Prasanna

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 7.1 Answer Key Parallel Lines Cut by a Transversal.

Texas Go Math Grade 8 Lesson 7.1 Answer Key Parallel Lines Cut by a Transversal

Texas Go Math Grade 8 Lesson 7.1 Explore Activity Answer Key

Explore Activity 1

Parallel Lines and Transversals

A transversal is a line that intersects two lines in the same plane at two different points. Transversal f and lines a and b form eight angles.

Use geometry software to explore the angles formed when a transversal intersects parallel lines.

A. Construct a line and label two points on the line A and B.
B. Create point C not on \(\overleftrightarrow{A B}\). Then construct a line parallel to \(\overleftrightarrow{A B}\) through point C. Create another point on this line and label it D.
C. Create two points outside the two parallel lines and label them E and F. Construct transversal \(\overleftrightarrow{E F}\). Label the points of intersection G and H.

D. Measure the angles formed by the parallel lines and the transversal. Write the angle measures in the table below.
E. Drag point E or point F to a different position. Record the new angle measures in the table.

Reflect

Make a Conjecture Identify the pairs of angles in the diagram. Then make a conjecture about their angle measures. Drag a point in the diagram to confirm your conjecture.

Question 1.
corresponding angles
Answer:
Corresponding angles ∠CGE = ∠AHG and ∠DGE = ∠BHG

Two Parallel Lines Cut by a Transversal Worksheet Answer Key Question 2.
alternate interior angles
Answer:
Alternate interior angles ∠CGH and ∠GHB, and the second pair ∠DGH and ∠AHG

Question 3.
alternate exterior angles
Answer:
Alternate exterior angles ∠CGE = ∠BHF and second pair ∠EGD = ∠AHF

Question 4.
same-side interior angles
Answer:
Same-side interior angles ∠CGH and angle ∠AHG, and second pair ∠DGH and ∠GHB

Explore Activity 2

Justifying Angle Relationships

You can use tracing paper to informally justify your conclusions from the first Explore Activity.

Lines a and b are parallel. (The black arrows on the diagram indicate parallel lines.)
A. Trace the diagram onto tracing paper.
B. Position the tracing paper over the original diagram so that ∠1 on the tracing is over ∠5 on the original diagram. Compare the two angles. Do they appear to be congruent?
C. Use the tracing paper to compare all eight angles in the diagram to each other. List all of the congruent angle pairs.

Your Turn

Find each angle measure.

Parallel Lines Cut by a Transversal Worksheet 8th Grade Pdf Question 5.
m∠GDE = __________
Answer:
Find m∠GDE
Angles m∠GDE and m∠DEH are vertical angles so
m∠GDE = m∠DEH = 4x°
Angle m∠GDE is supplementary to m∠DEH because they are same-side interior angles
m∠GDE + m∠DEH = 180° …………………… (1)
4x° + 6x° = 180° (Replace m∠GDE with 4x°) and m∠DEH with 6x° ………….. (2) & (3)
10x° = 180° (Combine like terms.) ……………. (4)
x° = \(\frac{1}{2}\) (Divide both sides with 10) …………… (5)
x = 18° (Simplify) ………….. (6)
m∠GDE = 4x° = (4 ∙ 18)° = 72°

Question 6.
m∠BEF = ___________
Answer:
Find m∠BEF.
Angles m∠GDE and m∠DEH are vertical angles so
m∠GDE = m∠DEH = 4x°
Angle m∠GDE is supplementary to m∠DEH because they are same-side interior angles.
m∠GDE + m∠DEH = 1800 …………. (1)
4x° + 6x° = 180° (RepLace m∠GDE with 4x°) and m∠DEH with 6x° …………… (2) & (3)
10x° = 180° (Combine like terms.) …………….. (4)
x° = \(\frac{180^{\circ}}{10}\) (Divide both sides with 10) ……………… (5)
x = 18° (Simplify) ……………. (6)
m∠BEF = 6x° = (6 ∙ 18)° = 108°

Question 7.
m∠CDG = ____________
Answer:
∠BEF is congruent to ∠DEH because they are vertical angles. Therefore,
m∠BEF = m∠DEH = 6x°
∠DEH is supplementary to ∠GDE because they are same-side interior angles. Therefore,
m∠DEH + m∠GDE = 180°
6x° + 4x° = 180°
10x = 180
\(\frac{10 x}{10}=\frac{180}{10}\)
x = 18
∠CDG is congruent to ∠DEH because they are corresponding angles. Therefore,
m∠CDG = m∠DEH = 6x° = (6 ∙ 18)° = 108°

Texas Go Math Grade 8 Lesson 7.1 Guided Practice Answer Key

Use the figure for Exercises 1-4. (Explore Activity 1 and Example 1)

Lesson 7.1 Interior and Exterior Angles Answer Key Question 1.
∠UVY and ___________ are a pair of corresponding angles.
Answer:
∠UVY and ∠VWZ are a pair of corresponding angles.

Question 2.
∠WVYand ∠VWT are ____________ angles.
Answer:
∠WVYand ∠VWT are alternate interior angles.

Question 3.
Find m∠SVW ___________.
Answer:
∠SVW is supplementary to ∠VWT because they are same-side interior angles. Therefore,
m∠SVW + m∠VWT = 180°
4x° + 5x° = 180°
9x = 180
\(\frac{9 x}{9}=\frac{180}{9}\)
x = 20
m∠SVW = 4x° = (4 ∙ 20)° = 80°

Question 4.
Find m∠VWT. ______________
Answer:
∠SVW is supplementary to ∠VWT because they are same-side interior angles Therefore.
m∠SVW + m∠VWT = 180°
4x° + 5x° = 180°
9x = 180
\(\frac{9 x}{9}=\frac{180}{9}\)
x = 20
m∠VWT = 5x° = (5 ∙ 20)° = 100°

Question 5.
Vocabulary When two parallel lines are cut by a transversal, ___________ angles are supplementary. (Explore Activity 1)
Answer:
If two parallel lines are cut by a transversal line, interior angles are formed. The pairs of consecutive interior angles are called same-side interior angles which are supplementary Another thing is when two parallel lines are cut by a transversal, the pairs of angles on either side of the transversal and inside the two lines are called alternate interior angles which are congruent.

By the description, when two parallel lines are cut by a transversal, same-side interior angle are supplementary.

Essential Question Check-In

Practice and Homework Lesson 7.1 Go Math 8th Grade Answer Key Question 6.
What can you conclude about the interior angles formed when two parallel lines are cut by a transversal?
Answer:
When two parallel lines are cut by a transversal, then four interior angles are formed. These four interior angles could be the alternate interior angles or the same-side interior angles.
Alternate interior angles lie on the alternate side of the transversal and they are congruent.
Same-side interior angles lie on the same side of the transversal which are not adjacent. Same-side interior angles are said to be supplementary.

Texas Go Math Grade 8 Lesson 7.1 Independent Practice Answer Key

Vocabulary Use the figure for Exercises 7-10.

Question 7.
Name all pairs of corresponding angles.
Answer:
The corresponding angles identified in the given diagram are:

• ∠1 and ∠5
• ∠3 and ∠7
• ∠2 and ∠6
• ∠4 and ∠8

Question 8.
Name both pairs of alternate exterior angles.
Answer:
Alternate exterior angles identified in the given diagram are:

• ∠1 and ∠8
• ∠2 and ∠7

Question 9.
Name the relationship between ∠3 and ∠6.
Answer:
∠3 and ∠6 are alternate interior angles.

Parallel Lines and Transversals Quiz Answer Key Pdf Question 10.
Name the relationship between ∠4 and ∠6.
Answer:
Angles ∠4 and ∠6 are same-side interior angles.

Find each angle measure.

