting the points to represent all the possible solutions

Question 9.
c. Explain why this relationship is not proportional.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 1: Representing Linear Nonproportional Relationships img 8
Type below:
____________

Answer:
It is not proportional as the graph does not pass through the origin

Explanation:
When a relationship is proportional, the graph of the equation passes through the origin.
It is not proportional as the graph does not pass through the origin

Question 9.
d. Does it make sense to connect the points on the graph with a solid line? Explain.
Type below:
____________

Answer:
No

Explanation:
No; The subscription is renewed for the entire year and cannot be done for a fraction of the year. The number of years must be a whole numb, so the total cost goes up in $8 increments.

Representing Linear Nonproportional Relationships – Page No. 100

Question 10.
Analyze Relationships
A proportional relationship is a linear relationship because the rate of change is constant (and equal to the constant of proportionality). What is required of a proportional relationship that is not required of a general linear relationship?
Type below:
____________

Answer:
The ratio between one quantity to the other quantity should be constant for a proportional linear relationship. The graph should be a straight line that passes through the origin.

Lesson 4.1 Representing Linear Nonproportional Relationships Answer Key Question 11.
Communicate Mathematical Ideas
Explain how you can identify a linear non-proportional relationship from a table, a graph, and an equation.
Type below:
____________

Answer:
In a table, the ratios y/x will not be equal. A graph will not pass through the origin. An equation will be in the form y = mx + b where b is not equal to 0.

Focus on Higher Order Thinking

Question 12.
Critique Reasoning
George observes that for every increase of 1 in the value of x, there is an increase of 60 in the corresponding value of y. He claims that the relationship represented by the table is proportional. Critique George’s reasoning.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 1: Representing Linear Nonproportional Relationships img 9
Type below:
____________

Answer:
The ratio is not constant, hence the relationship cannot be proportional.

Explanation:
Find y/x
90/1 = 90
150/2 = 75
210/3 = 70
270/4 = 67.5
330/5 = 66
The ratio is not constant, hence the relationship cannot be proportional.

Question 13.
Make a Conjecture
Two parallel lines are graphed on a coordinate plane. How many of the lines could represent proportional relationships? Explain.
Type below:
____________

Answer:
Maximum one

Explanation:
When there are two parallel lines, only one can pass through the origin and a line representing a proportional relationship must pass through the origin.
Maximum one

Guided Practice – Determining Slope and y-intercept – Page No. 104

Find the slope and y-intercept of the line in each graph.

Question 1.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 2: Determining Slope and y-intercept img 10
slope m = _____ y-intercept b = _____
m = ____________
b = ____________

Answer:
slope m = -2 y-intercept b = 1
m = -2
b = 1

Explanation:
Find the slope using two points from the grapgh by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (0, 1) and (x2, y2) = (2, -3)
Slope m = (y2 -y1)/(x2 – x1) = (-3 – 1)/(2 – 0) = -4/2 = -2
From the graph when x = 0
y-intercept (b) = 1

Lesson 4.2 Determining Slope and Y-Intercept Answer Key Pdf Question 2.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 2: Determining Slope and y-intercept img 11
slope m = _____ y-intercept b = _____
m = ____________
b = ____________

Answer:
slope m = 5 y-intercept b = -15
m = 5
b = -15

Explanation:
Find the slope using two points from the graph by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (3, 0) and (x2, y2) = (0, -15)
Slope m = (y2 -y1)/(x2 – x1) = (-15 – 0)/(0 – 3) = 15/3 = 5
From the graph when x = 0
y-intercept (b) = -15

Question 3.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 2: Determining Slope and y-intercept img 12
slope m = _____ y-intercept b = _____
Type below:
____________

Answer:
slope m = 3/2 y-intercept b = -2
m = 3/2
b = -2

Explanation:
Find the slope using two points from the graph by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (0, -2) and (x2, y2) = (2, 1)
Slope m = (y2 -y1)/(x2 – x1) = (1 – (-2))/(2 – 0) = 3/2
From the graph when x = 0
y-intercept (b) = -2

Question 4.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 2: Determining Slope and y-intercept img 13
slope m = _____ y-intercept b = _____
m = ____________
b = ____________

Answer:
slope m = -3 y-intercept b = 9
m = -3
b = 9

Explanation:
Find the slope using two points from the grapgh by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (3, 0) and (x2, y2) = (0, 9)
Slope m = (y2 -y1)/(x2 – x1) = (9 – 0))/(0 – 3) = -9/3 = -3
From the graph when x = 0
y-intercept (b) = 9

Find the slope and y-intercept of the line represented by each table.

Question 5.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 2: Determining Slope and y-intercept img 14
slope m = _____ y-intercept b = _____
m = ____________
b = ____________

Answer:
slope m = 3 y-intercept b = 1
m = 3
b = 1

Explanation:
Find the slope using two points from the grapgh by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (8, 25) and (x2, y2) = (6, 19)
Slope m = (y2 -y1)/(x2 – x1) = (19 – 25)/(6 – 8) = 6/2 = 3
From the graph when x = 0
y-intercept (b) = 1

Question 6.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 2: Determining Slope and y-intercept img 15
slope m = _____ y-intercept b = _____
m = ____________
b = ____________

Answer:
slope m = -4 y-intercept b = 140
m = -4
b = 140

Explanation:
Find the slope using two points from the grapgh by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (5, 120) and (x2, y2) = (15, 80)
Slope m = (y2 -y1)/(x2 – x1) = (80 – 120)/(15 – 5) = -40/10 = -4
From the graph when x = 0
y-intercept (b) = 140

Essential Question Check-In

Question 7.
How can you determine the slope and the y-intercept of a line from a graph?
Type below:
____________

Answer:
Choose any two points on the line from the graph and use it to find the slope. Determine the point where the line crosses the y-axis to find the y-intercept.

Independent Practice – Determining Slope and y-intercept – Page No. 105

Question 8.
Some carpet cleaning costs are shown in the table. The relationship is linear. Find and interpret the rate of change and the initial value for this situation.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 2: Determining Slope and y-intercept img 16
Type below:
_____________

Answer:
Find the slope using two points
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (1, 125) and (x2, y2) = (3, 225)
Slope m = (y2 -y1)/(x2 – x1) = (225 – 125)/(3 – 1) = 100/2 = 50
Find the initial value when the value of x is 0
Work backward from x = 1 to x = 0
(175 – 125)/(2 – 1) = 50/1 = 50
Subtract the difference of x and y from the first point.
x = 1 – 1 = 0
y = 125 – 50 = 75
y-intercept (b) = 75
The slope/rate of change represents the increase in the cost of cleaning the rooms for a unit increase in the number of rooms. The y-intercept shows the initial cost of carpet cleaning.

Question 9.
Make Predictions
The total cost to pay for parking at a state park for the day and rent a paddleboat are shown.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 2: Determining Slope and y-intercept img 17
a. Find the cost to park for a day and the hourly rate to rent a paddleboat.
Type below:
_____________

Answer:
$5

Explanation:
Find the slope using two points
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (1, 17) and (x2, y2) = (2, 29)
Slope m = (y2 -y1)/(x2 – x1) = (29 – 17)/(2 – 1) = 12/1 = 12
Find the initial value when the value of x is 0
Work backward from x = 1 to x = 0
(29 – 17)/(2 – 1) = 12/1 = 12
Subtract the difference of x and y from the first point.
x = 1 – 1 = 0
y = 17 – 12 = 5
The cost to park for a day is $5.

Question 9.
b. What will Lin pay if she rents a paddleboat for 3.5 hours and splits the total cost with a friend? Explain.
$ _____________

Answer:
$23.5

Explanation:
When Lin paddles for 3.5hr
Total Cost = 3.5(12) + 5 = 47
Lin’s cost = 47/2 = 23.5

Question 10.
Multi-Step
Raymond’s parents will pay for him to take sailboard lessons during the summer. He can take half-hour group lessons or half-hour private lessons. The relationship between cost and number of lessons is linear.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 2: Determining Slope and y-intercept img 18
a. Find the rate of change and the initial value for the group lessons.
Type below:
____________

Answer:
$25

Explanation:
Find the slope using two points
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (1, 55) and (x2, y2) = (2, 85)
Slope m = (y2 -y1)/(x2 – x1) = (85 – 55)/(2 – 1) = 30/1 = 30
Rate of change is $30 for per lesson
Find the initial value when the value of x is 0
Work backward from x = 1 to x = 0
(85 – 55)/(2 – 1) = 30/1 = 30
Subtract the difference of x and y from the first point.
x = 1 – 1 = 0
y = 55 – 30 = 25
The initial value of the group lesson is $25.

Question 10.
b. Find the rate of change and the initial value for the private lessons.
Type below:
_____________

Answer:
$25

Explanation:
Find the slope using two points
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (1, 75) and (x2, y2) = (2, 125)
Slope m = (y2 -y1)/(x2 – x1) = (125 – 75)/(2 – 1) = 50/1 = 50
The rate of change is $50 per lesson
Find the initial value when the value of x is 0
Work backward from x = 1 to x = 0
(125 – 75)/(2 – 1) = 50/1 = 50
Subtract the difference of x and y from the first point.
x = 1 – 1 = 0
y = 75 – 50 = 25
The initial value of the private lesson is $25.

Question 10.
c. Compare and contrast the rates of change and the initial values.
Type below:
_____________

Answer:
The initial value for both types of lessons is the same. The rate of change is higher for private lessons than group lesson

Explanation:
Compare the results of a and b
The initial value for both types of lessons is the same. The rate of change is higher for private lessons than group lesson

Vocabulary – Determining Slope and y-intercept – Page No. 106

Explain why each relationship is not linear.

Question 11.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 2: Determining Slope and y-intercept img 19
Type below:
_____________

Answer:
The rate of change is not constant, hence the relationship is not linear

Explanation:
Find the rate of change using two points Slope m = (y2 -y1)/(x2 – x1)
(6.5 – 4.5)/(2 – 1) = 2
(8.5 – 6.5)/(3 – 2) = 2
(11.5 – 8.5)/(4 – 3) = 3
The rate of change is not constant, hence the relationship is not linear

Question 12.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 2: Determining Slope and y-intercept img 20
Type below:
_____________

Answer:
The rate of change is not constant, hence the relationship is not linear

Explanation:
Find the rate of change using two points Slope m = (y2 -y1)/(x2 – x1)
(126 – 140)/(5 – 3) = -7
(110 – 126)/(7 – 5) = -8
(92 – 110)/(9 – 7) = -9
The rate of change is not constant, hence the relationship is not linear

Question 13.
Communicate Mathematical Ideas
Describe the procedure you performed to derive the slope-intercept form of a linear equation.
Type below:
_____________

Answer:
Express the slope m between a random point (x, y) on the line and the point (0, b) where the line crosses the y-axis. Then solve the equation for y.

H.O.T.

Focus on Higher Order Thinking

Question 14.
Critique Reasoning
Your teacher asked your class to describe a real-world situation in which a y-intercept is 100 and the slope is 5. Your partner gave the following description: My younger brother originally had 100 small building blocks, but he has lost 5 of them every month since.
a. What mistake did your partner make?
Type below:
_____________

Answer:
If the brother loses 5 blocks every month, the slope would be -5 and not 5.

Explanation:
When the initial value is decreasing, the slope is negative.
If the brother loses 5 blocks every month, the slope would be -5 and not 5.

Question 14.
b. Describe a real-world situation that does match the situation.
Type below:
_____________

Answer:
I bought a 100-card pack and buy 5 additional cards every month.

Explanation:
Real-world situation
I bought a 100-card pack and buy 5 additional cards every month.

Question 15.
Justify Reasoning
John has a job parking car. He earns a fixed weekly salary of $300 plus a fee of $5 for each car he parks. His potential earnings for a week are shown in the graph. At what point does John begin to earn more from fees than his fixed salary? Justify your answer.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 2: Determining Slope and y-intercept img 21
Type below:
_____________

Answer:
After parking 60 cars, John’s earnings become $600 double his initial base salary of $300.
Hence, after parking 61 cars, his earning from the fee becomes more than his fixed salary.

Explanation:
He earns the same in fees as his fixed salary for parking 300/5 = 60
After parking 60 cars, John’s earnings became $600 double his initial base salary of $300.
Hence, after parking 61 cars, his earning from the fee becomes more than his fixed salary.

Guided Practice – Graphing Linear Nonproportional Relationships Using Slope and Y-intercept – Page No. 110

Graph each equation using the slope and the y-intercept.

Question 1.
y = \(\frac{1}{2}\)x − 3
slope = _____ y-intercept = _____
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 3: Graphing Linear Nonproportional Relationships Using Slope and y-intercept img 22
Type below:
_____________

Answer:
slope = 1/2 y-intercept = -3
Grade 8 Chapter 4 image 1

Explanation:
y = 1/2 x – 3
The y-intercept is b = -3. Plot the point that contains the y-intercept (0, -3)
The slope m = 1/2. Use the slope to find a second point. From (0, -3) count 1 unit up and 2 unit right. The new point is (2, -2)
Draw a line through the points

Question 2.
y = −3x + 2
slope = _____ y-intercept = _____
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 3: Graphing Linear Nonproportional Relationships Using Slope and y-intercept img 23
Type below:
_____________

Answer:
slope = -3 y-intercept = 2
Grade 8 Chapter 4 image 2

Explanation:
y = -3x + 2
The y-intercept is b = 2. Plot the point that contains the y-intercept (0, 2)
The slope m = -3/1. Use the slope to find a second point. From (0, 2) count 3 unit down and 1 unit right. The new point is (1, -1)
Draw a line through the points

Nonproportional Relationships Worksheet Question 3.
A friend gives you two baseball cards for your birthday. Afterward, you begin collecting them. You buy the same number of cards once each week. The equation y = 4x + 2 describes the number of cards, y, you have after x weeks.
a. Find and interpret the slope and the y-intercept of the line that represents this situation. Graph y = 4x + 2. Include axis labels.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 3: Graphing Linear Nonproportional Relationships Using Slope and y-intercept img 24
Type below:
_____________

Answer:
Grade 8 Chapter 4 image 3

Explanation:
y = 4x + 2
The y-intercept is b = 2. Plot the point that contains the y-intercept (0, 2)
The slope m = 4. Use the slope to find a second point. From (0, 2) count 4 unit up and 1 unit right. The new point is (1, 6)
Draw a line through the points

Question 3.
b. Discuss which points on the line do not make sense in this situation. Then plot three more points on the line that do make sense.
Type below:
_____________

Answer:
Grade 8 Chapter 4 image 4

Explanation:
The points with a negative value of x or y do not make sense as the number of cards or weeks cannot be negative.

Essential Question Check-In

Question 4.
Why might someone choose to use the y-intercept and the slope to graph a line?
Type below:
_____________

Answer:
When the relationship is given in the form y = mx + b, the y-intercept (b) and the slope (m) are easily accessible and easily calculable. Therefore, it is a good practice to use them to graph the line.

Independent Practice – Graphing Linear Nonproportional Relationships Using Slope and Y-intercept – Page No. 111

Question 5.
Science
A spring stretches in relation to the weight hanging from it according to the equation y = 0.75x + 0.25 where x is the weight in pounds and y is the length of the spring in inches.
a. Graph the equation. Include axis labels.
Type below:
_____________

Answer:
grade 8 chapter 4 image 7

Explanation:
y = 0.75x + 0.25
Slope m = 0.75 and y-intercept = 0.25
Plot the point that contains the y-intercept (0, 0.25)
The slope is m = 0.75/1. Use the slope to find a second point. From (0,0.25) count 0.75 unit up and 1 unit right. The new point is (1, 1)

Question 5.
b. Interpret the slope and the y-intercept of the line.
Type below:
_____________

Answer:
The slope represents the increase in the length of spring in inches for each increase of pound of weight. y-intercept represents the unstretched length of the spring When there is no weight attached.

Question 5.
c. How long will the spring be if a 2-pound weight is hung on it? Will the length double if you double the weight? Explain
Type below:
_____________

Answer:
When there is a 2-pound weight hung, the length of the spring would be 1.75 inches. No, When there is a 4-pound weight hung, the length of the spring would be 3.25 inches and not 3.5 inches.

Look for a Pattern

Identify the coordinates of four points on the line with each given slope and y-intercept.

Question 6.
slope = 5, y-intercept = -1
Type below:
_____________

Answer:
(2, 9)
(3, 14)

Explanation:
slope = 5, y-intercept = -1
Plot the point that contains the y-intercept (0, -1)
The slope is m = 5/1. Use the slope to find a second point. From (0, -1) count 5 unit up and 1 unit right. The new point is (1, 4)
Follow the same procedure to find the remaining three points.
(2, 9)
(3, 14)

Question 7.
slope = -1, y-intercept = 8
Type below:
_____________

Answer:
(2, 6)
(3, 5)

Explanation:
slope = -1, y-intercept = 8
Plot the point that contains the y-intercept (0, 8)
The slope is m = -1/1. Use the slope to find a second point. From (0, 8) count 1 unit down and 1 unit right. The new point is (1, 7)
Follow the same procedure to find the remaining three points.
(2, 6)
(3, 5)

Question 8.
slope = 0.2, y-intercept = 0.3
Type below:
_____________

Answer:
(2, 0.7)
(3, 0.9)

Explanation:
slope = 0.2, y-intercept = 0.3
Plot the point that contains the y-intercept (0, 0.3)
The slope is m = 0.2/1. Use the slope to find a second point. From (0, 0.3) count 0.2 unit up and 1 unit right. The new point is (1, 0.5)
Follow the same procedure to find the remaining three points.
(2, 0.7)
(3, 0.9)

Question 9.
slope = 1.5, y-intercept = -3
Type below:
_____________

Answer:
(2, 0)
(3, 1.5)

Explanation:
slope = 1.5, y-intercept = -3
Plot the point that contains the y-intercept (0, -3)
The slope is m = 1.5/1. Use the slope to find a second point. From (0, -3) count 1.5 unit up and 1 unit right. The new point is (1, -1.5)
Follow the same procedure to find the remaining three points.
(2, 0)
(3, 1.5)

Question 10.
slope = −\(\frac{1}{2}\), y-intercept = 4
Type below:
_____________

Answer:
(4, 2)
(6, 1)

Explanation:
slope = −\(\frac{1}{2}\), y-intercept = 4
Plot the point that contains the y-intercept (0, 4)
The slope is m = −\(\frac{1}{2}\)/1. Use the slope to find a second point. From (0, 4) count 1 unit down and 2 unit right. The new point is (2, 3)
Follow the same procedure to find the remaining three points.
(4, 2)
(6, 1)

Question 11.
slope = \(\frac{2}{3}\), y-intercept = -5
Type below:
_____________

Answer:
(6, -1)
(9, 1)

Explanation:
slope = \(\frac{2}{3}\), y-intercept = -5
Plot the point that contains the y-intercept (0, -5)
The slope is m = \(\frac{2}{3}\). Use the slope to find a second point. From (0, -5) count 2 unit up and 3 unit right. The new point is (3, -3)
Follow the same procedure to find the remaining three points.
(6, -1)
(9, 1)

Question 12.
A music school charges a registration fee in addition to a fee per lesson. Music lessons last 0.5 hour. The equation y = 40x + 30 represents the total cost y of x lessons. Find and interpret the slope and y-intercept of the line that represents this situation. Then find four points on the line.
Type below:
_____________

Answer:
y = 40x + 30
Slope = 40
y-intercept = 30
The slope represents the fee of the classes per lesson and the y-intercept represents the registration fee.
Plot the point that contains the y-intercept (0, 30)
The slope is m = 40/1. Use the slope to find a second point. From (0, 30) count 40 units up and 1 unit right. The new point is (1, 70)
Follow the same procedure to find the remaining three points.
(2, 110)
(3, 150)

Graphing Linear Nonproportional Relationships Using Slope and Y-intercept – Page No. 112

Question 13.
A public pool charges a membership fee and a fee for each visit. The equation y = 3x + 50 represents the cost y for x visits.
a. After locating the y-intercept on the coordinate plane shown, can you move up three gridlines and right one gridline to find a second point? Explain.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 3: Graphing Linear Nonproportional Relationships Using Slope and y-intercept img 25
Type below:
_____________

Answer:
Yes

Explanation:
Yes; Since the horizontal and vertical gridlines each represent 25 units, hence moving up 3 gridlines and right 1 gridline represents a slope of 75/25 or 3

Question 13.
b. Graph the equation y = 3x + 50. Include axis labels. Then interpret the slope and y-intercept.
Type below:
_____________

Answer:
grade 8 chapter 4 image 8
The slope represents the fee per visit and the y-intercept represents the membership fee.

Explanation:
Slope = 3
y-intercept = 50
The slope represents the fee of the classes per lesson and the y-intercept represents the registration fee.
Plot the point that contains the y-intercept (0, 50)
The slope is m = 3/1. Use the slope to find a second point. From (0, 50) count 3 units up and 1 unit right. The new point is (1, 53)

Question 13.
c. How many visits to the pool can a member get for $200?
______ visits

Answer:
50 visits

Explanation:
You would get 50 visits for $200
grade 8 chapter 4 image 9

H.O.T.

Focus on Higher Order Thinking

Question 14.
Explain the Error
A student says that the slope of the line for the equation y = 20 − 15x is 20 and the y-intercept is 15. Find and correct the error.
Type below:
_____________

Answer:
The slope is -15 as it represents the change in y per unit change in x. The y-intercept is 20 when x = 0.

Explanation:
y = 20 − 15x
The slope is -15 as it represents the change in y per unit change in x. The y-intercept is 20 when x = 0.

Question 15.
Critical Thinking
Suppose you know the slope of a linear relationship and a point that its graph passes through. Can you graph the line even if the point provided does not represent the y-intercept? Explain.
Type below:
_____________

Answer:
Yes. You can plot the given point and use the slope to find a second point. Connect the points by drawing a line.

Question 16.
Make a Conjecture
Graph the lines y = 3x, y = 3x − 3, and y = 3x + 3. What do you notice about the lines? Make a conjecture based on your observation.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 3: Graphing Linear Nonproportional Relationships Using Slope and y-intercept img 26
Type below:
_____________

Answer:
grade 8 chapter 4 image 10

Explanation:
let’s tale the example
y = 3x
y = 3x – 3
y = 3x + 3
We notice that the lines are parallel to each other: the slopes of the lines are equal but the y-intersection point differs.

Guided Practice – Proportional and Nonproportional Situations – Page No. 117

Determine if each relationship is a proportional or nonproportional situation. Explain your reasoning.

Question 1.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 4: Proportional and Nonproportional Situations img 27
Look at the origin.
_____________

Answer:
Proportional relationship

Explanation:
Proportional relationship
The graph passes through the origin. Graph of a proportional relationship must pass through the origin

Question 2.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 4: Proportional and Nonproportional Situations img 28
_____________

Answer:
Non-proportional relationship

Explanation:
The graph does not pass through the origin. The graph of a proportional relationship must pass through the origin
Non-proportional relationship

Lesson 4 Skills Practice Proportional and Nonproportional Relationships Answer Key Question 3.
q = 2p + \(\frac{1}{2}\)
Compare the equation with y = mx + b.
_____________

Answer:
q = 2p + \(\frac{1}{2}\)
The equation is in the form y = mx + b, with p being used es the variable instead of x and q instead of y. The value of m is 2, and the value b is 1/2. Since b is not 0, the relationship presented through the above equation is non-proportional.

Question 4.
v = \(\frac{1}{10}\)u
_____________

Answer:
Proportional relationship

Explanation:
v = \(\frac{1}{10}\)u
Compare with the form of equation y = mx + b. The equation represents a proportional relationship if b = 0
Proportional relationship

Proportional and Nonproportional Situations – Page No. 118

The tables represent linear relationships. Determine if each relationship is a proportional or nonproportional situation.

Question 5.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 4: Proportional and Nonproportional Situations img 29
Find the quotient of y and x.
_____________

Answer:
proportional relationship

Explanation:
Find the ratio y/x
12/3 = 4
36/9 = 4
84/21 = 4
Since the ratio is constant, the relationship is proportional.

Lesson 4 Skills Practice Proportional and Nonproportional Relationships Question 6.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 4: Proportional and Nonproportional Situations img 30
_____________

Answer:
non-proportional

Explanation:
Find the ratio y/x
4/22 = 2/11
8/46 = 4/23
10/58 = 5/29
Since the ratio is not constant, the relationship is non-proportional.

Question 7.
The values in the table represent the number of households that watched three TV shows and the ratings of the shows. The relationship is linear. Describe the relationship in other ways.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 4: Proportional and Nonproportional Situations img 31
Type below:
_____________

Answer:
proportional relationship

Explanation:
Find the ratio y/x
12/15,000,000 = 0.0000008
16/20,000,000 = 0.0000008
20/25,000,000 = 0.0000008
Since the ratio is constant, the relationship is proportional.

Essential Question Check-In

Question 8.
How are using graphs, equations, and tables similar when distinguishing between proportional and nonproportional linear relationships?
Type below:
_____________

Answer:
The ratio between y to x is constant when the relationship is proportional. Graphs, tables, and equations all can be used to find the ratio. The ratio is not constant when the relationship is non-proportional.

Independent Practice – Proportional and Nonproportional Situations – Page No. 119

Question 9.
The graph shows the weight of a cross-country team’s beverage cooler based on how much sports drink it contains.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 4: Proportional and Nonproportional Situations img 32
a. Is the relationship proportional or nonproportional? Explain.
_____________

Answer:
Non-proportional

Explanation:
The graph does not pass through the origin. Graph of a proportional relationship must pass through the origin
Non-proportional

Question 9.
b. Identify and interpret the slope and the y-intercept.
Type below:
_____________

Answer:
Slope m = (y2 -y1)/(x2 – x1) = (12 – 10)/(4 – 0) = 0.5
y-intercept is the weight of the empty cooler, which is 10 lbs.

Explanation:
Find the slope using two points from the grapgh by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (0, 10) and (x2, y2) = (4, 12)
Slope m = (y2 -y1)/(x2 – x1) = (12 – 10)/(4 – 0) = 0.5
From the graph when x = 0
y-intercept (b) = 10
y-intercept is the weight of the empty cooler, which is 10 lbs.

In 10–11, tell if the relationship between a rider’s height above the first floor and the time since the rider stepped on the elevator or escalator is proportional or nonproportional. Explain your reasoning.

Question 10.
The elevator paused for 10 seconds after you stepped on before beginning to rise at a constant rate of 8 feet per second.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 4: Proportional and Nonproportional Situations img 33
_____________

Answer:
Non-proportional

Explanation:
As there is a pause of 10 seconds, it would be the y-intercept of the graph (when x = 0)
Non-proportional

Representing Linear Nonproportional Relationships Lesson 4.1 Answer Key Question 11.
Your height, h, in feet above the first floor on the escalator is given by h = 0.75t, where t is the time in seconds.
_____________

Answer:
Proportional

Explanation:
Comparing with y = mx + b, where b = 0
Proportional

Analyze Relationships

Compare and contrast the two graphs.

Question 12.
Graph A       Graph B
y = \(\frac{1}{3}\) x        y = \(\sqrt { x } \)
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 4: Proportional and Nonproportional Situations img 34
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 4: Proportional and Nonproportional Situations img 35
Type below:
_____________

Answer:
Graph A represents a linear relationship while Graph B represents an exponential relationship. They both pass through the origin and the value of y increases with an increase in x.