Question 11.
m∠AGE when m∠FHD = 30° _________
Answer:
∠AGE and ∠FHD are alternate exterior angles. Therefore,
m∠AGE = m∠FHD = 30°
m∠AGE = 30°

Question 12.
m∠AGH when m∠CHF = 150° __________
Answer:
m∠AGH and ∠CHF are corresponding angles. Therefore,
m∠AGH = m∠CHF = 150°
m∠AGH = 150°

Question 13.
m∠CHF when m∠BGE = 110° ___________
Answer:
∠CHF and ∠BGE are alternate exterior angles. Therefore,
m∠CHF = m∠BGE = 110°
m∠CHF = 110°

Question 14.
m∠CHG when m∠HGA = 120° _____________
Answer:
∠CHG is supplementary to ∠HGA because they are same-side interior angles. Therefore,
m∠CHG + m∠HGA = 180°
m∠CHG + 120° = 180°
m∠CHG + 120° – 120° = 180° – 120°
m∠CHG = 60°

Question 15.
m∠BGH = _____________
Answer:

∠BGH and ∠GHD are supplementary because they are same-side interior angles. So,
∠BGH + ∠GHD = 180°
3x + (2x + 50)° = 180°
5x = 180° – 50° = 130°
x = \(\frac{130}{5}\) = 26°
∠BGH = 3x° = 3 × 26° = 78°
∠GHD = (2x + 50)° = (2 × 26° + 50) = 102°

Lesson 7.1 How to Solve Parallel Lines Cut by a Transversal Question 16.
m∠GHD = __________
Answer:
∠BGH is supplementary to ∠GHD because they are same-side interior angles. Therefore,
m∠BGH + m∠GHD = 180°
3x° + (2x + 50)° = 180°
3x + 2x + 50 – 50 = 180 – 50
5x = 130
\(\frac{5 x}{5}=\frac{130}{5}\)
x = 26
m∠GHD = (2x + 50)° = (2 ∙ 26 + 50)° = (52 + 50)° = 102°
m∠GHD = 102°

Question 17.
The Cross Country Bike Trail follows a straight line where it crosses 350th and 360th Streets. The two streets are parallel to each other. What is the measure of the larger angle formed at the intersection of the bike trail and 360th Street? Explain.

Answer:
For easier calculations, we are going to use the following schema. It is given that m∠3 = 48°

The larger angle formed at the intersection of the bike trail and 360th Street is the angle 5 in our schema. ∠5 is supplementary to ∠3 because they are same-side interior angles. Therefore,
m∠5 + m∠3 = 180°
m∠5 + 48°= 180°
m∠5 + 48° – 48° = 180° – 48°
m∠5 = 132°
The larger angle formed at the intersection of the bike trail and 360th Street is 132°

Question 18.
Critical Thinking How many different angles would be formed by a transversal intersecting three parallel lines? How many different angle measures would there be?
Answer:

As we can see from the above schema, there are 12 different angles formed by a transversal intersecting three parallel lines.
There are 2 different angle measures:

• m∠1 = m∠4 = m∠5 = m∠8 = m∠9 = m∠12
• m∠2 = m∠3 = m∠6 = m∠7 = m∠10 = m∠11

Question 19.
Communicate Mathematical Ideas In the diagram at the right, suppose m∠6 = 125°. Explain how to find the measures of each of the other seven numbered angles.

Answer:

• m∠2 = m∠6 = 125° because ∠2 and ∠6 are corresponding angles.
• m∠3 = m∠2 = 125° because ∠3 and ∠2 are vertical angles.
• m∠7 = m∠3 = 125° because ∠7 and ∠3 are corresponding angles.
• ∠4 is supplementary to ∠6 because they are same-side interior angles. Therefore,

m∠4 + m∠6 = 180°
m∠4 + 125° = 180°
m∠4 + 125° – 125° = 180° – 125°
m∠4 = 55°

• m∠8 = m∠4 = 55° because ∠8 and ∠4 are corresponding angles.
• m∠1 = m∠4 = 55° because ∠1 and ∠4 are vertical angles
• m∠5 = m∠1 = 55° because ∠5 and ∠1 are corresponding angles

H.O.T. Focus on Higher Order Thinking

Question 20.
Draw Conclusions In a diagram showing two parallel lines cut by a transversal, the measures of two same-side interior angles are both given as 3x°. Without writing and solving an equation, can you determine the measures of both angles? Explain. Then write and solve an equation to find the measures.
Answer:
We are given that the measures of both angles are equal. We also know that the sum of two same-side interior angles is 180°. Therefore, without writing and solving any equation, we divide 180 by 2 and we find that each angle measures 90°.
If we use an equation to find the measures of the angle, we have:
m∠1 + m∠2 = 180°
3x + 3x = 180°
6x = 180°
\(\frac{6 x}{6}=\frac{180}{6}\)
x = 30
m∠1 = m∠2 = 3x = 3 ∙ 30 = 90°

Question 21.
Make a Conjecture Draw two parallel lines and a transversal. Choose one of the eight angles that are formed. How many of the other seven angles are congruent to the angle you selected? How many of the other seven angles are supplementary to your angle? Will your answer change if you select a different angle?
Answer:
Select angle ∠1, for example. There are 3 congruent angles to ∠1 : ∠3, ∠5, and ∠7
∠3 = ∠1 because they are vertical angles,
∠7 = ∠1 because they are alternate exterior angles.
And, there are 4 supplementary angles: ∠2, ∠4, ∠6 and ∠8
No matter what angle we select, we will always have the same answer – 3 congruent angles, and 4 supplementary.

Three of the other seven are congruent to the selected angle.

• Four are supplementary.
• The answer will not change.

Lesson 7.1 How to Find Angles of Parallel Lines Cut by a Transversal Question 22.
Critique Reasoning In the diagram at the right, ∠2, ∠3, ∠5, and ∠8 are all congruent, and ∠1, ∠4, ∠6, and ∠7 are all congruent. Aiden says that this is enough information to conclude that the diagram shows two parallel lines cut by a transversal. Is he correct? Justify your answer.

Answer:
In the diagram, congruent angles are ∠2, ∠3, ∠5, ∠8., and also congruent are ∠1, ∠4, ∠6, ∠7. Angle ∠2 is supplementary to angle ∠1, so we can say that ∠2 is supplementary to angle ∠4, because ∠4 = ∠1 . If this is a transversal angle ∠2 and ∠4 would be equal. So, the only option is that all the angles are right angles, it mean that transversal is normal on two parallel. We don’t have enough information to claim that, so we can not tell if this is transversal.

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Module 6 Answer Key Functions.

Texas Go Math Grade 8 Module 6 Answer Key Functions

Texas Go Math Grade 8 Module 6 Are You Ready? Answer Key

Complete these exercises to review skills you will need for this chapter.

Evaluate each expression for the given value of x.

Question 1.
2x + 3 for x = 3 ____
Answer:
2x + 3 for x = 3 Given
2(3) + 3 Substitute the given value of x for x
6 + 3 Multiply
9 Add

Go Math Grade 8 Module 6 Answer Key Question 2.
-4x + 7 for x = -1 _____
Answer:
To evaluate an expression for the given value of x, let us substitute the value of x in the expression.

Evaluating -4x + 7 for x = -1
-4x + 7 = -4(-1) + 7 Substitute the value of x.
= 4 + 7 Simplify.
= 11

Question 3.
1.5x – 2.5 for x = 3 _____
Answer:
1.5x – 2.5 for x = 3 Given
1.5(3) – 2.5 Substitute the given value of x for x
4.5 – 2.5 Multiply
2 Subtract

Question 4.
0.4x + 6.1 for x = -5 ______
Answer:
0.4x + 6.1 for x = -5 Given
0.4(-5) + 6.1 Substitute the given value of x for x
-2 + 6.1 Multiply
4.4 Add

Question 5.
\(\frac{2}{3}\)x – 12 for x = 18 _____
Answer:
\(\frac{2}{3}\)x – 12 for x = 18 Given
\(\frac{2}{3}\)(18) – 12 Substitute the given value of x for x
12 – 12 Multiply
0 subtract

Go Math Grade 8 Module 6 Answer Key Pdf Question 6.
–\(\frac{5}{8}\)x + 10 for x = -8 ______
Answer:
To evaluate an expression for the given value of x, let us substitute the value of x in the expression.

Evaluating –\(\frac{5}{8}\)x + 10 for x = -8
–\(\frac{5}{8}\)x + 10 = –\(\frac{5}{8}\)(-8) + 10 Substitute the value of x.
= \(\frac{40}{8}\) + 10 Simplify.
= 5 + 10
= 15

Define the variables for each situation. Then write an equation.

Question 7.
Jana’s age plus 5 equals her sister’s age.
Answer:
j = Janas age Define the variables used in the situation
s = Jana’s sister age

s = j + 5 Identify the operation involved. Plus indicated addition

Question 8.
Andrew’s class has 3 more students than Lauren’s class.
Answer:
A = Number in Andrew’s class Define the variables used in the situation
L = Number in Lauren’s class

A = L + 3 Identify the operation involved. more indicated addition

Question 9.
The bank is 50 feet shorter than the firehouse.
Answer:
Given the statement The bank is 50 feet shorter than the firehouse’ the variables can be defined by letting x be the height of the bank; and y be the height of the firehouse.

Since x represents the height of the bank that is 50 feet shorter than the firehouse represented as variable y, we
can equate x to y – 50 gives an equation
x = y – 50

Grade 8 Module 6 Answer Key Go Math Question 10.
The pencils were divided into 6 groups of 2.
Answer:
g = number in pencils in groups Define the variables used in the situation
n = number of groups

\(\frac{12}{n}\) = g Identify the operation involved. divided indicated division

Texas Go Math Grade 8 Module 6 Reading Start-Up Answer Key

Visualize Vocabulary
Use the ✓ words to complete the diagram. You can put more than one word in each section of the diagram.