Proportional and Nonproportional Situations – Page No. 120

Question 13.
Represent Real-World Problems
Describe a real-world situation where the relationship is linear and nonproportional.
Type below:
_____________

Answer:
The entrance fee to the amusement park is $8 and there is a fee of $2 per ride.

H.O.T.

Focus on Higher Order Thinking

Question 14.
Mathematical Reasoning
Suppose you know the slope of a linear relationship and one of the points that its graph passes through. How can you determine if the relationship is proportional or nonproportional?
Type below:
_____________

Answer:
Use the graph and the given point to determine the second point. Connect the two points by a straight line. If the graph passes through the origin, the relationship is proportional and if the graph does not pass through the origin, the relationship is non-proportional.

Lesson 4 Proportional and Nonproportional Relationships Question 15.
Multiple Representations
An entrant at a science fair has included information about temperature conversion in various forms, as shown. The variables F, C, and K represent temperatures in degrees Fahrenheit, degrees Celsius, and kelvin, respectively.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Lesson 4: Proportional and Nonproportional Situations img 36
a. Is the relationship between kelvins and degrees Celsius proportional? Justify your answer in two different ways.
_____________

Answer:
No, the relationship is not proportional.

Explanation:
Compare the equation B to the form: y = mx + b. Since b is not equal to 0, the relationship is non-proportional.
Find the ratio between the Kelvin and Degrees Celsius. Since the ratio is not constant, the relationship is non-proportional.
281.15/8 = 35.14
288.15/15 = 19.21
309.15/36 = 8.59
No, the relationship is not proportional.

Question 15.
b. Is the relationship between degrees Celsius and degrees Fahrenheit proportional? Why or why not?
_____________

Answer:
No, the relationship is not proportional.

Explanation:
Compare the equation A to the form: y = mx + b. Since b is not equal to 0, the relationship is non-proportional.
No, the relationship is not proportional.

4.1 Representing Linear Nonproportional Relationships – Model Quiz – Page No. 121

Lesson 4.1 Representing Linear Nonproportional Relationships Question 1.
Complete the table using the equation y = 3x + 2.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Model Quiz img 37
Type below:
_____________

Answer:
grade 8 chapter 4 image 11

Explanation:
Given y = 3x + 2
grade 8 chapter 4 image 11
x = -1; y = 3(-1) + 2 = -3 + 2 = -1
x = 0; y = 3(0) +2 = 2
x = 1; y = 3(1) + 2 = 3 + 2 = 5
x = 2; y = 3(2) + 2 = 6 + 2 = 8
x = 3: y = 3(3) + 2 = 9 + 2 = 11

4.2 Determining Slope and Y-intercept

Question 2.
Find the slope and y-intercept of the line in the graph.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Model Quiz img 38
Type below:
_____________

Answer:
Slope = 3
y-intercept (b) = 1

Explanation:
Find the slope using two points from the grapgh by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (0, 1) and (x2, y2) = (1, 4)
Slope m = (y2 -y1)/(x2 – x1) = (4 – 1)/(1 – 0) = 3/1
From the graph when x = 0
y-intercept (b) = 1

4.3 Graphing Linear Nonproportional Relationships

Question 3.
Graph the equation y = 2x − 3 using slope and y-intercept.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Model Quiz img 39
Type below:
_____________

Answer:
grade 8 chapter 4 image 12

Explanation:
Slope = 2
y-intercept = -3
Plot the point that contains the y-intercept (0, -3)
The slope is m = 2/1. Use the slope to find a second point. From (0, -3) count 2 unit up and 1 unit right. The new point is (1, -1)
Draw a line through the points

4.4 Proportional and Nonproportional Situations

Question 4.
Does the table represent a proportional or a nonproportional linear relationship?
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Model Quiz img 40
_____________

Answer:
Since the ratio is constant, the table represents a proportional linear relationship.

Explanation:
Find the ratio y/x
4/1 = 4
8/2 = 4
12/3 = 4
16/4 = 4
20/5 = 4
Since the ratio is constant, the table represents a proportional linear relationship.

Question 5.
Does the graph in Exercise 2 represent a proportional or a nonproportional linear relationship?
_____________

Answer:
It represents a non-proportional linear relationship

Explanation:
The line of the graph does not pass through the origin. The graph of a proportional relationship must pass through the origin.
It represents a non-proportional linear relationship

Question 6.
Does the graph in Exercise 3 represent a proportional or a nonproportional relationship?
_____________

Answer:
It represents a non-proportional linear relationship

Explanation:
The line of the graph does not pass through the origin. The graph of a proportional relationship must pass through the origin
It represents a non-proportional linear relationship

Essential Question

Question 7.
How can you identify a linear nonproportional relationship from a table, a graph, and an equation?
Type below:
_____________

Answer:
In a table, the ratio of y/x is not constant for a non-proportional relationship.
In a graph, the line of the graph does not pass through the origin for a non-proportional relationship.
In an equation, the b is not equal to y = mx +b for a non-proportional relationship.

Selected Response – Mixed Review – Page No. 122

Question 1.
The table below represents which equation?
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Mixed Review img 41
Options:
a. y = −x − 10
b. y = −6x
c. y = −4x − 6
d. y = −4x + 2

Answer:
c. y = −4x − 6

Explanation:
From the table, you can see that the y-intercept (when x = 0) is b = -6. Comparable to y = mx + b
The table is represented by Option C y = -4x – 6

Question 2.
The graph of which equation is shown below?
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Mixed Review img 42
Options:
a. y = −2x + 3
b. y = −2x + 1.5
c. y = 2x + 3
d. y = 2x + 1.5

Answer:
a. y = −2x + 3

Explanation:
From the table, you can see that the y-intercept (when x = 0) is b = 3. Comparable to y = mx + b
The Option B and D are rejected.
Since the graph is slanting downwards, the slope is negative.
Option C is rejected
The graph represents y = -2x + 3

Question 3.
The table below represents a linear relationship.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Mixed Review img 43
What is the y-intercept?
Options:
a. -4
b. -2
c. 2
d. 3

Answer:
b. -2

Explanation:
Find the rate of change
(7 – 4)/(3 – 2) = (10 – 7)/(4 – 3) = 3
Find the value of y for x = 0
Works backward from x = 2 to x = 1
x = 2 – 1 = 1
y = 4 – 3 = 1
x = 1 – 1 = 0
y = 1 – 3 = -2
y-intercept = -2

Question 4.
Which equation represents a nonproportional relationship?
Options:
a. y = 3x + 0
b. y = −3x
c. y = 3x + 5
d. y = \(\frac{1}{3}\)x

Answer:
c. y = 3x + 5

Explanation:
For a non-proportional relationship, the equation is y = mx + b and b is not equal to 0.
Option C represents a non-proportional relationship y = 3x + 5

Question 5.
The table shows a proportional relationship. What is the missing y-value?
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Mixed Review img 44
Options:
a. 16
b. 20
c. 18
d. 24

Answer:
c. 18

Explanation:
Find the ratio y/x
6/4 = 3/2
Since the relationship is proportional, the ratio is constant.
Using the ratio to find missing y
3/2 = y/12
y = 3/2 × 12 = 18

Question 6.
What is 0.00000598 written in scientific notation?
Options:
a. 5.98 × 10-6
b. 5.98 × 10-5
c. 59.8 × 10-6
d. 59.8 × 10-7

Answer:
c. 59.8 × 10-6

Explanation:
0.00000598
Move the decimal 6 points
59.8 × 10-6

Mini-Task

Question 7.
The graph shows a linear relationship.
Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships Mixed Review img 45
a. Is the relationship proportional or nonproportional?
____________

Answer:
It represents a non-proportional linear relationship

Explanation:
The line of the graph does not pass through the origin. The graph of a proportional relationship must pass through the origin.
It represents a non-proportional linear relationship

Question 7.
b. What is the slope of the line?
_______

Answer:
Slope m = -2

Explanation:
Find the slope using two points from the graph by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (0, -2) and (x2, y2) = (2, 1)
Slope m = (y2 -y1)/(x2 – x1) = (-3 -1)/(0 + 2) = -4/2 = -2

Question 7.
c. What is the y-intercept of the line?
_______

Answer:
y-intercept (b) = -3

Explanation:
From the graph when x = 0
y-intercept (b) = -3

Question 7.
d. What is the equation of the line?
Type below:
____________

Answer:
y = -2x – 3

Explanation:
Substitute m and b in the form: y = mx + b
y = -2x – 3

Conclusion:

Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships for Download. All the beginners can easily start their practice and learn the maths in an easy way. Quickly start your practice with Go Math Grade 8 Answer Key.

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Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships

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Go Math Grade 8 Chapter 3 Proportional Relationships Answer Key

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Lesson 1: Representing Proportional Relationships

Lesson 2: Rate of Change and Slope

Lesson 3: Interpreting the Unit As Slope

Lesson 4: Representing Proportional Relationships – Model Quiz

Mixed Review

Guided Practice – Representing Proportional Relationships – Page No. 74

Question 1.
Vocabulary
A proportional relationship is a relationship between two quantities in which the ratio of one quantity to the other quantity is/is not constant.
______ constant

Answer:
is constant

Explanation:
The rent would be proportional so therefore it is.

Question 2.
Vocabulary
When writing an equation of a proportional relationship in the form y = kx, k represents the __________________________.
______________

Answer:
constant of proportionality

Explanation:
When writing an equation of a proportional relationship in the form y = kx, k represents the constant of proportionality.

Representing Proportional Relationships With Equations Answer Key Question 3.
Write an equation that describes the proportional relationship between the number of days and the number of weeks in a given length of time.
a. Complete the table.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 1: Representing Proportional Relationships img 1
Type below:
______________

Answer:
Grade 8 Chapter 3 image 1

Question 3.
b. Let x represent _____.
Let y represent _____.
The equation that describes the relationship is _____.
Type below:
______________

Answer:
Let x represent the time in weeks.
Let y represent the time in days.
The equation that describes the relationship is y = 7x.

Each table or graph represents a proportional relationship. Write an equation that describes the relationship.

Question 4.
Physical Science
The relationship between the numbers of oxygen atoms and hydrogen atoms in water.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 1: Representing Proportional Relationships img 2
Type below:
______________

Answer:
y = 2x
Grade 8 Chapter 3 image 2

Explanation:
x represents the Oxygen atoms
y represents the Hydrogen atoms
For every point of the x-axis, the y-axis is varying with 2x times.
y = 2x

Question 5.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 1: Representing Proportional Relationships img 3
Type below:
______________

Answer:
y = 30x

Explanation:
x represents the Distance in inches
y represents the Actual Distance in miles
For every point of the x-axis, the y-axis is varying with 30x times.
y = 30x

Essential Question Check-In

Question 6.
If you know the equation of a proportional relationship, how can you draw the graph of the equation?
Type below:
______________

Answer:
Use the equation to make a table with x-values and y-values. Then graph the points (x, y) and draw a line through the points.

Independent Practice – Representing Proportional Relationships – Page No. 75

The table shows the relationship between temperatures measured on the Celsius and Fahrenheit scales.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 1: Representing Proportional Relationships img 4

Question 7.
Is the relationship between the temperature scales proportional? Why or why not?
______________

Answer:
No. The ratios of the numbers in each column are not equal.

Explanation:
Fahrenheit Temperature/Celsius Temperature = 50/10 = 5
86/30 = 2.87
122/50 = 2.44
The relationship is not propotional as the ratio is not constant.

Question 8.
Describe the graph of the Celsius-Fahrenheit relationship.
Type below:
______________

Answer:
A line starting at (0, 32) and slanting upward to the right.

Explanation:
The graph is a straight line with a y-intercept of 32.

Representing Proportional Relationships Worksheet Question 9.
Analyze Relationships
Ralph opened a savings account with a deposit of $100. Every month after that, he deposited $20 more.
a. Why is the relationship described not proportional?
Type below:
______________

Answer:
The account had a balance of $100, to begin with.

Question 9.
b. How could the situation be changed to make the situation proportional?
Type below:
______________

Answer:
Have Ralph open the account with no money, to begin with, and then put $20 every month.

Question 10.
Represent Real-World Problems
Describe a real-world situation that can be modeled by the equation y = \(\frac{1}{20}\)x. Be sure to describe what each variable represents.
Type below:
______________

Answer:
If x is the number of nickels you have, y = \(\frac{1}{20}\)x is the amount of money you have in dollars.

Look for a Pattern

The variables x and y are related proportionally.

Question 11.
When x = 8, y = 20. Find y when x = 42.
_______

Answer:
y = 105

Explanation:
x = 8, y = 20
y/x = 20/8
y = 20x/8
when x = 42
y = (20 × 42)/8
y = 105

Using Proportional Relationships Answer Key Question 12.
When x = 12, y = 8. Find x when y = 12.
_______

Answer:
x = 18

Explanation:
x/y = 12/8
x = 12y/8
when y = 12
x = (12 × 12)/8
x = 18

Representing Proportional Relationships – Page No. 76

Question 13.
The graph shows the relationship between the distance that a snail crawls and the time that it crawls.
a. Use the points on the graph to make a table.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 1: Representing Proportional Relationships img 5
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 1: Representing Proportional Relationships img 6
Type below:
______________

Answer:
Grade 8 Chapter 3 image 10

Question 13.
b. Write the equation for the relationship and tell what each variable represents.
Type below:
______________

Answer:
y = 1/10 x, where y is the time in minutes and x is the distance in inches

Question 13.
c. How long does it take the snail to crawl 85 inches?
_______ minutes

Answer:
8.5 minutes

H.O.T.

Focus on Higher Order Thinking

Question 14.
Communicate Mathematical Ideas
Explain why all of the graphs in this lesson show the first quadrant but omit the other three quadrants.
Type below:
______________

Answer:
All of the graphs represent real-world data for which both x and y take on only nonnegative values, which graph in the first quadrant or on the axes. If either x or y or both could be negative, then other quadrants would be needed.

Representing Proportional Relationships Question 15.
Analyze Relationships
Complete the table.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 1: Representing Proportional Relationships img 7

Grade 8 Chapter 3 image 10
a. Are the length of a side of a square and the perimeter of the square related proportionally? Why or why not?
______________

Answer:
Yes. The ratio of the perimeter of a square to its side length is always 4.

Question 15.
b. Are the length of a side of a square and the area of the square related proportionally? Why or why not?
______________

Answer:
No. The ratio of the area of a square to its side length is not constant

Question 16.
Make a Conjecture
A table shows a proportional relationship where k is the constant of proportionality. The rows are then switched. How does the new constant of proportionality relate to the original one?
Type below:
______________

Answer:
It is the reciprocal of the original constant of proportionality

Guided Practice – Rate of Change and Slope – Page No. 80

Tell whether the rates of change are constant or variable.

Question 1.
building measurements _____
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 2: Rate of Change and Slope img 8
______________

Answer:
constant

Explanation:
Input variable: Feet
Output variable: Yard
For every point of the Yard, the Feet is increasing 3 times.
So, the answer is constant.

Question 2.
computers sold _____
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 2: Rate of Change and Slope img 9
______________

Answer:
variable

Explanation:
Identify the input and output variables
Input variable: Week
Output variable: The number sold
x-axis and y-axis points are not varying constantly. So, the answer is variable.

Question 3.
distance an object falls _____
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 2: Rate of Change and Slope img 10
______________

Answer:
variable

Explanation:
Identify the input and output variables
Input variable: Time
Output variable: Distance
x-axis and y-axis points are not varying constantly. So, the answer is variable.

Question 4.
cost of sweaters _____
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 2: Rate of Change and Slope img 11
______________

Answer:
constant

Explanation:
Identify the input and output variables
Input variable: Number
Output variable: Cost
x-axis and y-axis points are varying constantly. So, the answer is constant.

Erica walks to her friend Philip’s house. The graph shows Erica’s distance from home over time.

Question 5.
Find the rate of change from 1 minute to 2 minutes.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 2: Rate of Change and Slope img 12
\(\frac{\text { change in distance }}{\text { change in time }}=\frac{400-?}{2-?}=\frac{?}{?}=?\) ft per min
________ feet per minute

Answer:
200 feet per minute

Explanation:
(400 – 200)/(2 – 1)
200/1
200 feet per minute

Question 6 (request help)
Find the rate of change from 1 minute to 4 minutes.
______ feet per minute

Answer:
200 ft per min

Explanation:
change in distance/change in time
(800 – 200)/(4 – 1)
600/3 = 200 ft per min

Find the slope of each line.

Question 7.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 2: Rate of Change and Slope img 13
slope = _____
________

Answer:
slope = -2

Explanation:
From the given image, the x-axis is separated by 1 unit.
the y-axis is separated with 1 unit.
For every 1 unit of the x-axis, the slope is taken -2 units on the y-axis
The slope of the line is -2

Lesson 8 Graph Proportional Relationships and Define Slope Question 8.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 2: Rate of Change and Slope img 14
slope = _____
\(\frac{□}{□}\)

Answer:
\(\frac{3}{2}\)

Explanation:
From the given image, the slope is 3/2

Essential Question Check-In

Question 9.
If you know two points on a line, how can you find the rate of change of the variables being graphed?
Type below:
_____________

Answer:
Find the coordinates of two points on the line. Then divide the change in y-values from one point to the next by the change in x-values.

Independent Practice – Rate of Change and Slope – Page No. 81

Question 10.
Rectangle EFGH is graphed on a coordinate plane with vertices at E(-3, 5), F(6, 2), G(4, -4), and H(-5, -1).
a. Find the slopes of each side.
Type below:
_____________

Answer:
Slope EF = 1/3
slope FG = 3
slope GH = -1/3
slope HE =3

Question 10.
b. What do you notice about the slopes of opposite sides?
Type below:
_____________

Answer:
They are the same.

Question 10.
c. What do you notice about the slopes of adjacent sides?
Type below:
_____________

Answer:
They are negative reciprocals of one another.

Question 11.
A bicyclist started riding at 8:00 A.M. The diagram below shows the distance the bicyclist had traveled at different times. What was the bicyclist’s average rate of speed in miles per hour?
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 2: Rate of Change and Slope img 15
_______ miles per hour

Answer:
15 miles per hour

Explanation:
Total distance traveled by bicyclist = 4.5 mile + 7.5 mile = 12 mile
Total time taken by bicyclist = (8:48 A.M – 8:00 A.M) = 48 min = 0.8 hr
conversion used : ( 1 hour = 60 minute)
Average rate of speed = total distance/total time = 15 mile/hr

Lesson 3.2 Rate of Change and Slope Answer Key Question 12.
Multistep
A line passes through (6, 3), (8, 4), and (n, -2). Find the value of n.
_______

Answer:
n = -4

Explanation:
A line passes through (6, 3), (8, 4), and (n, -2).
From the given information, for every 2 points on the x-axis, the y-values are changing one point.
(4, 2), (2, 1), (0, 0), (-2, -1), (-4, -2)

Question 13.
A large container holds 5 gallons of water. It begins leaking at a constant rate. After 10 minutes, the container has 3 gallons of water left.
a. At what rate is the water leaking?
_______ gallons per minute

Answer:
1 gallon every 5 minutes, or 0.2 gal/min

Explanation:
Rate = (5 – 3)/(0 – 10)
= 2/-10
= -0.2
The rate of water leaking is 1 gallon every 5 minutes, or 0.2 gal/min

Question 13.
b. After how many minutes will the container be empty?
_______ minutes

Answer:
25 minutes

Explanation:
Number of minutes = 5/0.2 = 25
It will take 25 minutes for the container to be empty.

Question 14.
Critique Reasoning
Billy found the slope of the line through the points (2, 5) and (-2, -5) using the equation \(\frac{2-(-2)}{5-(-5)}=\frac{2}{5}\). What mistake did he make?
Type below:
_____________

Answer:
He used the change in x over the change in y instead of the change in y over the change in x.

Rate of Change and Slope – Page No. 82

Question 15.
Multiple Representations
Graph parallelogram ABCD on a coordinate plane with vertices at A(3, 4), B(6, 1), C(0, -2), and D(-3, 1).
a. Find the slope of each side.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 2: Rate of Change and Slope img 16
Type below:
_____________

Answer:
slope AB = -1;
slope BC = 1/2
slope CD = -1;
slope DA = 1/2

Explanation:
Grade 8 Chapter 3 image 5

Question 15.
b. What do you notice about the slopes?
Type below:
_____________

Answer:
The slopes of the opposite sides are the same.

Question 15.
c. Draw another parallelogram on the coordinate plane. Do the slopes have the same characteristics?
Type below:
_____________

Answer:
Yes; opposite sides still have the same slope.

H.O.T.

Focus on Higher Order Thinking

Question 16.
Communicate Mathematical Ideas
Ben and Phoebe are finding the slope of a line. Ben chose two points on the line and used them to find the slope. Phoebe used two different points to find the slope. Did they get the same answer? Explain.
_____________

Answer:
Yes. The slope of a line is constant. Therefore, the slope that you calculate will be the same no matter which two points you choose.

Proportional Relationships and Slope Question 17.
Analyze Relationships
Two lines pass through the origin. The lines have slopes that are opposites. Compare and contrast the lines.
Type below:
_____________

Answer:
One line has a positive slope and one has a negative slope. The lines are equally steep, but one slants upward left to right while the other slants downward left to right. The lines cross at the origin.

Question 18.
Reason Abstractly
What is the slope of the x-axis? Explain.
_____________

Answer:
Zero. The rise along the x-axis is zero, while the run along the x-axis is not zero. The slope zero/run or zero.

Guided Practice – Interpreting the Unit As Slope – Page No. 86

Give the slope of the graph and the unit rate.

Question 1.
Jorge: 5 miles every 6 hours
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 3: Interpreting the Unit As Slope img 17
\(\frac{□}{□}\) miles per hour

Answer:
\(\frac{5}{6}\) miles per hour

Explanation:
Grade 8 Chapter 3 image 11
Slope = rise/run = 5/6
The unit rate a the distance traveled and the slope a the graph of the relationship is equal, 5/6 miles per hour
Calculate miles PER hour
5/6

Interpreting the Unit Rate as Slope Lesson 3.3 Answer Key Question 2.
Akiko
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 3: Interpreting the Unit As Slope img 18
\(\frac{□}{□}\) miles per hour

Answer:
\(\frac{5}{4}\) miles per hour

Explanation:
Calculate miles PER hour
5 miles/4hours = 5/4 miles per hour

Question 3.
The equation y = 0.5x represents the distance Henry hikes, in miles, over time, in hours. The graph represents the rate which Clark hikes. Determine which hiker is faster. Explain.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 3: Interpreting the Unit As Slope img 19
___________

Answer:
Clark is faster.

Explanation:
From the equation, Henry’s rate is equal to 0.5, or 1/2 mile per hour. Clark’s rate is the slope of the line, which is 3/2 or 1.5 miles per hour.

Write an equation relating the variables in each table.

Question 4.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 3: Interpreting the Unit As Slope img 20
Type below:
___________

Answer:
y = 15x

Explanation:
y/x = 15/1
y = 15x
Multiply 15 with the x values to get the y values.
y = 15x

Constant of Proportionality Worksheet With Answers Question 5.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 3: Interpreting the Unit As Slope img 21
Type below:
___________

Answer:
y = 3/8 x

Explanation:
y/x = 6/16
y/x = 3/8
y = 3/8 x

Essential Question Check-In

Question 6.
Describe methods you can use to show a proportional relationship between two variables, x and y. For each method, explain how you can find the unit rate and the slope.
Type below:
___________

Answer:
The ratio of y to x gives the unit rate and slope.

Explanation:
If the equation can be written as y = mx, then m is the unit rate and the slope. Graph: When the line passes through the origin, then the value of r at the point (1, r) is the unit rate and the slope.

Independent Practice – Interpreting the Unit As Slope – Page No. 87

Question 7.
A Canadian goose migrated at a steady rate of 3 miles every 4 minutes.
a. Fill in the table to describe the relationship.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 3: Interpreting the Unit As Slope img 22
Type below:
___________

Answer:
Grade 8 Chapter 3 image 6

Explanation:
Canadian goose migrated at a steady rate of 3 miles every 4 minutes
y/x = 3/4; x/y = 4/3
y = 3/4 x; x = 4/3 y
If x = 8, y = 3/4 × 8 = 6
If y = 9, x = 4/3 × 9 = 12
If y = 12, x = 4/3 × 12 = 16
If x = 20, y = 3/4 × 20 = 15

Question 7.
b. Graph the relationship.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 3: Interpreting the Unit As Slope img 23
Type below:
___________

Answer:
Grade 8 Chapter 3 image 7

Explanation:
The points are (4, 3), (8, 6), (12, 9), (16, 12), (20, 15)

Question 7.
c. Find the slope of the graph and describe what it means in the context of this problem.
Slope: \(\frac{□}{□}\) miles per minute

Answer:
Slope: \(\frac{3}{4}\) miles per minute

Explanation:
The unit rate of migration of the goose and the slope of the graph both equal 3/4 mi/min

Question 8.
Vocabulary
A unit rate is a rate in which the first quantity / second quantity in the comparison is one unit.
___________

Answer:
second quantity

Explanation:
A unit rate is a rate in which the “second quantity” in the comparison is one unit

Question 9.
The table and the graph represent the rate at which two machines are bottling milk in gallons per second.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 3: Interpreting the Unit As Slope img 24
a. Determine the slope and unit rate of each machine.
Type below:
___________

Answer:
Machine 1: slope = unit rate = 0.6/1 = 0.6 gal/s
Machine 2: slope = unit rate = 3/4 = 0.75 gal/s

Question 9.
b. Determine which machine is working at a faster rate.
___________

Answer:
Machine 2 is working at a faster rate since 0.75 > 0.6

Interpreting the Unit As Slope – Page No. 88

Question 10.
Cycling
The equation y = \(\frac{1}{9}\) x represents the distance y, in kilometers, that Patrick traveled in x minutes while training for the cycling portion of a triathlon. The table shows the distance y Jennifer traveled in x minutes in her training. Who has the faster training rate?
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 3: Interpreting the Unit As Slope img 25
___________

Answer:
Jennifer has a faster training rate

Explanation:
Patrick’s rate is 1/9 kilometer per minute. Jennifer’s rate is 1/8 kilometer per minute. 1/9 < 1/8. So, Jennifer has a faster training rate.

H.O.T.

Focus on Higher Order Thinking

Question 11.
Analyze Relationships
There is a proportional relationship between minutes and dollars per minute, shown on a graph of printing expenses. The graph passes through the point (1, 4.75). What is the slope of the graph? What is the unit rate? Explain.
______ dollars per minute

Answer:
$4.75/min

Explanation:
slope = unit rate = 4.75.
If the graph of a proportional relationship passes through the point (1, r), then r equals the slope and the unit rate, which is $4.75/min.

Question 12.
Draw Conclusions
Two cars start at the same time and travel at different constant rates. A graph for Car A passes through the point (0.5, 27.5), and a graph for Car B passes through (4, 240). Both graphs show distance in miles and time in hours. Which car is traveling faster? Explain.
___________

Answer:
Car B

Explanation:
The slope and unit rate of speed of Car A is (27.5 – 0)/(0.5 – 0) = 27.5/0.5 = 55 mi/h.
The slope and unit rate of speed of Car B is (240 – 0)/(4 – 0) = 240/4 = 60 mi/h.
60 > 55, so Car B is traveling faster.