Understand Vocabulary

Complete the sentences using the preview words.

Question 1.
A rule that assigns exactly one output to each input is a ___.
Answer:
A rule that assigns exactly one output to each input is a function.

Explanation:
A function is defined as the relationship in which the input value is paired with exactly one output value.
By definition, a rule that assigns exactly one output to each input is a function.
Thus, the blank part of the statement is a function

Grade 8 Module 6 Answer Key Texas Go Math Question 2.
The value that is put into a function is the ___________
Answer:
The value that is put into a function is the input

Explanation:
In a function, we have two types of values the input and the output values. Input value is defined as the independent variable that is being put into a function. It affects the output because output is the dependent variable or the result in a function.

By definition, the value that is put into a function is input

Thus, the blank part of the statement is input.

Question 3.
The result after applying the function machine’s rule is the ____.
Answer:
The result after applying the function machine’s rule is the <output

Explanation:
In a function, we have two types of values the input and the output values. Input values is defined as the independent variable that is being put into function. It affects the output because output is the dependent variable or the result in a function.

By definition, the result after applying the function machine’s rule is the output.

Thus, the blank part of the statement is output.

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 5.2 Answer Key Writing Linear Equations from a Table.

Texas Go Math Grade 8 Lesson 5.2 Answer Key Writing Linear Equations from a Table

Essential Question
How do you write an equation to model a linear relationship given a table?

Your Turn

Question 1.
The table shows the volume of water released by Hoover Dam over a certain period of time. Graph the data, and find the slope and y-intercept from the graph. Then write the equation for the graph in slope-intercept form.

Answer:

Graph the ordered pairs from the table and draw a line through the points.
Slope = \(\frac{150000-75000}{10-5}\) = \(\frac{75000}{5}\) = 15000
Find the slope using two points on the graph by
Slope(m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
where
(x₂, y₂) = (10, 150000) and (x₁, y₁) = (5, 75000)
b = 0 Read the y-intercept from the graph
y = 15000x Substituting the value of slope (m) and y intercept in y = mx + b

Example 2.
Elizabeth’s cell phone plan lets her choose how many minutes are included each month. The table shows the plan’s monthly cost y for a given number of included minutes x. Write an equation in slope-intercept form to represent the situation.

Step 1
Notice that the change in cost is the same for each increase of 100 minutes. So, the relationship is linear. Choose any two ordered pairs from the table to find the slope.

Step 2
Find the y-intercept. Use the slope and any point from the table.

Step 3’
Substitute the slope and y-intercept.
y = mx + b Slope-intercept form
y = 0.06x + 8 Substitute 0.06 for m and 8 for b.

Reflect

Math Grade 8 Answer Key Pdf Lesson 5.2 Using Intercepts Question 2.
What is the base price for the cell phone plan, regardless of how many minutes are included? What is the cost per minute? Explain.
Answer:
y = 0.06x + 8 From the example 2
The base price = $8 It is the y-intercept, the cost of plan when 0 minutes are included

Cost per minute = $0.06 It is the slope, the cost of call per minute

Question 3.
What If? Elizabeth’s cell phone company changes the cost of her plan as shown below. Write an equation in slope-intercept form to represent the situation. How did the plan change?

Answer:
Slope = \(\frac{35-30}{200-100}\) = \(\frac{5}{100}\) = 0.05
Find the slope using two points on from the table by
Slope(m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
where
(x₂, y₂) = (200, 35)) and
(x₁, y₁) = (100, 30)

30 = 0.05(100) + b Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b):
30 = 5 + b y = mx + b
b = 30 – 5 = 25

y = 0.05x+25 . Substituting the value of slope (m) and y-intercept in slope-intercept form.
y = mx + b
The base price increases to $\$25$ and the cost per minute decreases to $\$0.05$ per minute.

Your Turn

Question 4.
A salesperson receives a weekly salary plus a commission for each computer sold. The table shows the total pay, p, and the number of computers sold, n. Write an equation in slope-intercept form to represent this situation.

Answer:
Slope = \(\frac{700-550}{6-4}\) = \(\frac{150}{2}\) = 75
Find the slope using two points from the table by
where
(x₂, y₂) = (6, 700) and (x₁, y₁) = (4, 550)

700 = 75(6) + b Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b):
700 = 450 + b y = mx + b
b = 700 – 450 = 250
p = 75n + 250 Substituting the value of slope (m) and y-intercept in slope-intercept form. y = mx + b
Note that p is represented by the axis and n is represented by the axis

Lesson 5.2 Writing Linear Equations from a Table Answer Key Question 5.
To rent a van, a moving company charges a daily fee plus a fee per mile. The table shows the total cost, c, and the number of miles driven, d. Write an equation in slope-intercept form to represent this situation.

Answer:
Slope = \(\frac{50-45}{20-10}\) = \(\frac{5}{10}\) = 0.5
Find the slope using two points on from the table by
Slope(m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
where
(x₂, y₂) (20, 50) and (x₁, y₁) = (10, 45)
50 = 0.5(20) + b Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b): y = mx + b
b = 50 – 10 = 40

c = 0.5d + 40 Substituting the value of slope (m) and y-intercept in slope-intercept form. y = mx + b
Note that c is represented by the axis and d is represented by the axis

Texas Go Math Grade 8 Lesson 5.2 Guided Practice Answer Key 

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. Then write the equation for the graph in slope-intercept form. (Example 1)

Answer:
Graph the ordered pairs from the table and draw a line through the points.
Slope(m) = \(\frac{20-15}{0-4}\) = \(\frac{5}{-4}\) = -1.25 Find the slope using two points on the graph by where slope(m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where (x₂, y₂) = (4, 15) and (x₁, y₁) = (0, 20)
b = 20 Read the y-intercept from the graph
y = -2.5x + 20 Substituting the value of slope (m) and y-intercept in slope-intercept form.
y = mx + b

The table shows the temperature (y) at different altitudes (x). This is a linear relationship. (Example 2)

Writing Equations from a Table Worksheet Answer Key Question 2.
Find the slope for this relationship.
Answer:
From the table we can see that the rate of change is constant, so we can use the ratio \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) to find the slope.
The relationship is linear. Choose two points from the table and put them in the calculation.

Question 3.
Find the y-intercept for this relationship.
Answer:
b = 9 Read the y-intercept from the table (when x = 0)

Question 4.
Write an equation in slope-intercept form that represents this relationship.
Answer:
y = -0.004x + 59 Substituting the value of slope (m) and y intercept in slope-intercept form.
y = mx + b

Question 5.
Use your equation to determine the temperature at an altitude of 5,000 feet.
Answer:
The equation for the given table is y = -0.004x + 59.

We need to determine the temperature (y), measured in degrees Fahrenheit, at an altitude of 5000 feet (x). Put the given value of x into the equation.
y = -0.004 . 5000 + 59
y = -20 + 59
y = 39

So the temperature at an altitude of 5000 feet is 39 degrees Fahrenheit.

Essential Question Check-In

Answer Key Math Grade 8 Lesson 5.2 Writing Linear Equations from a Table 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.
Answer:
Use any two points from the table to find the slope using,
Slope(m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

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 form:
y = mx + b to find b.

Texas Go Math Grade 8 Lesson 5.2 Independent Practice Answer Key 

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.


Answer:
Plot the given values of x and y on the graph

From the table we can see that the rate of change is constant, so we can use the ratio \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) to find the slope.
The relationship is linear. Choose two points from the table and put it in the calculation,
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
m = \(\frac{12-10}{2-1}\)
m = \(\frac{2}{1}\)
m = 2
The starting point on the graph is (0, 8), i.e. (0, b). So the value of the y-intercept is 8. → b = 8

The standard form of the linear equation is y = mx + b, so include the obtained values for the slope and y-intercept:
y = 2x + 8

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.


Answer:
plot the given values of x and y on the graph.

From the table we can see that the rate of change is constant, so we can use the ratio \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) to find the slope.
The relationship is linear. Choose two points from the table and put it in the calculation.
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
m = \(\frac{150-100}{2-1}\)
m = \(\frac{50}{1}\)
m = 50
The starting point on the graph is (0, 50), i.e. (0, b). So the value of the y-intercept is 50. → b = 50

The standard form of the linear equation is y = mx + b, so include the obtained values for the slope and y-intercept:
y = 50x + 50

Go Math 8th Grade Pdf Lesson 5.2 Answer Key Question 9.
A friend gave Ms. Morris a gift card for a number of car local car wash. The table shows the linear relationship of how the value left on the Amount 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.
Answer:
Slope = \(\frac{18-30}{8-0}\) = \(\frac{-12}{8}\) = -1.5
Find the slope using two points from the table by Slope(m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where (x₂, y₂) = (8, 18) and (x₁, y₁) = (0, 30)
b = 30 Read the y-intercept from the table (when x = 0)
y = -1.50x + 30 Substituting the value of slope (m) and y-intercept in slope-intercept form. y = mx + b

b. Explain the meaning of the negative slope in this situation.
Answer:
The negative slope means that the amount on the card is decreased by $1.5 per car wash. Meaning of negative slope

c. What is the maximum value of x that makes sense in this context? Explain.
Answer:
0 = -1.50x + 30 The maximum value of s that makes sense in this context is when $0 is left on the card i.e. y = 0
1.5x = 30
x = \(\frac{30}{1.5}\) = 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.

Answer:
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Put the given values of x and y from the table into the calculation. Choose two points.
m = \(\frac{0-(-1)}{-1-(-2)}\)
m = \(\frac{0+1}{-1+2}\)
m = \(\frac{1}{1}\)
m = 1
The standard form of the linear equation is y = mx + b, so include the obtained values of x, y, and the slope to get y-intercept.
For x and y, choose one point from the table.
y = mx + b
-1 = 1 • (-2) + b
-1 = -2 + b
-1 + 2 = b
b = 1

Equation in slope-intercept form is: y = 1x + 1 → y = x + 1

Writing Equations from a Table Worksheet Answer Key Pdf Question 11.

Answer:
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Put the given values of x and y from the table into the calculation. Choose two points.
m = \(\frac{4-14}{1-(-4)}\)
m = \(\frac{-10}{1+4}\)
m = \(\frac{-10}{5}\)
m = \(-\frac{10}{5}\)
m = -2

The standard form of the linear equation is y = mx + b, so include the obtained values of x, y, and the slope to get y-intercept.
For x and y, choose one point from the table.
y = mx + b
4 = -2 • 1 + b
4 = -2 + b
4 + 2 = b
b = 6
The equation in slope-intercept form is: y = 2x + 6

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.

Answer:
complete the table

b. Write an equation in slope-intercept form that shows how much money Desiree has in her savings account after x months.
Answer:
y = 53.5x + 125 Substituting the value of slope (m) and y-intercept in slope-intercept form. y = mx + b where, m = 53.5 and b = 125

c. Use the equation to find how much money Desiree will have in savings after 11 months.
Answer:
y = 53.5(11) + 123 Substitute x = 11 months
y = 588.5 + 125 = 713.5
Desiree will have $713.5 after 11 months

Go Math Answer Key Grade 8 Linear Equations Answer Key Question 13.
Monty documented the amount of rain his farm received on a monthly basis, as shown in the table.

a. Is the relationship linear? Why or why not?
Answer:
No
The change in the months is constant but the change in rainfall is not constant

b. Can an equation be written to describe the amount of rain? Explain.
Answer:
No
There is no apparent pattern in the given data.

Texas Go Math Grade 8 Lesson 5.2 H.O.T. Focus On Higher Order Thinking Answer Key 

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

Lesson 5.2 Answer Key 8th Grade Graphing Equations from a Table Answer Key Question 15.
What If? Jaime graphed linear data given in the form (of 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.
Answer:
Jamie’s graph contained (0, 0). Since Jayla’s data were the same, y intercept is 0 but x and y are switched.

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 6.3 Answer Key Comparing Functions.

Texas Go Math Grade 8 Lesson 6.3 Answer Key Comparing Functions

Essential Question
How can you use tables, graphs, and equations to compare functions?

Your Turn

Question 1.
Quentin is choosing between buying books at the bookstore or buying online versions of the books for his tablet. The cost, y dollars, of ordering books online for x books is y = 6.95x + 1.50. The cost of buying the books at the bookstore is shown in the table. Which method of buying books is more expensive if Quentin wants to buy 6 books?

Answer:
y = 6.95x + 1.5 Buying the books online
y = 6.95(6) + 1.5 = $43.2 When Quentin wants to buy 6 books, substitute x = 6
Slope = \(\frac{15-7.5}{2-1}\) = \(\frac{7.5}{1}\) = 7.5 Writing the equation for buying the books at the bookstore. Find the slope using given points by Slope (m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where (x₂, y₂) = (2, 15) and (x₁, y₁) = (1, 7.5)
15 = 7.5(2) + b Substituting the value of slope (m) and (x, y) in the
15 = 15 + b slope intercept form to find y intercept (b): y = mx + b
b = 15 – 15 = 0
y = 7.5x Substituting the value of slope (m) and y intercept (b) in slope intercept form: y = mx + b
y = 7.5(6) = $45 When Quentin wants to buy 6 books, substitute x = 6
Buying books at the bookstore is more expensive. Compare the cost of buying 6 books $43.2 < $45

Texas Go Math Grade 8 Lesson 6.3 Explore Activity Answer Key

Explore Activity 1

Comparing a Table and a Graph
The table and graph show how many words Morgan and Brian typed correctly on a typing test. For both students, the relationship between words typed correctly and time is linear.

A. Find Morgan’s unit rate.

B. Find Brian’s unit rate.

C. Which student types more correct words per minute?

Reflect

Lesson 6.3 Comparing Functions Answer Key Go Math 8th Grade Question 2.
Katie types 17 correct words per minute. Explain how a graph of Katie’s test results would compare to Morgan’s and Brian’s.
Answer:
Katie types 17 correct words per minute; Brian types 20 correct words per minute and Morgan types 15 correct words per minute. Therefore, the graph of Katie’s test results will be a straight line that passes through the origin (0, 0). The value of its slope is greater than the slope of Morgan’s graph of test results and smaller than the slope of Brian’s graph of test results. So, the graph will be a straight line that lies between the line of Morgan and Brian.

Explore Activity 2
Comparing a Graph and a Description
Jamal wants to buy a new game system that costs $200. He does not have enough money to buy it today, so he compares layaway plans at different stores.

The plan at Store A is shown on the graph.
Store B requires an initial payment of $60 and weekly payments of $20 until the balance is paid in full.

A. Write an equation in slope-intercept form for Store A’s layaway plan. Let x represent the number of weeks and y represent the balance owed.

B. Write an equation in slope-intercept form for Store B’s layaway plan. Let x represent the number of weeks and y represent the balance owed.

C. Sketch a graph of the plan at Store B on the same grid as Store A.

D. How can you use the graphs to tell which plan requires the greater down payment? How can you use the equations?

E. How can you Use the graphs to tell which plan requires the greater weekly payment? How can you use the equations?

F. Which plan allows Jamal to pay for the game system faster? Explain.

Texas Go Math Grade 8 Lesson 6.3 Guided Practice Answer Key

Doctors have two methods of calculating maximum heart rate. With the first method, the maximum heart rate,y, in beats per minute is y = 220 – x, where x is the person’s age. The maximum heart rate with the second method is shown in the table. (Example 1)

Question 1.
Which method gives the greater maximum heart rate for a 70-year-old?
Answer:
y = 220 – x First method
y = 220 – 70 = 150 Substitute x = 70 yrs
slope = \(\frac{187-194}{30-20}\) = \(\frac{-7}{10}\) = -0.7
Writing the equation for the second method. Find the slope using given points by slope (m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where (x₂, y₂) = (30, 187) and (x₁, y₁) = (20, 194)
194 = -0.7(20) + b Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b): y = mx + b
194 = -14 + b
b = 194 + 14 = 208

y = 208 – 0.7x Substituting the value of slope (m) and y-intercept (b) in slope-intercept form: y = mx + b
y = 208 – 0.7(70) = 159 Substitute x = 70 yrs
The second method gives the greater maximum heart rate for a 70-year-old methods

Compare the maximum heart rate for the two methods
150 < 159

Question 2.
Are heart rate and age proportional or nonproportional for each method?
Answer:
For method 1, the relationship is non-proportional. Comparing with the general linear form of an equation: y = mx + b.
For method 2, the relationship is non-proportional. Since b ≠ 0, the relationship is not proportional.