Question 13.
Critical Thinking
The table shows the rate at which water is being pumped into a swimming pool. Use the unit rate and the amount of water pumped after 12 minutes to find how much water will have been pumped into the pool after 13 \(\frac{1}{2}\) minutes. Explain your reasoning.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Lesson 3: Interpreting the Unit As Slope img 26
______ gallons

Answer:
243 gallons

Explanation:
The unit rate is 36/2 = 18gal/min.
So, 1 1/2 minutes after 12 minutes, an additional 18 × 1 1/2 = = 27 gallons will be pumped in.
So, the total is 216 + 27 = 243 gal.

3.1 Representing Proportional Relationships – Model Quiz – Page No. 89

Question 1.
Find the constant of proportionality for the table of values.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Model Quiz img 27
______

Answer:
1.5

Explanation:
constant of proportionality
k = y/x = 3/2 = 1.5
k = y/x = 4.5/3 = 1.5
k = y/x = 6/4 = 1.5
k = y/x = 7.5/5 = 1.5

Proportional Relationship 8th Grade Question 2.
Phil is riding his bike. He rides 25 miles in 2 hours, 37.5 miles in 3 hours, and 50 miles in 4 hours. Find the constant of proportionality and write an equation to describe the situation.
Type below:
___________

Answer:
The constant of proportionality is 12.5 miles per hour.

Explanation:
The equation is d = 12.5 × t
25 miles ÷ 2 hours = 12.5 miles/hour
A direct proportionality d = 12.5 × t

3.2 Rate of Change and Slope

Find the slope of each line.

Question 3.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Model Quiz img 28
______

Answer:
Slope = 3

Question 4.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Model Quiz img 29
______

Answer:
Slope = -5

3.3 Interpreting the Unit Rate as Slope

Question 5.
The distance Train A travels is represented by d = 70t, where d is the distance in kilometers and t is the time in hours. The distance Train B travels at various times is shown in the table. What is the unit rate of each train? Which train is going faster?
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Model Quiz img 30
____________

Answer:
Train A: 70 km per hour; Train B: 75 km per hour; Train B is faster.

Explanation:
The distance Train A travels is represented by d = 70t
where d is the distance in kilometers and t is the time in hours
The speed of train A is 70 kilometer per hour
To find the speed of train B use the table
Slope = (300 – 150)/(4 – 2) = 75
The speed of train B is 75 km per hour
The speed of train B is more

Essential Question

Question 6.
What is the relationship among proportional relationships, lines, rates of change, and slope?
Type below:
____________

Answer:
The relationship between the x-axis and y-axis of any graph is a proportional Relationship which is defined by slope i.e. calculating the rate of change of the plotted line.
1) Here proportional change and rate of change are algebraic quantities which specify how one quantity changes with respect to another.
2) Line and Slope are geometric quantities which describe the graph of any equation.

Selected Response – Mixed Review – Page No. 90

Question 1.
Which of the following is equivalent to 5-1?
Options:
a. 4
b. \(\frac{1}{5}\)
c. −\(\frac{1}{5}\)
d. -5

Answer:
b. \(\frac{1}{5}\)

Explanation:
5-1
1/5

Question 2.
Prasert earns $9 an hour. Which table represents this proportional relationship?
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Mixed Review img 31
Options:
a. A
b. B
c. C
d. D

Answer:
a. A

Explanation:
a. 36/4 = 9; 54/6 = 9; 72/8 = 9
b. 36/4 = 9; 45/6 = 7.5; 54/8 = 6.75
c. 9/2 = 4.5
d. 18/2 = 9; 27/3 = 9; 54/4 = 13.5

Chapter 3 Math Test 8th Grade Answers Question 3.
A factory produces widgets at a constant rate. After 4 hours, 3,120 widgets have been produced. At what rate are the widgets being produced?
Options:
a. 630 widgets per hour
b. 708 widgets per hour
c. 780 widgets per hour
d. 1,365 widgets per hour

Answer:
c. 780 widgets per hour

Explanation:
A factory produces widgets at a constant rate. After 4 hours, 3,120 widgets have been produced.
3,120/4 = 780 widgets per hour

Question 4.
A full lake begins dropping at a constant rate. After 4 weeks it has dropped 3 feet. What is the unit rate of change in the lake’s level compared to its full level?
Options:
a. 0.75 feet per week
b. 1.33 feet per week
c. -0.75 feet per week
d. -1.33 feet per week

Answer:
c. -0.75 feet per week

Explanation:
A full lake begins dropping at a constant rate. After 4 weeks it has dropped 3 feet.
(-3 ft)/(4 weeks) = -3/4 ft/wk = -0.75 ft/wk

Question 5.
What is the slope of the line below?
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Mixed Review img 32
Options:
a. -2
b. −\(\frac{1}{2}\)
c. \(\frac{1}{2}\)
d. 2

Answer:
c. \(\frac{1}{2}\)

Explanation:
(-1.5, 1.5) & (1.5, 0)
(0 – 1..5)/(1.5 – (-1.5))
1.5/3
= 1/2

Question 6.
Jim earns $41.25 in 5 hours. Susan earns $30.00 in 4 hours. Pierre’s hourly rate is less than Jim’s, but more than Susan’s. What is his hourly rate?
Options:
a. $6.50
b. $7.75
c. $7.35
d. $8.25

Answer:
b. $7.75

Explanation:
Jim earns $41.25 in 5 hours.
$41.25/5 = 8.25
Jim’s unit rate is $8.25 per hour
30/4 = 7.5
Pierre’s hourly rate is is less than $8.25 but more than $7.50
$7.75

Mini-Task

Question 7.
Joelle can read 3 pages in 4 minutes, 4.5 pages in 6 minutes, and 6 pages in 8 minutes.
a. Make a table of the data.
Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships Mixed Review img 33
Type below:
______________

Answer:
Grade 8 Chapter 3 image 8

Explanation:
Joelle can read 3 pages in 4 minutes,
y/x = 3/4
y = 3/4 x
If x = 6, y = 3/4 × 6 = 4.5
If x = 8, y = 3/4 × 8 = 6

Question 7.
b. Use the values in the table to find the unit rate.
_______ pages per minute

Answer:
0.75 pages per minute

Explanation:
3/4 = 0.75
4.5/6 = 0.75
6/8 = 0.75
The unit rate is 0.75 pages per minute

Question 7.
c. Graph the relationship between minutes and pages read.
Type below:
______________

Answer:
Grade 8 Chapter 3 image 9

Conclusion:

All the students refer to Go Math Grade 8 Answer Key Chapter 3 Proportional Relationships PDF to practice maths. Students can definitely score good marks in the exam with the help of the Go Math Grade 8 Chapter 3 Proportional Relationships Answer Key. Practice all the questions and finish your learning of Chapter 3 Proportional Relationships.

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Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations

Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations PDF is available for free. Learn all the best maths tactics and ways of solving with the help of the Go Math Grade 8 Chapter 7 Solving Linear Equations Solution Key. You can practice all the questions to have a perfect grip on the Grade 8 maths subject. Go Math Grade 8 Answer Key is the best source to practice maths. Don’t leave your home to learn maths in the best way with the help of Go Math Grade 8 Solution Key.

Go Math Grade 8 Chapter 7 Solving Linear Equations Answer Key

Students who use Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations will enjoy solving math questions. Every student is entitled not only to download but also practice them online. Don’t need to use a single paper to practice. Just practice online by looking into the questions and answers available on Go Math Grade 8 Answer Key.

Lesson 1: Equations with the Variable on Both Sides

Lesson 2: Equations with Rational Numbers

Lesson 3: Equations with the Distributive Property

Lesson 4: Equations with Many Solutions or No Solution

Lesson 5: Equations with the Variable on Both Sides

Reviews

Guided Practice – Equations with the Variable on Both Sides – Page No. 200

Use algebra tiles to model and solve each equation.

Question 1.
x + 4 = -x – 4
x = ________

Answer:
x = -4

Explanation:
Model x + 4 on the left side of the mat and -x -4 on the right side.
grade 8 chapter 7 image 1
Add one c-tile to both sides. This represents adding x to both sides of the equation. Remove zero pairs.
grade 8 chapter 7 image 2
Place four -1-tiles on both sides. This represents subtracting -4 from both sides of the equation. Remove zero pairs.
grade 8 chapter 7 image 3
Separate each side into 2 equal groups. One x-tile is equivalent to four -1-tiles.
grade 8 chapter 7 image 4
x = -4

Question 2.
2 – 3x = -x – 8
x = ________

Answer:

Explanation:
Given 2 – 3x = -x – 8
Model 2-3x on the left side of the mat and -x-8 on the right side.
grade 8 chapter 7 image 5
Place one x tile to both sides. This represents subtracting from both sides of the equation.
grade 8 chapter 7 image 6
Remove 2 1 tiles from sides. This represents subtracting from both sides of the equation.
grade 8 chapter 7 image 7
Separate each side into 2 equal groups. One -x tile is equivalent to 5 – 1 tile.
grade 8 chapter 7 image 8
The solution is -x = -5 or x = 5

Solving Linear Equations Grade 8 Question 3.
At Silver Gym, membership is $25 per month, and personal training sessions are $30 each. At Fit Factor, membership is $65 per month, and personal training sessions are $20 each. In one month, how many personal training sessions would Sarah have to buy to make the total cost at the two gyms equal?
________ sessions

Answer:
4 sessions

Explanation:
At Silver Gym, membership is $25 per month, and personal training sessions are $30 each.
Membership + Personal training session = 25 + 30x
At Fit Factor, membership is $65 per month, and personal training sessions are $20 each.
Membership + Personal training session = 65 + 20x
Membership at Silver Gym = Membership at Fit Factor
25 + 30x = 65 + 20x
30x – 20x = 65 – 25
10x = 40
x = 4
Sarah would have to buy 4 sessions for the total cost at the two gyms to be equal.

Question 4.
Write a real-world situation that could be modeled by the equation 120 + 25x = 45x.
Type below:
_______________

Answer:
120 + 25x = 45x
Sarah offers a plan to tutor a student at $25 per her plus a one-time registration fee of $ 120.
Surah offers an alternative plan to tutor a student at $45 per hour and no registration fee.
120 + 25x = 45x

Question 5.
Write a real-world situation that could be modeled by the equation 100 – 6x = 160 – 10x.
Type below:
_______________

Answer:
100 – 6x = 160 – 10x
The initial water in Tank A is 100 gallons and leaks at 6 gallons per week.
The initial water in Tank B is 160 gallon and leaks at 10 gallons per week
100 – 6x = 160 – 10x

Essential Question Check-In

Question 6.
How can you solve an equation with the variable on both sides?
Type below:
_______________

Answer:
Isolate the variable on one side. Add/subtract the variable with a lower coefficient from both sides. Add/subtract the constant (with the variable) from both sides. Divide both sides by coefficient of the isolated variable.

Independent Practice – Equations with the Variable on Both Sides – Page No. 201

Question 7.
Derrick’s Dog Sitting and Darlene’s Dog Sitting are competing for new business. The companies ran the ads shown.
Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations Lesson 1: Equations with the Variable on Both Sides img 1
a. Write and solve an equation to find the number of hours for which the total cost will be the same for the two services.
________ hours

Answer:
3 hours

Explanation:
Hourly rate + One time fee = 5x + 12
Hourly rate + One time fee = 3x + 18
5x + 12 = 3x + 18
5x – 3x = 18 – 12
2x = 6
x = 3
The cost of the two dog sitting would be the same for 3 hrs.

Question 7.
b. Analyze Relationships
Which dog-sitting service is more economical to use if you need 5 hours of service? Explain.
____________

Answer:
Darlene’s Dog Sitting would be cheaper

Explanation:
Let y be the cost of dog sitting after x hours for both companies
y = 5x +12
y = 3x +18
Substitute x = 5
y = 5(5) + 12 = 37
y = 3 (5) + 18 = 33
Compare the cost for both companies for x = 5hr.
$37 > $33
Darlene’s Dog Sitting would be cheaper

Linear Equations 8th Grade Question 8.
Country Carpets charges $22 per square yard for carpeting and an additional installation fee of $100. City Carpets charges $25 per square yard for the same carpeting and an additional installation fee of $70.
a. Write and solve an equation to find the number of square yards of carpeting for which the total cost charged by the two companies will be the same.
_______ square yards

Answer:
10 square yards

Explanation:
Unit square rate + One time installation fee = 22x + 100
Unit square rate + One time installation fee = 25x + 70
22x + 100 = 25x + 70
25x – 22x = 100 – 70
3x = 30
x = 10
The total cost charged by the two companies will be the same for 10 square yards of carpeting.

Question 8.
b. Justify Reasoning
Mr. Shu wants to hire one of the two carpet companies to install carpeting in his basement. Is he more likely to hire Country Carpets or City Carpets? Explain your reasoning.
___________

Answer:
City Carpets are cheaper when x < 10
y = 25(9) + 70 = 295
y = 22(9) + 100 = 298
Country Carpets are cheaper when x > 10
y = 25(11) + 70 = 345
y = 25(11) + 100 = 342
If Mr.Shu needs the carpeting done for less than 10 square yards, he will hire City Carpets and if he needs carpeting for more than 10 square yards, he will hire Country Carpets.

Write an equation to represent each relationship. Then solve the equation.