Aisha runs a tutoring business. With Plan 1, students may choose to pay $15 per hour. With Plan 2, they may follow the plan shown on the graph. (Explore Activity 1 and 2)

Question 3.
Describe the plan shown on the graph.
Answer:
We choose two points on the graph to find the slope. We substitute (0, 40) for (x₁, y₁) and (4, 60) for (x₂, y₂).
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) = \(\frac{60-40}{4-0}\) = \(\frac{20}{4}\) = 5
We read the y-intercept from the graph, which is 40.
b = 40
We use the slope and y-intercept values that we found to write an equation in slope-intercept form.
y = mx + b
y = 5x + 40
This means that
Plan 2 has an initial cost of $40 and a rate of $5 per hour

Question 4.
Sketch a graph of Plan 1 on the same grid as Plan 2.
Answer:

Plan 1 is represented by red and Plan 2 is represented by blue

8th Grade Comparing Functions Answer Key Lesson 6.3 Question 5.
What does the intersection of the two graphs mean?
Answer:
The intersection of the two graphs represents the number of hours for which both plans will cost the same.

Question 6.
Which plan is cheaper for 10 hours of tutoring?
Answer:
Knowing that with Plan 1 student pays 15 dollars per hour, we can calculate the cost of 10 hours of tutoring. Variable y is the cost and variable æ is the number of hours. Therefore:
y₁ = 15x
y₁ = 15 . 10
y₁ =150

To calculate the cost of Plan 2, we will use the following formula:
y₂ = 5x + 40,
where variable y is the cost and variable x is the number of hours. Therefore:
y₂ = 5 . 10 + 40
y₂ = 90
Hence, we can conclude that Plan 2 is cheaper for 10 hours of tutoring.

Question 7.
Are cost and time proportional or nonproportional for each plan?
Answer:
The cost and time are proportional for Plan 1 Compared with the general linear form of equation: y = mx + b. Since b = 0, the relationship is proportional.

The cost and time are not proportional for Plan 2 Compared with the general linear form of equation: y = mx + b.
Since b ≠ 0, the relationship is not proportional.

Essential Question Check-In

Go Math Grade 8 Lesson 6.3 Answer Key Question 8.
When using tables, graphs, and equations to compare functions, why do you find the equations for tables and graphs?
Answer:
The tables and graphs represent a part of the solution of the function By writing the equation, any value can be substituted to evaluate the function and compare it with the equations.

Texas Go Math Grade 8 Lesson 6.3 Independent Practice Answer Key

The table and graph show the miles driven and gas used for two scooters.

Question 9.
Which scooter uses fewer gallons of gas when 1350 miles are driven?
Answer:
Looking at the given table for scooter A, we can conclude that scooter uses 2 gallons for every 150 miles. Therefore, we can write the equation where variable y represents used gallons of gas and variable x represents driven miles.
y_a = \(\frac{2 x}{150}\)
y_a = \(\frac{x}{75}\)
Considering the fact that we are testing the consumption of gas for 1350 driven miles, we can use the equation for y_a and insert x = 1350 as follows:
y_a = \(\frac{1350}{75}\)
y_a = 18
Given graph represents driven miles and gas used for scooter B. We will find value of driven miles for gas
consumption of 1 gallon by drawing a horizontal line on the value of 1 gallon and looking for an intersection with the
function. Lines intersect on x = 90, meaning that scooter B uses 1 gallon of gas for 90 driven miles. Therefore:
y_b = \(\frac{x}{90}\),
where y_b is the consumption of gas and variable x represents driven miles.

Using the equation obtained in the previous step, we can calculate gas consumption for 1350 driven miles.
y_b = \(\frac{1350}{90}\)
y_b = 15
Hence, we can conclude that scooter B uses fewer gallons of gas when driving 1350 miles.

Lesson 6.3 Answer Key 8th Grade Comparing Functions Question 10.
Are gas used and miles proportional or nonproportional for each scooter?
Answer:
The gas used and miles are proportional for both scooters.

Comparing with the general linear form of an equation: y = mx + b. Since b = 0, the relationship is proportional.

A cell phone company offers two texting plans to its customers. The monthly cost, y dollars, of one plan, is y = 0.10x + 5, where x is the number of texts. The cost of the other plan is shown in the table.

Question 11.
Which plan is cheaper for under 200 texts? ______
Answer:
y = 0.10x + 5 Plan 1
y = 0.10(199) + 5 = $24.90 Substitute x = 199
slope = \(\frac{25-20}{200-100}\) = \(\frac{5}{100}\) = 0.05 Writing the equation for plan 2. Find the slope using given points by Slope (m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
where (x₂, y₂) = (200, 25) (x₁, y₁) = (100, 20)
20 = 0.05(100) + b Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b):
20 = 5 + b y = mx + b
b = 20 – 5 = 15
y = 0.05x + 15 Substituting the value of slope (m) and y-intercept (b) in slope-intercept form: y = mx + b
y = 0.05(199) + 15 = $24.95 Substitute x = 199

Plan 1 is cheaper Compare the cost for two plans for text < 200
$24.90 < $24.95

Question 12.
The graph of the first plan does not pass through the origin. What does this indicate?
Answer:
y = 0.10x + 5 Plan 1
The graph that does not pass through the origin indicates that there is a base price of $\$5$ for the plan.

Go Math 8th Grade Lesson 6.3 Comparing Functions Question 13.
Brianna wants to buy a digital camera for a photography class. One store offers the camera for $50 down and a payment plan of $20 per month. The payment plan for a second store is described by y = 15x + 80, where y is the total cost in dollars and x is the number of months. Which camera is cheaper when the camera is paid off in 12 months? Explain.
Answer:
For us to determine which camera is cheaper when the camera is paid off in 12 months, let us determine the total cost of the digital camera offered by the two stores.

The first store offers the camera for $50 down and a payment of $20 per month. By applying the slope-intercept form y = mx + b. Let y be the total cost of the camera for the first store x be the number of months the camera needs to be paid off m be the payment every month b be the down payment

Substituting the values of m = 20, b = 50, and x = 12, the total cost of the camera offered by the first store is
\begin{aligned}
y&=mx+b\
y&=20(12)+50\
y&=240+50\
y&=290
\end{aligned}

Thus, the total cost of the camera offered by the first store is $290

The payment plan for the second store is described by y = 15x + 80
where y is the total. cost of the camera in dollars and x is the number of months

Thus, the total cost of the camera offered by the second store is $260

Now, let us compare the total costs of the camera offered by the two stores. The total cost of the camera offered by the first store is $290 while the second store camera’s total cost is $260. Since $260 is less than $290, then we can conclude that **the camera of the second store is cheaper than the camera of the first store**.

Question 14.
The French club and soccer team are washing cars to earn money. The amount earned, y dollars, for washing x cars is a linear function. Which group makes the most money per car? Explain.

Answer:
slope = \(\frac{20-10}{4-2}\) = \(\frac{10}{2}\) = 5 Find the slope/rate of change for the French club. Find the slope using the given points by
Slope (m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
French club makes $\$5$ per car
where
(x₂, y₂) = (4, 20) and (x₁, y₁) = (2, 10)
slope = \(\frac{16-0}{2-0}\) = \(\frac{16}{2}\) = 8 Find the slope/rate of change for a soccer club. Find the slope using the given points by
Slope (m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where
(x₂, y₂) = (2, 16) and (x₁, y₁) = (0, 0)
Soccer club makes the most money per car Compare the money earned for washing one car $5 < $8

Texas Go Math Grade 8 Lesson 6.3 H.O.T. Focus On Higher Order Thinking Answer Key

Question 15.
Draw Conclusions Gym A charges $60 a month plus $5 per visit. The monthly cost at Gym B is represented by y = 5x + 40, where x is the number of visits per month. What conclusion can you draw about the monthly costs of gyms?
Answer:
Since the rate per visit is the same, the monthly cost of Gym A is always more than Gym B

Comparing Functions Worksheet 8th Grade Answers Question 16.
Justify Reasoning Why will the value of y for the function y = 5x + 1 always be greater than that for the function y = 4x + 2 when x > 1?
Answer:
y₁ = 5x + 1 and y₂ = 4x + 2 Subtracting y₂ from y₁ we get
y₁ – y₂ = 5x + 1 – (4x + 2)
y₁ – y₂ = x – 1
For x ≥ 1 we get x – 1 ≥ 0
So, y₁ – y₂ ≥ 0
or y₁ ≥ y₂
Hence proved.

Question 17.
Analyze Relationships The equations of two functions are y = -21x + 9 and y = -24x + 8. Which function is changing more quickly? Explain.
Answer:
To determine among the given equations of functions changing more quickly, we have to determine the absolute
value of slope of the equations and then compare them. If the absolute value of the slope of one equation is greater than the slope of the other, then this equation changing more quickly than the other.

Determining the slopes of the two equations below.
y = -21x + 9
y = -24x + 8
Since, the two equations are already in the form of y = mx + b, where m is the slope, it is easier to determine their slope and these are
| Equation | Slope |
|–|–|
|y = -21x + 9 | -21|
|y = -24x + 81 -24|
Now, Let us get the absolute values of the two slopes.
|- 21| = 21
|-24| = 24

The absolute value of the slope of the equation y = -21x + 9 is 21 while of the equation y = -24x + 8 is 24. Since 24 is greater than 21, then the equation y = -24x + 8 changes more quickly than the equation y = -21x + 9.
See the explanation

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Module 4 Quiz Answer Key.