Question 9.
Two less than 3 times a number is the same as the number plus 10.
________

Answer:
3x – 2 = x + 10
x = 6

Explanation:
Two less than 3 times a number is the same as the number plus 10.
Two less than 3 times x is the same as the x plus 10.
Two less than 3x is the same as the x + 10
3x – 2 is the same as x + 10
3x – 2 = x + 10
3x – x = 10 + 2
2x = 12
x = 6

Question 10.
A number increased by 4 is the same as 19 minus 2 times the number.
______

Answer:
x + 4 = 19 – 2x
x = 5

Explanation:
A number increased by 4 is the same as 19 minus 2 times the number.
x increased by 4 is the same as 19 minus 2x.
x + 4 is the same as 19 – 2x
x + 4 = 19 – 2x
x + 2x = 19 – 4
3x = 15
x = 15/3
x = 5

Question 11.
Twenty less than 8 times a number is the same as 15 more than the number.
Type below:
____________

Answer:
8x – 20 = x + 15
x = 5

Explanation:
Twenty less than 8 times a number is the same as 15 more than the number.
Twenty less than 8 times x is the same as 15 more than the x.
Twenty less than 8x is the same as 15 more than the x
8x – 20 is the same as x + 15
8x – 20 = x + 15
8x – x = 15 + 20
7x = 35
x = 35/7 = 5
x = 5

Equations with the Variable on Both Sides – Page No. 202

Question 12.
The charges for an international call made using the calling card for two phone companies are shown in the table.
Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations Lesson 1: Equations with the Variable on Both Sides img 2
a. What is the length of a phone call that would cost the same no matter which company is used?
_______ minutes

Answer:
10 minutes

Explanation:
Cost of minutes + One time fee = 3x + 35
Cost of minutes + One time fee = 2x + 45
3x + 35 = 2x + 45
3x – 2x = 45 – 35
x = 10
The cost would be same for 10 minutes.

Question 12.
b. Analyze Relationships
When is it better to use the card from Company B?
Type below:
____________

Answer:
y = 3x + 35
y = 3(11) + 35 = $68
y = 2x + 45
y = 2(11) + 45 = $67
Since when x > 10, Company B is cheaper so it should be better to use when the length of the call is greater than 10 minutes.

H.O.T.

Focus on Higher Order Thinking

Question 13.
Draw Conclusions
Liam is setting up folding chairs for a meeting. If he arranges the chairs in 9 rows of the same length, he has 3 chairs left over. If he arranges the chairs in 7 rows of that same length, he has 19 left over. How many chairs does Liam have?
______ chairs

Answer:
75 chairs

Explanation:
Number of chairs in 9 row + leftover chairs = 9x + 3
Number of chairs in 7 row + leftover chairs = 7x + 19
9x + 3 = 7x + 19
9x – 7x = 19 – 3
2x = 16
x = 16/2
x = 8
Total number of chairs = 9(8) + 3 = 75

Solving Linear Equations 8th Grade Question 14.
Explain the Error
Rent-A-Tent rents party tents for a flat fee of $365 plus $125 a day. Capital Rentals rents party tents for a flat fee of $250 plus $175 a day. Delia wrote the following equation to find the number of days for which the total cost charged by the two companies would be the same:
365x + 125 = 250x + 175
Find and explain the error in Delia’s work. Then write the correct equation.
Type below:
____________

Answer:
Delia’s equation
365x + 125 = 250x + 175
The error is that she attached the variable with the flat fee (which is constant) and put the daily rent as a constant (which is variable).
Correct equation
125x + 365 = 175x + 250

Question 15.
Persevere in Problem-Solving
Lilliana is training for a marathon. She runs the same distance every day for a week. On Monday, Wednesday, and Friday, she runs 3 laps on a running trail and then runs 6 more miles. On Tuesday and Sunday, she runs 5 laps on the trail and then runs 2 more miles. On Saturday, she just runs laps. How many laps does Lilliana run on Saturday?
________ laps

Answer:
2 laps

Explanation:
Miles covered by lap + Addition number of miles = 3x + 6
Miles covered by lap + Addition number of miles = 5x + 2
3x + 6 = 5x + 2
5x – 3x = 6 – 2
2x = 4
x = 4/2
x = 2

Guided Practice – Equations with Rational Numbers – Page No. 206

Question 1.
Sandy is upgrading her Internet service. Fast Internet charges $60 for installation and $50.45 per month. Quick Internet has free installation but charges $57.95 per month.
a. Write an equation that can be used to find the number of months at which the Internet service would cost the same.
Type below:
____________

Answer:
50.45x + 60 = 57.95x

Explanation:
Write an equation for Fast Internet, where x is the number of months.
Charge per Month × Number of Months + Installation Fee
50.45x + 60
Write an equation for Quick Internet, where x is the number of months.
Charge per Month × Number of Months + Installation Fee
57.95x
50.45x + 60 = 57.95x

Question 1.
b. Solve the equation.
_______ hours

Answer:
8

Explanation:
50.45x + 60 = 57.95x
57.95x – 50.45x = 60
7.5x = 60
x = 60/7.5
x = 8
The total cost will be the same for 8 months.

Solve.

Question 2.
\(\frac{3}{4}\) n – 18 = \(\frac{1}{4}\) n – 4
______

Answer:
n = 28

Explanation:
3/4 . n – 18 = 1/4 . n – 4
Determine the least common multiple of the denominators
LCM is 4
Multiply both sides of the equation by the LCM
4(3/4 . n – 18) = 4(1/4 . n – 4)
3n – 72 = n – 16
3n – n = -16 + 72
2n = 56
n = 56/2
n = 28

Question 3.
6 + \(\frac{4}{5}\) b = \(\frac{9}{10}\) b
_______

Answer:
b = 60

Explanation:
6 + \(\frac{4}{5}\) b = \(\frac{9}{10}\) b
LCM is 10
10(6 + \(\frac{4}{5}\) b) = 10(\(\frac{9}{10}\) b)
60 + 8b = 9b
9b – 8b = 60
b = 60

Solving Linear Equations Worksheets Grade 8 Pdf Question 4.
\(\frac{2}{11}\) m + 16 = 4 + \(\frac{6}{11}\) m
_______

Answer:
m = 33

Explanation:
\(\frac{2}{11}\) m + 16 = 4 + \(\frac{6}{11}\) m
The LCM is 11
11(\(\frac{2}{11}\) m + 16) = 11(4 + \(\frac{6}{11}\) m)
2m + 176 = 44 + 6m
6m – 2m = 176 – 44
4m = 132
m = 132/4
m = 33

Question 5.
2.25t + 5 = 13.5t + 14
_______

Answer:
t = -0.8

Explanation:
2.25t + 5 = 13.5t + 14
13.5t – 2.25t = 5 – 14
11.25t = -9
t = -9/11.25
t = -0.8

Question 6.
3.6w = 1.6w + 24
_______

Answer:
w = 12

Explanation:
3.6w = 1.6w + 24
3.6w – 1.6w = 24
2w = 24
w = 24/2
w = 12

Question 7.
-0.75p – 2 = 0.25p
_______

Answer:
p = -2

Explanation:
-0.75p – 2 = 0.25p
-2 = 0.25p + 0.75p
-2 = p
p = -2

Question 8.
Write a real-world problem that can be modeled by the equation 1.25x = 0.75x + 50.
Type below:
______________

Answer:
1.25x = 0.75x + 50.
Cell offers Plan A for no base fee and $1.25 per minute.
Cell offers Plan B for a $50 base fee and $0.75 per minute.
The equation shows when the total cost of the plan would be equal.

Essential Question Check-In

Question 9.
How does the method for solving equations with fractional or decimal coefficients and constants compare with the method for solving equations with integer coefficients and constants?
Type below:
______________

Answer:
When solving equations with fractional or decimal coefficients, the equations need to be multiplied by the multiple of the denominator such that the equations have integer coefficients and constants.

Independent Practice – Equations with Rational Numbers – Page No. 207

Question 10.
Members of the Wide Waters Club pay $105 per summer season, plus $9.50 each time they rent a boat. Nonmembers must pay $14.75 each time they rent a boat. How many times would a member and a non-member have to rent a boat in order to pay the same amount?
_______ times

Answer:
20 times

Explanation:
Members of the Wide Waters Club pay $105 per summer season, plus $9.50 each time they rent a boat.
9.5x + $105
Nonmembers must pay $14.75 each time they rent a boat.
9.5x + $105 = 14.75x
9.5x – 14.75x = $105
5.25x = 105
x = 105/5.25
x = 20
The cost for members and non-members will be the same for 8 visits.

Question 11.
Margo can purchase tile at a store for $0.79 per tile and rent a tile saw for $24. At another store, she can borrow the tile saw for free if she buys tiles there for $1.19 per tile. How many tiles must she buy for the cost to be the same at both stores?
_______ tiles

Answer:
60 tiles

Explanation:
Margo can purchase tile at a store for $0.79 per tile and rent a tile saw for $24.
0.79x + 24
At another store, she can borrow the tile saw for free if she buys tiles there for $1.19 per tile.
1.19x
0.79x + 24 = 1.19x
1.19x – 0.79x = 24
0.4x = 24
x = 24/0.4
x = 60
Margo should buy 60 tiles for the cost to be the same at both stores.

Question 12.
The charges for two shuttle services are shown in the table. Find the number of miles for which the cost of both shuttles is the same.
Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations Lesson 2: Equations with Rational Numbers img 3
_______ miles

Answer:
40 miles

Explanation:
0.1x + 10
0.35x
0.1x + 10 = 0.35x
0.35x – 0.1x = 10
0.25x = 10
x = 10/0.25
x = 40
The cost of shuttles would be the same for 40 miles.

Question 13.
Multistep
Rapid Rental Car charges a $40 rental fee, $15 for gas, and $0.25 per mile driven. For the same car, Capital Cars charges $45 for rental and gas and $0.35 per mile.
a. For how many miles is the rental cost at both companies the same?
_______ miles

Answer:
100 miles

Explanation:
0.25x + 40 + 15 = 0.35x + 45
0.35x – 0.25x = 55 – 45
0.1x = 10
x = 10/0.1
x = 100
The cost of car rentals would be the same for 100 miles.

Question 13.
b. What is that cost?
$ _______

Answer:
$80

Explanation:
Let y be the total cost. Substitute 100 miles in any one of the two equations
y = 0.35x + 45
y = 0.35(100) + 45 = $80
Total cost would be $80.

Question 14.
Write an equation with the solution x = 20. The equation should have the variable on both sides, a fractional coefficient on the left side, and a fraction anywhere on the right side.
Type below:
______________

Answer:
4/3x + 10 = 50/3 + x

Explanation:
Write an equation with the solution x = 20. The equation should have the variable on both sides, a fractional coefficient on the left side, and a fraction anywhere on the right side.
1/3 . x = 1/3 . 20
1/3 . x + x = 1/3 . 20 + x
4/3x = 20/3 + x
4/3x + 10 = 20/3 + x + 10
4/3x + 10 = 50/3 + x

Question 15.
Write an equation with the solution x = 25. The equation should have the variable on both sides, a decimal coefficient on the left side, and a decimal anywhere on the right side. One of the decimals should be written in tenths, the other in hundredths.
Type below:
______________

Answer:
x=25
divide both sides by 25
x/25 = 1
convert 1/25 to decimal form 0.04
0.04x = 1
add x on both sides
1.04x = 1 + x
add 0.1 on both sides
1.04x + 0.1 = x + 1.1

Question 16.
Geometry
The perimeters of the rectangles shown are equal. What is the perimeter of each rectangle?
Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations Lesson 2: Equations with Rational Numbers img 4
Perimeter = _______

Answer:
Perimeter = 3.2

Explanation:
Perimeter of the first rectangle
P = 2(n + n + 0.6) = 2(2n + 0.6) = 4n + 1.2
Perimeter of the second rectangle
P = 2(n + 0.1 + 2n) = 2(3n + 0.1) = 6n + 0.2
the perimeter is equal
4n + 1.2 = 6n + 0.2
6n – 4n = 1.2 – 0.2
2n = 1
n = 1/2
n = 0.5
P = 4n + 1.2 = 4(0.5) + 1.2 = 3.2

Question 17.
Analyze Relationships
The formula F = 1.8C + 32 gives the temperature in degrees Fahrenheit (F) for a given temperature in degrees Celsius (C). There is one temperature for which the number of degrees Fahrenheit is equal to the number of degrees Celsius. Write an equation you can solve to find that temperature and then use it to find the temperature
Type below:
______________

Answer:
x = 1.8x + 32

Explanation:
F = 1.8C +32
let x be the temperature such that it is the same in both Celsius and in Fahrenheit
Then the required equation is
x = 1.8x + 32
subtract 1.8x from both sides
-0.8x = 32
divide by -0.8 on both sides
x = -40
So -40 degree celsius

Equations with Rational Numbers – Page No. 208

Question 18.
Explain the Error
Agustin solved an equation as shown. What error did Agustin make? What is the correct answer?
Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations Lesson 2: Equations with Rational Numbers img 5
x = _______

Answer:
x = -12

Explanation:
Agustin did not multiply by 12 on both sides in step 2. He only partially multiplied the variable and left the constants as such, which doesn’t make any sense.
The correct solution is
12(x/3 – 4) = 12(3x/4 + 1)
4x – 48 = 9x + 12
subtract 12 on both sides
4x – 60 = 9x
subtract 4x on both sides
-60 = 5x
x = -12

H.O.T.

Focus on Higher Order Thinking

Question 19.
Draw Conclusions
Solve the equation \(\frac{1}{2} x-5+\frac{2}{3} x=\frac{7}{6} x+4\). Explain your results.
Type below:
_____________

Answer:
\(\frac{1}{2} x-5+\frac{2}{3} x=\frac{7}{6} x+4\)
The least common multiple of the denominators: LCM(2, 3, 6) = 6
6(\(\frac{1}{2} x-5+\frac{2}{3} x=\frac{7}{6} x+4\))
6.1/2x – 6.5 + 6.2/3x = 6.7/6x +6.4
3x – 30 + 4x = 7x + 24
7x – 30 = 7x + 24
-30 = 24
This is not true. The equation has no solution.

Question 20.
Look for a Pattern
Describe the pattern in the equation. Then solve the equation.
0.3x + 0.03x + 0.003x + 0.0003x + .. = 3
x = ______

Answer:
x = 9

Explanation:
0.3x + 0.03x + 0.003x + 0.0003x + .. = 3
0.3x = 3
0.9x = 9
x = 9

Question 21.
Critique Reasoning
Jared wanted to find three consecutive even integers whose sum was 4 times the first of those integers. He let k represent the first integer, then wrote and solved this equation : k + (k + 1) + (k + 2) = 4k. Did he get the correct answer? Explain.
__________

Answer:
No, it is wrong on two accounts.
First, he has not specified if k is even or not. An easy way of doing so would assume x to be any integer and k=2a
This ensures that k is an even integer.
Nest the question asks for 3 consecutive even integers, Jared just took 3 consecutive integers, and thus at least 1 of them is odd.
So correct representation would be
k + (k+2) + (k + 4) = 4k
which upon solving yields k=6

Guided Practice – Equations with the Distributive Property – Page No. 212

Solve each equation.