Texas Go Math Grade 8 Module 4 Quiz Answer Key

Texas Go Math Grade 8 Module 4 Ready to Go On? Answer Key

4.1 Representing Linear Nonproportional Relationships

Question 1.
Complete the table using the equation y = 3x + 2.

Answer:
In the given equation y = 3x + 2 include the given values of x.
x₁ = -1 → y = 3 • (-1) + 2 = -3 + 2 → y = -1
x₂ = 0 → y = 3 • 0 + 2 = 0 + 2 → y₁ = 2
x₃ = 1 → y = 3 • 1 + 2 = 3 + 2 → y₁ = 5
x₄ = 2 → y = 3 • 2 + 2 = 6 + 2 → y₁ = 8
x₅ = 3 → y = 3 • 3 + 2 = 9 + 2 → y₁ = 11
Obtained values of y put in the tabla

4.2 Determining Slope and Y-intercept

Grade 8 Module 4 End of Module Assessment Answer Key Question 2.
Find the slope and y-intercept of the line in the graph.

Answer:
The following equation for finding the slope and y-intercept is y = mx + b
The slope presents the ‘m’ in the equation, and the y-intercept represents b
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
From the given graph, find two points that will help you calculate the slope.
I’ll take (0, 1) for (x₁, y₁) and (1, 4) for (x₂, y₂).
Thus: m = \(\frac{4-1}{1-0}\) = \(\frac{43}{1}\) = 3
To find the y-intercept, we need to include values of x and y from one of these two points and include the obtained slope. So:
y = mx + b
1 = 3 . 0 + b
1 = 0 + b
b = 1
m = 3, b = 1

4.3 Graphing Linear Nonproportional Relationships

Question 3.
Graph the equation y = 2x – 3 using slope and y-intercept.

Answer:
y = 2x – 3
Slope = 2
y-intercept = -3

Plot the point that contains the y-intercept:
(0, -3)
The slope is m = \(\frac{2}{1}\)
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?

Answer:
To see if some linear relationship is proportional or nonproportional, we have to use ratio \(\frac{y}{x}\), which needs to be constant for proportionality.

From the given table, we have values for x and y, so just put them into the calculation.
\(\frac{y}{x}\) = \(\frac{4}{1}\) = 4
= \(\frac{8}{2}\) = 4
= \(\frac{12}{3}\) = 4
= \(\frac{16}{4}\) = 4
= \(\frac{20}{5}\) = 4
Thus, the obtained values are constant so this table represents a proportional relationship.

4.5 Solving Systems of Linear Equations by Graphing

Question 5.
A school band ordered hats for $3 and large T-shirts for $5. They bought 150 items in all for $590. Graph a system of equations to find how many hats and T-shirts the band ordered.
____________
Answer:
The graph represents the nonproportional relationship because the obtained b, i.e. y-intercept, is bigger than 0 (b /
= 0) and the line does not pass the origin. When the b = 0 and the line passes the origin, the linear relationship is proportional.

Essential Question

Grade 8 Math Module 4 Answer Key Question 6.
How can you identify a linear nonproportional relationship from a table, a graph, and an equation?
Answer:
The graph represents the nonproportional relationship because the b, i.e. y-intercept from the given equation, is
bigger than 0 (b ≠ 0) and the line does not pass the origin. When the b = 0 and the line passes the origin, the linear relationship is proportional.

Texas Go Math Grade 8 Module 4 Mixed Review Texas Test Prep Answer Key

Selected Response

Question 1.
The table below represents which equation?

(A) y = -x – 10
(B) y = -6x
(C) y = 4x – 6
(D) y = -4x + 2
Answer:
(C) y = 4x – 6

Explanation:
The table is represented by Option C From the table, you can see that the y-intercept (when x = 0) is b = -6. Comparable to y = mx + b
y = 4x – 6

Question 2.
The graph of which equation is shown below?

(A) y = -2x + 3
(B) y = -2x + 1.5
(C) y = 2x + 3
(D) y = 2x + 1.5
Answer:
(A) y = -2x + 3

Explanation:
Option C is rejected
Since the graph is slanting downwards, the slope is negative.
The graph represents
y = -2x + 3

Texas Go Math 8th Grade Module 4 Answer Key Pdf Question 3.
The table below represents a linear relationship.

What is the y-intercept?
(A) -4
(B) -2
(C) 2
(D) 3
Answer:

The y-intercept is -2.
Option B is the correct answer.

Question 4.
Which equation represents a nonproportional relationship?
(A) y = 3x + 0
(B) y = -3x
(C) y = 3x + 5
(D) y = \(\frac{1}{3}\)x
Answer:
(C) y = 3x + 5

Explanation:
Option C represents a non-proportional relationship
y = 3x + 5

For a non-proportional relationship, the equation is y = mx + b and b ≠ 0.

Go Math 8th Grade Pdf Module 4 Quiz Answer Key Question 5.
Which statement describes the solution of a system of linear equations for two lines with the same slope and the same y-intercept?
(A) one nonzero solution
(B) infinitely many solutions
(C) no solution
(D) solution of 0
Answer:
(B) infinitely many solutions

Gridded Response

Question 6.
The table shows a proportional relationship. What is the missing y-value?

Answer:

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 5.3 Answer Key Linear Relationships and Bivariate Data.

Texas Go Math Grade 8 Lesson 5.3 Answer Key Linear Relationships and Bivariate Data

Essential Question
How can you contrast linear and nonlinear sets of bivariate data?

Your Turn

Find the equation of each linear relationship.

Question 1.

Answer:
Slope = \(\frac{60-40}{10-5}\) = \(\frac{20}{5}\) = 4
Find the slope using two points from the graph by Slope(m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where (x₂, y₂) = (10, 60) and (x₁, y₁) = (5, 40)
b = 20 Read the y-intercept from the graph (when x = 0)
y = 4x + 20 Substituting the value of slope (m) and y-intercept in slope-intercept form.
y = mx + b
y = 4x + 20

Lesson 5.3 Answer Key Go Math 8th Grade Pdf Question 2.

Answer:
Slope = \(\frac{3600-480}{15-2}\) = \(\frac{3120}{13}\) = 240
Find the slope using two points from the graph by Slope(m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where (x₂, y₂) = (15, 3600) and (x₁, y₁) = (2, 480)
b = 20 Read the y-intercept from the graph (when x = 0)

y = 240x Substituting the value of slope (m) and y-intercept in slope-intercept form. y = mx + b
y = 240x

Reflect

Question 3.
What If? Suppose a regulation changes the cost of the taxi ride to $1.80 per mile, plus a fee of $4.30. How does the price of the 6.5-mile ride compare to the original price?
Answer:
y = 1.8x + 4.3 Substituting the value of slope (m) and y-intercept in slope-intercept form.
y = mx + b
where m = 1.8 and b = 4.3

y = 1.8(6.5) + 4.3 Substitute x = 6.5 mile
y = 11.7 + 4.3 = 16
The cost remains the same.

Lesson 5.3 Answer Key 8th Grade Question 4.
How can you use a graph of a linear relationship to predict a value for a new input?
Answer:
Use the two points from the graph to determine the slope. Determine the y-intercept from the graph Substitute the m and b in slope-intercept form: y = mx + b
Substitute x in the equation to find y.

Question 5.
How can you use a table of linear data to predict a value?
Answer:
Use the two points from the table to determine the slope. Determine the y-intercept from slope and given point by substituting in slope-intercept form: y = mx + b
Write the equation. Substitute x in the equation to find y.

Your Turn

Paulina’s income from a job that pays her a fixed amount per hour is shown in the graph. Use the graph to find the predicted value.