Question 1.
4(x + 8) – 4 = 34 – 2x
________

Answer:
x = 1

Explanation:
4(x + 8) – 4 = 34 – 2x
4x + 32 – 4 = 34 – 2x
4x + 2x = 34 – 28
6x = 6
x = 6/6
x = 1

7.3 Equations with the Distributive Property Question 2.
\(\frac{2}{3}\)(9 + x) = -5(4 – x)
________

Answer:
x = 6

Explanation:
\(\frac{2}{3}\)(9 + x) = -5(4 – x)
2/3(9 + x) = -5(4 – x)
3 (2/3(9 + x)) = 3(-5(4 – x))
2(9 + x ) = -15 (4 – x)
18 + 2x = -60 + 15x
15x – 2x = 18 + 60
13x = 78
x = 78/13
x = 6

Question 3.
-3(x + 4) + 15 = 6 – 4x
________

Answer:
x = 3

Explanation:
-3(x + 4) + 15 = 6 – 4x
-3x – 12 + 15 = 6 – 4x
-3x + 3 = 6 – 4x
-3x + 4x = 6 – 3
x = 3

Question 4.
10 + 4x = 5(x – 6) + 33
________

Answer:
x = 7

Explanation:
10 + 4x = 5(x – 6) + 33
10 + 4x = 5x – 30 + 33
10 + 4x = 5x + 3
5x – 4x = 10 – 3
x = 7

Question 5.
x – 9 = 8(2x + 3) – 18
________

Answer:
x = -1

Explanation:
x – 9 = 8(2x + 3) – 18
x – 9 = 16x + 24 – 18
x – 9 = 16x + 6
16x – x = -9 – 6
15x = – 15
x = -15/15
x = -1

Question 6.
-6(x – 1) – 7 = -7x + 2
________

Answer:
x = 3

Explanation:
-6(x – 1) – 7 = -7x + 2
-6x + 6 – 7 = -7x + 2
-6x – 1 = -7x + 2
-7x + 6x = -1 -2
-x = -3
x = 3

Question 7.
\(\frac{1}{10}\)(x + 11) = -2(8 – x)
________

Answer:
x = 9

Explanation:
\(\frac{1}{10}\)(x + 11) = -2(8 – x)
10(\(\frac{1}{10}\)(x + 11)) = 10 (-2(8 – x))
x + 11 = -20(8 – x)
x + 11 = -160 + 20x
20x – x = 11 + 160
19x = 171
x = 171/19 = 9

Question 8.
-(4 – x) = \(\frac{3}{4}\)(x – 6)
________

Answer:
x = -2

Explanation:
-(4 – x) = \(\frac{3}{4}\)(x – 6)
4(-(4 – x)) = 4 (3/4(x – 6))
-16 + 4x = 3x – 18
4x – 3x = -18 + 16
x = -2

Question 9.
-8(8 – x) = \(\frac{4}{5}\)(x + 10)
________

Answer:
x = 10

Explanation:
-8(8 – x) = \(\frac{4}{5}\)(x + 10)
5(-8(8 – x)) = 5(\(\frac{4}{5}\)(x + 10))
-40(8 – x) = 4(x + 10)
-320 + 40x = 4x + 40
40x – 4x = 40 + 320
36x = 360
x = 360/36
x = 10

Question 10.
\(\frac{1}{2}\)(16 – x) = -12(x + 7)
________

Answer:
x = 8

Explanation:
\(\frac{1}{2}\)(16 – x) = -12(x + 7)
2 (\(\frac{1}{2}\)(16 – x)) = 2 (-12(x + 7))
16 – x = -24 (x + 7)
16 – x = -24x – 168
24x – x = -168 – 16
23x = 184
x = 184/23
x = 8

Lesson 7.3 Equations with the Distributive Property Answer Key Question 11.
Sandra saves 12% of her salary for retirement. This year her salary was $3,000 more than in the previous year, and she saved $4,200.What was her salary in the previous year?
Write an equation _____
Sandra’s salary in the previous year was _____
Salary = $ _____

Answer:
Write an equation 0.12x + 360 = 4200
Sandra’s salary in the previous year was $32000
Salary = $3000

Explanation:
0.12(x + 3000) = 4200
0.12x + 360 = 4200
0.12x = 4200 – 360
0.12x = 3840
x = 3840/0.12
x = 32000
Sandra’s salary in the previous year was $32000

Essential Question Check-In

Question 12.
When solving an equation using the Distributive Property, if the numbers being distributed are fractions, what is your first step? Why?
Type below:
___________

Answer:
Multiply both sides by the denominator of the fraction

Independent Practice – Equations with the Distributive Property – Page No. 213

Question 13.
Multistep
Martina is currently 14 years older than her cousin Joey. In 5 years she will be 3 times as old as Joey. Use this information to answer the following questions.
a. If you let x represent Joey’s current age, what expression can you use to represent Martina’s current age?
Type below:
___________

Answer:
y = x + 14

Explanation:
y = x + 14
where x is Joey’s current age and t is Martna’s current age.

Question 13.
b. Based on your answer to part a, what expression represents Joey’s age in 5 years? What expression represents Martina’s age in 5 years?
Type below:
___________

Answer:
Ages in 5 years
Joey’s age = x + 5
Martina’s age = x + 14 + 5 = x + 19

Question 13.
c. What equation can you write based on the information given?
Type below:
___________

Answer:
3(x + 5) = x + 19

Explanation:
In 5 years, Martina will be three times as old as Joey
3(x + 5) = x + 19

Question 13.
d. What is Joey’s current age? What is Martina’s current age?
Joey’s current age ___________
Martina’s current age ___________

Answer:
Joey’s current age 2
Martina’s current age 16

Explanation:
3(x + 5) = x + 19
3x + 15 = x + 19
3x – x = 19 – 15
2x = 4
x = 2

Question 14.
As part of a school contest, Sarah and Luis are playing a math game. Sarah must pick a number between 1 and 50 and give Luis clues so he can write an equation to find her number. Sarah says, “If I subtract 5 from my number, multiply that quantity by 4, and then add 7 to the result, I get 35.” What equation can Luis write based on Sarah’s clues and what is Sarah’s number?
Type below:
___________

Answer:
x = 12

Explanation:
As part of a school contest, Sarah and Luis are playing a math game. Sarah must pick a number between 1 and 50 and give Luis clues so he can write an equation to find her number. Sarah says, “If I subtract 5 from my number, multiply that quantity by 4, and then add 7 to the result, I get 35.”
4 (x – 5) + 7 = 35
4x – 20 + 7 = 35
4x – 13 = 35
4x = 35 + 13
4x = 48
x = 48/4
x = 12

Question 15.
Critical Thinking
When solving an equation using the Distributive Property that involves distributing fractions, usually the first step is to multiply by the LCD to eliminate the fractions in order to simplify computation. Is it necessary to do this to solve \(\frac{1}{2}\)(4x + 6) = 13(9x – 24)? Why or why not?
___________

Answer:
It is not necessary. In this case, distributing the fractions directly results in whole-number coefficients and constants, however, if the results are not in whole-number coefficients and constants it is harder to solve fractions.

Question 16.
Solve the equation given in Exercise 15 with and without using the LCD of the fractions. Are your answers the same?
___________

Answer:
x = 11

Explanation:
\(\frac{1}{2}\)(4x + 6) = 13(9x – 24)
6(\(\frac{1}{2}\)(4x + 6)) = 6(13(9x – 24))
3(4x + 6) = 2(9x – 24)
12x + 18 = 18x – 48
18x – 12x = 18 + 48
6x = 66
x = 66/6
x = 11

Equations with the Distributive Property – Page No. 214

Question 17.
Represent Real-World Problems
A chemist mixed x milliliters of 25% acid solution with some 15% acid solution to produce 100 milliliters of a 19% acid solution. Use this information to fill in the missing information in the table and answer the questions that follow.
Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations Lesson 3: Equations with the Distributive Property img 6
a. What is the relationship between the milliliters of acid in the 25% solution, the milliliters of acid in the 15% solution, and the milliliters of acid in the mixture?
Type below:
_____________

Answer:
The milliliters of acid in the 25% solution plus the milliliters of acid in the 15% solution equals the milliliters of acid in the mixture

Explanation:
grade 8 chapter 7 image 9

Question 17.
b. What equation can you use to solve for x based on your answer to part a?
Type below:
_____________

Answer:
0.25x + 0.15(100 – x) = 19

Question 17.
c. How many milliliters of the 25% solution and the 15% solution did the chemist use in the mixture?
Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations Lesson 3: Equations with the Distributive Property img 7
Type below:
_____________

Answer:
0.25x + 0.15(100 – x) = 19
0.25x + 15 – 0.15x = 19
0.1x + 15 = 19
0.1x = 4
x = 4/0.1
x = 40
The chemist used 40ml of the 25% solution and 100 – 40 = 60ml of the 15% solution.

H.O.T.

Focus on Higher Order Thinking

Question 18.
Explain the Error
Anne solved 5(2x) – 3 = 20x + 15 for x by first distributing 5 on the left side of the equation. She got the answer x = -3. However, when she substituted -3 into the original equation for x, she saw that her answer was wrong. What did Anne do wrong, and what is the correct answer?
x = ________

Answer:
x = -1.8

Explanation:
Dado que 5 solo se multiplica por 2x, no tiene sentido usar la distribución aquí. Básicamente, distribuir 5 fue el problema
Solución correcta:
5 (2x) – 3 = 20x + 15
10x -3 = 20x + 15
restar 15 en ambos lados
10x – 18 = 20x
restar 10x de ambos lados
-18 = 10x
x = -1.8

Question 19.
Communicate Mathematical Ideas
Explain a procedure that can be used to solve 5[3(x + 4) – 2(1 – x)] – x – 15 = 14x + 45. Then solve the equation.
x = ________

Answer:
x = 1

Explanation:
5[3(x + 4) – 2(1 – x)] – x – 15 = 14x + 45
5[3x + 12 – 2 + 2x] – x – 15 = 14x + 45
5[5x + 10] – x – 15 = 14x + 45
25x + 50 – x – 15 = 14x + 45
24x + 35 = 14x + 45
24x – 14x = 45 – 35
10x = 10
x = 1

Guided Practice – Equations with Many Solutions or No Solution – Page No. 218

Use the properties of equality to simplify each equation. Tell whether the final equation is a true statement.

Question 1.
Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations Lesson 4: Equations with Many Solutions or No Solution img 8
The statement is: _______

Answer:
The statement is: true

Explanation:
3x – 2 = 25 – 6x
3x + 6x -2 = 25 -6x + 6x
9x – 2 = 25
9x -2 + 2 = 25 + 2
9x = 27
x = 27/9
x = 3
The statement is true.

Solving Equations by Clearing Fractions Worksheet Question 2.
Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations Lesson 4: Equations with Many Solutions or No Solution img 9
____________

Answer:
The statement is false.

Explanation:
2x – 4 = 2(x – 1) + 3
2x – 4 = 2x – 2 + 3
2x – 4 = 2x + 1
2x – 4 – 2x = 2x + 1 – 2x
-4 not equal to 1
The statement is false.

Question 3.
How many solutions are there to the equation in Exercise 2?
____________

Answer:
There is no solution to exercise 2.

Question 4.
After simplifying an equation, Juana gets 6 = 6. Explain what this means.
____________

Answer:
When 6 = 6, there are infinite solutions.

Write a linear equation in one variable that has infinitely many solutions.

Question 5.
Start with a _____ statement.
Add the _____ to both sides.
Add the _____ to both sides.
Combine _____ terms.
Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations Lesson 4: Equations with Many Solutions or No Solution img 10
Type below:
____________

Answer:
Start with a “true” statement
Add the “same variable” to both sides
Add the “same constant” to both sides
Combine “like” terms

Explanation:
Start with a “true” statement
10 = 10
Add the “same variable” to both sides
10 + x = 10 + x
Add the “same constant” to both sides
10 + x + 5 = 10 + x + 5
Combine “like” terms
15 + x = 15 + x

Essential Question Check-In

Question 6.
Give an example of an equation with an infinite number of solutions. Then make one change to the equation so that it has no solution.
Type below:
____________

Answer:
An equation with infinitely many solutions
x – 2x + 3 = 3 – x
-x + 3 = 3 – x
+x/3 = +x/3
An equation for no solution
x – 2x + 3 = 3 – x + 4
-x + 3 = 7 – x
-x/3 = -x/7

Independent Practice – Equations with Many Solutions or No Solution – Page No. 219

Tell whether each equation has one, zero, or infinitely many solutions.

Question 7.
-(2x + 2) – 1 = -x – (x + 3)
____________

Answer:
The statement is true

Explanation:
-(2x + 2) – 1 = -x – (x + 3)
-2x – 2 – 1 = -x – x + 3
-2x – 3 = -2x + 3
-3 = -3
The statement is true

Question 8.
-2(z + 3) – z = -z – 4(z + 2)
____________

Answer:
The statement is false.

Explanation:
-2(z + 3) – z = -z – 4(z + 2)
-3z – 6 = -3z -8
-3z -6 + 3z = -3z – 8 + 3z
-6 not equal to -8
The statement is false.

Create an equation with the indicated number of solutions.

Question 9.
No solution:
3(x – \(\frac{4}{3}\)) = 3x + _____
Type below:
______________

Answer:
3(x – \(\frac{4}{3}\)) = 3x + ?
3x – 4 = 3x + ?
3x – 4 = 3x + 2
When there is no solution, the statement should be false. Any number except -4 would make the equation have no solutions.