Question 6.
Income earned for working 2 hours ______
Answer:
From the graph, we can read that in 2 hours, her income is 30.Butlet’strytogetitbycalculatingtheslopeand y$-intercept

From the graph, we can see that the line passes through the origin, which means that this linear relationship is proportional.
Therefore our starting point is (0, 0), and what else we can conclude is that our y-intercept (b) in this case is 0. → b = 0
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Choose two points from the graph and put them into the calculation. I’ll take (4, 60) for (x₁, y₁) and (6, 90) for (x₂, y₂). Thus
m = \(\frac{90-60}{6-4}\)
m = \(\frac{30}{2}\)
m = 15

We need to find the value of y, income earned for working 2 hours. x represents the time in hours. The standard form of the linear equation is y = mx + b, so include the obtained values of x, the slope, and the y-intercept.
y = 15 . 2 +0
y = 30 + 0
y = 30
To conclude, for 2 hours of work Paulina gets $30

Go Math Grade 5 Lesson 5.3 Answer Key Question 7.
Income earned for working 3.25 hours ______
Answer:
From the graph, we can see that the line passes through the origin, which means that this linear relationship is proportional.
Therefore our starting point is (0, 0), and what else we can conclude is that our y-intercept (b) in this case is 0. → b = 0
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Choose two points from the graph and put them into the calculation. I’ll take (4, 60) for (x₁, y₁) and (6, 90) for (x₂, y₂). Thus
m = \(\frac{90-60}{6-4}\)
m = \(\frac{30}{2}\)
m = 15

We need to find the value of y, income earned for working 3.5 hours. x represents the time in hours. The standard form of the linear equation is y = mx + b, so include the obtained values of x, the slope, and the y-intercept.
y = 15 . 3.5 + 0
y = 52.5 + 0
y = 52.5
$
To conclude, for 3.5 hours of work Paulina gets $52.5

Question 8.
Total income earned for working for five 8-hour days all at the standard rate ______
Answer:
Income after 8 hr = $120 Estimated from the graph, when x = 8hrs
Income after 5 – 8hrs = 120 * 5 = $600

Texas Go Math Grade 8 Lesson 5.3 Explore Activity Answer Key 

Contrasting Linear and Nonlinear Data
Bivariate data is a set of data that is made up of two paired variables. If the relationship between the variables is linear, then the rate of change (slope) is constant. If the graph shows a nonlinear relationship, then the rate of change varies between pairs of points.

Andrew has two options in which to invest $200. Option A earns a simple interest of 5%, while Option B earns an interest of 5% compounded annually. The table shows the amount of the investment for both options over 20 years. Graph the data and describe the differences between the two graphs.

Step 1
Graph the data from the table for Options A and B on the same coordinate grid.

Step 2
Find the rate of change between pairs of points for Option A and classify the relationship.

The rate of change between the data values is ____, so the graph of Option A shows a __________ relationship.

Step 3
Find the rate of change between pairs of points for Option B and classify the relationship.

The rate of change between the data values is ____,
so the graph of Option B shows a ____ relationship.

Reflect

Question 9.
Why are the graphs drawn as lines or curves and not discrete points?
Answer:
The number of years can be in fractions, hence graphs are drawn as lines or curves and not discrete points

Lesson 5.3 Answer Key 8th Grade Linear Relationships and Bivariate Data Question 10.
Can you determine by viewing the graph if the data have a linear or nonlinear relationship? Explain.
Answer:
Yes. Linear relationship is represented by a straight line and non-linear relationship is represented by a curve.

Question 11.
Draw Conclusions Find the differences in the account balances to the nearest dollar at 5-year intervals for Option B. Flow does the length of time that money is in an account affect the advantage that compound interest has over simple interest?
Answer:
Difference (0 – 5) = 255.26 – 200 = 55.26
Difference (5 – 10) = 325.78 – 255.26 = 70.52
Difference (10 – 15) = 415.79 – 325.78 = 90.01
Difference (15 – 20) = 530.66 – 415.79 = 114.87
The amount of interest earned increases with the number of years for compound interest.

Texas Go Math Grade 8 Lesson 5.3 Guided Practice Answer Key 

Use the following graphs to find the equation of the linear relationship. (Example 1)

Question 1.

Answer:

Use the given points to show that it is a linear relationship
Slope = \(\frac{60-30}{2-1}\) = \(\frac{30}{1}\) = 30 Find the slope using two points from the graph by Slope(m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where (x₂, y₂) = (2, 60) and (x₁, y₁) = (1, 30)
b = 0 Read the y-intercept from the graph (when x = 0)
y = 30x Substituting the value of slope (m) and y-intercept in slope-intercept form.
y = mx + b
y = 30x

Question 2.

Answer:

Use the given points to show that it is a linear relationship.
Slope = \(\frac{12-7}{4-2}\) = \(\frac{5}{2}\) = 2.5 Find the slope using two points from the graph by Slope(m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where (x₂, y₂) = (4, 12) and (x₁, y₁) = (2, 7)
b = 2 Read the y-intercept from the graph (when x = 0)
y = 2.5x + 2 Substituting the value of slope (m) and y-intercept in slope-intercept form. y = mx + b
y = 2.5x + 2

Linear Relationships and Bivariate Data Answer Key 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. (Example 2)


Answer:
Write the equation of the linear relationship.
(2, 70) and (4, 110)(Select two points)
m = \(\frac{110-70}{4-2}\) (Calculate the rate of change)
= \(\frac{40}{20}\) (Simplify)
= 20
y = mx + b
70 = 20(2) + b (Fill in values for x, y and m)
70 = 40 + b (Simplify)
30 = b (Solve for b)
The equation of the linear relationship is y = 20x + 30.

Use your equation from Step 1 to predict the cost of rental that last 5.5 hours.
y = 20x + 30 (Substitute x=5.5)
y = 20(5.5) + 30
y = 110 + 30 (Solve for y)
y = 140
The cost of a rental that lasts 5.5 hours is \140$.

Does each of the following tables represent a linear relationship? Why or why not? (Explore Activity)

Question 4.

Answer:
Yes The graph has a constant rate of change

8th Grade Go Math Lesson 5.3 Answer Key Question 5.

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

Essential Question Chk-In

Question 6.
How can you tell if a set of bivariate data shows a linear relationship?
Answer:
It is a linear relationship if the rate of change is constant or the graph is a straight line

Texas Go Math Grade 8 Lesson 5.3 Independent Practice Answer Key 

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

Question 7.

Answer:
Rate of change = \(\frac{45-15}{9-3}\) = \(\frac{30}{6}\) = 5 Find the rate of change using
Rate of change = \(\frac{105-45}{21-9}\) = \(\frac{60}{12}\) = 5
Slope (m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
It is a linear relationship as the rate of change is constant.

Question 8.

Answer:
Rate of change = \(\frac{76.8-30}{8-5}\) = \(\frac{46.8}{3}\) = 15.6 Find the rate of change using Slope (m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Rate of change = \(\frac{235.2-76.8}{14-8}\) = \(\frac{158.4}{6}\) = 26.4
It is a linear relationship as the rate of change is constant.

Explain whether or not you think each relationship is linear.

Question 9.
the cost of equal-priced DVDs and the number purchased
Answer:
Explanation A:
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 e 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

Explanation B:
Linear relationship The cost of DVDs increases at a constant rate with the number of purchases.

Go Math Grade 8 Lesson 5.3 Answer Key Question 10.
the height of a person and the person’s age
Answer:
Non – Linear relationship 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:
We can say that a relationship is linear if the rate of change or the slope is constant

In the relationship between the area of a square quilt and its side length, the independent variable is the side length and the dependent variable is the area. Now, say, for example, the side of the quilt is 4 feet 3 feet 2 feet its length will be 42, 32, and 22 square feet, respectively.

Now, we have these set of ordered pairs: {(2, 4), (3, 9), (4, 16)}. Based on this example, we have the following as possible solutions for the slope:
when we use {(2, 4), (3, 9)}, slope is
\(\frac{9-4}{3-2}[latex] = 5
when we use {(3, 9), (4, 16)}, slope is
[latex]\frac{16-9}{4-3}[latex] = 7
Looking at the slope, they are not constant, as the area and the sides increase, the rate of change also increases, thus, this is a nonlinear relationship

See the explanation.

Question 12.
the number of miles to the next service station and the number of kilometers
Answer:
We can say that a relationship is linear if the rate of change or the slope is constant.

In the relationship of the number of miles to the next service station and the number of kilometers, the independent variable is a number of kilometers and the dependent variable is the number of miles.

Now, for these values of the number of kilometers, {1, 2, 3, 4}, the corresponding number of miles are:
{0.621, 1.21, 1.86, 2.485}
when we use {(1, 0.61), (2, 1.243)}, slope is
[latex]\frac{1.24-0.62}{2-1}\) = 0.62
when we use {(3, 1.86), (4, 2.48)}, slope is
\(\frac{2.48-1.86}{4-3}\) = 0.62
Looking at the slope, they are constant, as the area and the sides increase, the rate of change remains the same, thus, this is a linear relationship

See the explanation.

Lesson 5.3 Go Math 8th Grade Answer Key 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.
Answer:
The black points on the chart are the default. The green point is what we were supposed to plot

To see if our linear relationship ¡s proportional, calculate the rate of change from the given information. To know if some linear relationship is proportional or nonproportional, we have to use ratio m = \(\frac{y}{x}\), which needs to be constant for proportionality.