Question 10.
Infinitely many solutions:
2(x – 1) + 6x = 4( _____ – 1) + 2
Type below:
______________

Answer:
2(x – 1) + 6x = 4( _____ – 1) + 2
2(x – 1) + 6x = 4( ? – 1) + 2
2x – 2 + 6x = 4(? – 1) + 2
8x – 2 = 4(? – 1) + 2
8x – 2 = 4(2x – 1) + 2
8x – 2 = 8x – 4 + 2
8x – 2 = 8x – 2
When there are infinitely many solutions, the statement should be true

Question 11.
One solution of x = -1:
5x – (x – 2) = 2x – ( _____ )
Type below:
______________

Answer:
Put x = -1 in the equation
-5 – (-1 – 2) = -2 – blank
simplifying
-2 = -2 – blank
add 2 on both sides
0 = blank

Question 12.
Infinitely many solutions:
-(x – 8) + 4x = 2( _____ ) + x
Type below:
______________

Answer:
-(x – 8) + 4x = 2( ?) + x
-x + 8 + 4x = 2(?) + x
3x + 8 = 2(?) + x
3x + 8 = 2 (x + 4) + x
3x + 8 = 2x + 8x + x
3x + 8 = 3x + 8
When there are infinitely many solutions, the statement should be true.

Question 13.
Persevere in Problem Solving
The Dig It Project is designing two gardens that have the same perimeter. One garden is a trapezoid whose nonparallel sides are equal. The other is a quadrilateral. Two possible designs are shown at the right.
Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations Lesson 4: Equations with Many Solutions or No Solution img 11
a. Based on these designs, is there more than one value for x? Explain how you know this.
______________

Answer:
There are more than one value of x

Explanation:
Perimeter of the trapezoid
P = 2x – 2 + x + 1 + x + x + 1 = 5x
Perimeter of the quadrilateral
P = 2x – 9 + x + x + 8 + x + 1 = 5x
5x = 5x
There are more than one value of x

Question 13.
b. Why does your answer to part a make sense in this context?
Type below:
______________

Answer:
The condition was that the two perimeters are to be equal. However, a specific number was not given, so there are an infinite number of possible perimeters

Explanation:
Interpretation of part a in this context
The condition was that the two perimeters are to be equal. However, a specific number was not given, so there are an infinite number of possible perimeters

Question 13.
c. Suppose the Dig It Project wants the perimeter of each garden to be 60 meters. What is the value of x in this case? How did you find this?
______ meters

Answer:
12 meters

Explanation:
2x – 2 + x + 1 + x + x + 1 = 60
5x = 60
x = 60/5
x = 12

Equations with Many Solutions or No Solution – Page No. 220

Question 14.
Critique Reasoning
Lisa says that the indicated angles cannot have the same measure. Marita disagrees and says she can prove that they can have the same measure. Who do you agree with? Justify your answer.
Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations Lesson 4: Equations with Many Solutions or No Solution img 12
I agree with: ______________

Answer:
I agree with: Marita

Explanation:
9x – 25 + x = x + 50 + 2x – 12
10x – 25 = 3x + 38
10x – 3x = 38 + 25
7x = 63
x = 63/7
x = 9
When x = 9 the angles will be same and for any other value of x, the angles will not be the same.

Question 15.
Represent Real-World Problems
Adele opens an account with $100 and deposits $35 a month. Kent opens an account with $50 and also deposits $35 a month. Will they have the same amount in their accounts at any point? If so, in how many months and how much will be in each account? Explain.
______________

Answer:
Adele’s amount after x months
A = 100 + 35x
Kent’s amount after x months
A = 50 + 35x
100 + 35x = 50 + 35x
100 is not equal to 50
The statement is false, the amounts in two accounts would never be equal.

H.O.T.

Focus on Higher Order Thinking

Question 16.
Communicate Mathematical Ideas
Frank solved an equation and got the result x = x. Sarah solved the same equation and got 12 = 12. Frank says that one of them is incorrect because you cannot get different results for the same equation. What would you say to Frank? If both results are indeed correct, explain how this happened.
Frank is: ____________

Answer:
Both of them can be correct as both equations give the same result i.e. there are infinitely many solutions. Frank eliminated the constant from both sides while Sarah eliminated the variable from both sides.

Question 17.
Critique Reasoning
Matt said 2x – 7 = 2(x – 7) has infinitely many solutions. Is he correct? Justify Matt’s answer or show how he is incorrect.
Matt is: ____________

Answer:

Explanation:
2x – 7 = 2(x – 7)
2x – 7 = 2x – 14
-7 not equal to -14
The statement is false, there is no solution. Matt is incorrect.

7.1 Equations with the Variable on Both Sides – Model Quiz – Page No. 221

Solve.

Question 1.
4a – 4 = 8 + a
_______

Answer:
a = 4

Explanation:
4a – 4 = 8 + a
4a – a = 8 + 4
3a = 12
a = 12/3
a = 4

Question 2.
4x + 5 = x + 8
_______

Answer:
x = 1

Explanation:
4x + 5 = x + 8
4x – x = 8 – 5
3x = 3
x = 3/3
x = 1

Homework and Practice Solving Linear Equations 7.1 Answer Key Question 3.
Hue is arranging chairs. She can form 6 rows of a given length with 3 chairs left over, or 8 rows of that same length if she gets 11 more chairs. Write and solve an equation to find how many chairs are in that row length.
_______ chairs

Answer:
7 chairs

Explanation:
Hue is arranging chairs. She can form 6 rows of a given length with 3 chairs left over, or 8 rows of that same length if she gets 11 more chairs.
6x + 3 = 8x – 11
8x – 6x = 3 + 11
2x = 14
x = 14/2
x = 7
There are 7 chairs in each row.

7.2 Equations with Rational Numbers

Solve.

Question 4.
\(\frac{2}{3} n-\frac{2}{3}=\frac{n}{6}+\frac{4}{3}\)
_______

Answer:
n = 4

Explanation:
\(\frac{2}{3} n-\frac{2}{3}=\frac{n}{6}+\frac{4}{3}\)
The LCM is 6.
6(2/3n – 2/3) = 6(n/6 + 4/3)
6(2/3n) -6(2/3) = 6(n/6) + 6(4/3)
4n – 4 = n + 8
4n – n = 8 + 4
3n = 12
n = 12/3
n = 4

Question 5.
1.5d + 3.25 = 1 + 2.25d
_______

Answer:
d = 3

Explanation:
1.5d + 3.25 = 1 + 2.25d
2.25d – 1.5d = 3.25 – 1
0.75d = 2.25
d = 2.25/0.75
d = 3

Question 6.
Happy Paws charges $19.00 plus $1.50 per hour to keep a dog during the day. Woof Watchers charges $14.00 plus $2.75 per hour. Write and solve an equation to find for how many hours the total cost of the services is equal.
_______ hours

Answer:
3.2 hours

Explanation:
Happy Paws charges $19.00 plus $1.50 per hour to keep a dog during the day.
1.5x + 19
Woof Watchers charges $14.00 plus $2.75 per hour.
2.75x + 15
1.5x + 19 = 2.75x + 15
2.75x – 1.5x = 19 – 15
1.25x = 4
x = 4/1.25
x = 3.2
The total cost of the services is equal after 3.2 hrs.

7.3 Equations with the Distributive Property

Solve.

Question 7.
14 + 5x = 3(-x + 3) – 11
_______

Answer:
x = -2

Explanation:
14 + 5x = 3(-x + 3) – 11
14 + 5x = -3x + 9 – 11
14 + 5x = -3x – 2
5x + 3x = -2 –  14
8x = – 16
x = -16/8
x = -2

Question 8.
\(\frac{1}{4}\)(x – 7) = 1 + 3x
_______

Answer:
x = -1

Explanation:
\(\frac{1}{4}\)(x – 7) = 1 + 3x
4(\(\frac{1}{4}\)(x – 7)) = 4(1 + 3x)
(x – 7) = 4 + 12x
12x – x = -7 – 4
11x = -11
x = -11/11
x = -1

Question 9.
-5(2x – 9) = 2(x – 8) – 11
_______

Answer:
x = 6

Explanation:
-5(2x – 9) = 2(x – 8) – 11
-10x + 45 = 2x – 16 – 11
-10x + 45 = 2x – 27
2x + 10x = 45 + 27
12x = 72
x = 72/12
x = 6

Question 10.
3(x + 5) = 2(3x + 12)
_______

Answer:
x = -3

Explanation:
3(x + 5) = 2(3x + 12)
3x + 15 = 6x + 24
6x – 3x = 15 – 24
3x = -9
x = -9/3
x = -3

7.4 Equations with Many Solutions or No Solution

Tell whether each equation has one, zero, or infinitely many solutions.

Question 11.
5(x – 3) + 6 = 5x – 9
____________

Answer:
There are infinitely many solutions

Explanation:
5(x – 3) + 6 = 5x – 9
5x – 15 + 6 = 5x – 9
5x – 9 = 5x – 9
The statement is true. There are infinitely many solutions.

Question 12.
5(x – 3) + 6 = 5x – 10
____________

Answer:
There are no solutions

Explanation:
5(x – 3) + 6 = 5x – 10
5x – 15 + 6 = 5x – 10
5x – 9 = 5x – 10
-9 not equal to -10
The statement is false. There are no solutions.

Question 13.
5(x – 3) + 6 = 4x + 3
____________

Answer:
There is one solution

Explanation:
5(x – 3) + 6 = 4x + 3
5x – 15 + 6 = 4x + 3
5x – 9 = 4x + 3
5x – 4x = 3 + 9
x = 12
There is one solution

Selected Response – Mixed Review – Page No. 222

Question 1.
Two cars are traveling in the same direction. The first car is going 40 mi/h, and the second car is going 55 mi/h. The first car left 3 hours before the second car. Which equation could you solve to find how many hours it will take for the second car to catch up to the first car?
Options:
a. 55t + 3 = 40t
b. 55t + 165 = 40t
c. 40t + 3 = 55t
d. 40t + 120 = 55t

Answer:
d. 40t + 120 = 55t

Explanation:
Two cars are traveling in the same direction. The first car is going 40 mi/h, and the second car is going 55 mi/h. The first car left 3 hours before the second car.
3 × 40 + 40t = 120 + 40t
55t
40t + 120 = 55t

Question 2.
Which linear equation is represented by the table?
Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations Mixed Review img 13
Options:
a. y = -x + 5
b. y = 2x – 1
c. y = x + 3
d. y = -3x + 11

Answer:
a. y = -x + 5

Explanation:
Find the slope using
m = (y2 – y1)/(x2 – x1)
where (x1, y1) = (3, 2), (x2, y2) = (1, 4)
Slope = (4 – 2)/(1 – 3) = -2/2 = -1

Question 3.
Shawn’s Rentals charges $27.50 per hour to rent a surfboard and a wetsuit. Darla’s Surf Shop charges $23.25 per hour to rent a surfboard plus $17 extra for a wetsuit. For what total number of hours are the charges for Shawn’s Rentals the same as the charges for Darla’s Surf Shop?
Options:
a. 3
b. 4
c. 5
d. 6

Answer:
b. 4

Explanation:
Shawn’s Rentals charges $27.50 per hour to rent a surfboard and a wetsuit.
27.5x
Darla’s Surf Shop charges $23.25 per hour to rent a surfboard plus $17 extra for a wetsuit.
23.25x + 17
23.25x + 17 = 27.5x
27.5x – 23.25x = 17
4.25x = 17
x = 17/4.25
x = 4
The charge would be equal after 4 hrs

Question 4.
Which of the following is irrational?
Options:
a. -8
b. 4.63
c. \(\sqrt { x } \)
d. \(\frac{1}{3}\)

Answer:
c. \(\sqrt { x } \)

Explanation:
\(\sqrt { x } \) is irrational

Question 5.
Greg and Jane left a 15% tip after dinner. The amount of the tip was $9. Greg’s dinner cost $24. Which equation can you use to find x, the cost of Jane’s dinner?
Options:
a. 0.15x + 24 = 9
b. 0.15(x + 24) = 9
c. 15(x + 24) = 9
d. 0.15x = 24 + 9

Answer:
b. 0.15(x + 24) = 9

Explanation:
Let x be the cost of Jane’s dinner. The amount of tip is the 15% of the total cost of dinner.
0.15(x + 24) = 9

Question 6.
For the equation 3(2x − 5) = 6x + k, which value of k will create an equation with infinitely many solutions?
Options:
a. 15
b. -5
c. 5
d. -15

Answer:
d. -15

Explanation:
3(2x – 5) = 6x + k
6x – 15 = 6x + k
6x – 15 = 6x – 15
The statement is true. k = -15

Question 7.
Which of the following is equivalent to 2−4?
Options:
a. \(\frac{1}{16}\)
b. \(\frac{1}{8}\)
c. -2
d. -16

Answer:
a. \(\frac{1}{16}\)

Explanation:
2−4
1/24
1/16

Mini-Task

Question 8.
Use the figures below for parts a and b.
Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations Mixed Review img 14
a. Both figures have the same perimeter. Solve for x.
_______

Answer:
x=12

Explanation:
4x+10=3x+22
4x – 3x = 22 – 10
x = 12
Answer: x=12

Question 8.
b. What is the perimeter of each figure?
_______

Answer:
Both are 58

Explanation:
x + x + 5 + x + x + 5
12 + 12 + 5 + 12 + 12 + 5
58
x + 7 + x + 4 + x + 11
12 + 7 + 12 + 4 + 12 + 11
58

Conclusion:

Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations are available along with the all practice questions here. Get all the questions, answers and also Go Math Grade 8 text books for free online. Refer to Go Math Grade 8 Chapter 7 Solving Linear Equations Answer Key to learn the quick maths in an easy way.

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