We have 3 points on the graph: (4, 0.5); (6, 0.75) and (12, 1.5). Put those in the calculation:
m = \(\frac{0.5}{4}\) = 0.125
m = \(\frac{0.75}{6}\) = 0.125
m = \(\frac{1.5}{12}\) = 0.125

The rate of change is constant so this relationship is proportional. If the relationship is proportional we can
immediately conclude that our y-intercept is equal to 0 (b = 0).

The standard form of the linear equation is y = mx + b.
Our equation in slope-intercept form is:
y = 0.125x + 0 → y = 0.125x
y represents the distance (ft), and x represents the time (s).

We need to determine the distance that the Mars Rover would travel in 1 minute. Since the time is measured in seconds, we need to convert the minute to the default unit So 1 minute has 60 seconds. Put it in the equation to
get the distance.
y = 0.125 • 60
y = 7.5
In one minute, the Mars Rover would travel 7.5 ft.

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 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:
Zefram used y = 3.5x + 16 for the prediction of y for a given value of æ. Eventually, he found out that he made an error calculating the y-intercept and that it was actually 12

Now, if he wants to make a correction to his prediction knowing that the slope is correct, he can simply subtract the predicted value by 4. Look at these illustrations.
y = 3.5x + 16 → y = (3.5x – 16) 4 → y = 3.5x + 12

Since subtracting 4 from the predicted value of y would make the y-intercept equal to 12(as shown in the illustrations), then Zefram can subtract 4 from the predicted value of y. So the answer is Yes.

See the explanation.

Texas Go Math Grade 8 Lesson 5.3 H.O.T. Focus On Higher Order Thinking Answer Key 

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?

Answer:
The only calculation in which we have values for both x and y is the slope calculation.
Since y is unknown and we have to find its value, first we must calculate the slope to include its value as well in the
calculation. Otherwise, both the slope and y would be unknown.

So, to get the slope, choose two points from the table:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
m = \(\frac{114-76}{12-8}\)
m = \(\frac{38}{4}\)
m = 9.5

To get y, when X = 6, choose one point from the table, include it into the calculation, include value of x and the slope.

The obtained value of y is 57 when x = 6. → (6, 57).

Texas Go Math Grade 8 Lesson 5.3 Linear Relationships and Bivariate Data 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:
If the relationship between the variables is linear, then the rate of change(slope) is constant The slope is simply the difference between the y values divided by the difference of the corresponding x values.

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. I do not agree with their idea of Louis since some of the values of x may not be the same difference as the other points. For example, x values of a Linear equation are {1, 3, 7, 8}, the difference between 1 and 3 is different from the difference between 7 and 8, and so on.

See the explanation.

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

Go Math Grade 8 Pdf Linear Relationships Unit Test Answer Key Question 18.
Explain the Errors 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
45 = 79 • 18 + b
45 = 1422 + b, so b = -1377
The equation is y = 79x – 1377.
Answer:
Thomas used (7, 17.5) and (18, 45) from the graph to find the linear relationship. Let us examine his mistake in his solution below.
m = \(\frac{45-7}{18-17.5}\) = \(\frac{38}{0.5}\) = 79
y = 79x + b
45 = 79 • 18 + b
45 = 1422 + b, so b = -1377
The equation is y = 79x – 1377

As we can notice, there is a mistake on the substitution of values of the variables in getting the slope m. The formula for finding the slope is
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
In the problem, the points are (7, 17.5) and (18, 45), wherein
x₁ = 7, y₁ = 17.5, x₂ = 18, y₂ = 45
Instead of substituting 17.5 as the value of y, Thomas substituted 7 and instead of substituting 7 as the value of x₁, he substituted 17.5. In short, Thomas interchanged the values of x₁ and y₁.

Since Thomas made a mistake in the first step of the solution, the proceeding solutions are also incorrect.

Here is the correct answer for the given problem in the statement.
Substituting the values of the variables gives the slope of

Solving for the y-intercept, where in = 2.5 using the point (18, 45)
y = 2.5x + b
45 = 2.5(18)+b
45 = 45 + b
b = 45 – 45
b = 0

Now, having the slope m = 2.5 and y-intercept b = 0, the equation is
y = 2.5x + 0
y = 2.5x

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 4.2 Answer Key Determining Slope and y-Intercept.

Texas Go Math Grade 8 Lesson 4.2 Answer Key Determining Slope and y-Intercept

Essential Question
How can you determine the slope and the y-intercept of a line?

Texas Go Math Grade 8 Lesson 4.2 Explore Activity Answer Key 

Investigating Slope and y-intercept
The graph of every nonvertical line crosses the y-axis. The y-intercept is the y-coordinate of the point where the graph intersects the y-axis. The x-coordinate of this point is always 0.

The graph represents the linear equation y = –\(\frac{2}{3}\) + 4.

Step 1
Find the slope of the line using the points (0, 4) and (-3, 6)

Step 2
The line also contains the point (6, 0). What is the slope using (0, 4) and (6, 0)? Using (-3, 6) and (6, 0). What do you notice?

Step 3
Compare your answers in Steps 1 and 2 with the equation of the graphed line.

Step 4
Find the value of y when x = 0 using the equation y = –\(\frac{2}{3}\)x + 4. Describe the point on the graph that corresponds to this solution.

Step 5
Compare your answer in Step 3 with the equation of the line.

Your Turn

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

Question 1.

Answer:
\(\frac{32-22}{4-2}\) = 5 Finding the slope using given points by
\(\frac{42-32}{6-4}\) = 5 slope (m) = (y₂ – y₁) ÷ (x₂ – x₁)
\(\frac{52-42}{8-6}\) = 5
Slope(m) = $5$
Work backward from x = 2 to x = 0 Find the initial value when the value of x is 0
\(\frac{32-22}{4-2}\) = \(\frac{10}{2}\)
x = 2 – 2 = 0 Subtract the difference of x and y from the first point
y = 22 – 10 = 12
y-intercept (b) 12

Lesson 4.2 Answer Key Go Math Grade 8 Pdf Question 2.

Answer:
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Thus:

So m = 7
To get the y-intercept (“b”) of the line, we need to include values of x and y in the equation:
y = mx + b
Include values of x and y from one point, I will take the first one.
8 = 7 . 1 + b
b = 8 – 7
b = 1

Explore Activity 2

Deriving the Slope-intercept Form of an Equation

In the following Explore Activity, you will derive the slope-intercept form of an equation.

Step 1
Let L be a line with slope m and y-intercept b. Circle the point that must be on the line. Justify your choice.
(b, 0) (0, b) (0, m) (m, 0)

Step 2
Recall that slope is the ratio of change in y to change in x. Complete the equation for the slope m of the line using the y-intercept (0, b) and another point (x, y) on the line.

Step 3
In an equation of a line, we often want y by itself on one side of the equation. Solve the equation from Step 2 for y.

Reflect

Question 3.
Critical Thinking Write the equation of a line with slope m that passes through the origin. Explain your reasoning.
Answer:
y= x When the line passes through the origin, y-intercept = 0 in
y = mx + b

Texas Go Math Grade 8 Lesson 4.2 Guided Practice Answer Key 

Find the slope and y-intercept of the line in each graph. (Explore Activity 1)

Question 1.

Slope m = ___ y-intercept b = ___
Answer:
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
We have point one (0, 1) and point two (2, -3). Include them in the calculation:
m = \(\frac{-3-1}{2-0}\) = \(\frac{-4}{2}\) = -2
m = -2

According to the equation y = mx + b we can calculate the y-intercept, that is b. Include the values of x and y from one point and the obtained slope value
y = mx + b
-1 = -2 . 0 + b
b = 1 – 0
b = 1

Determining Slope and Y-Intercept Lesson 4.2 Answer Key Question 2.

Slope m = ___ y-intercept b = ___
Answer:
The scope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
We have point one (0,-15) and point two (3,0). Include them into the calculation:
m = \(\frac{0-(-15)}{3-0}\) = \(\frac{0+15}{3}\) = \(\frac{15}{3}\) = 5
m = 5
According to the equation y = mx + b we can calculate the y-intercept, that is b. Include the values of x and y from one point and the obtained slope value.
y = mx + b
-15 = 5 . 0 + b
b = -15

Question 3.

Slope m = ___ y-intercept b = ___
Answer:
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
First, read the points on the graph, the places where the graph intersects a certain values of the x and y axes. The first one is (0, -2) and the second one is (2, 1). Include them into the calculation:
m = \(\frac{1-(-2)}{2-0}\) = \(\frac{1+2}{2}\) = \(\frac{3}{2}\)
m = \(\frac{3}{2}\)

According to the equation y = mx + b we can calculate the y-intercept, that is b. Include the values of x and y from one point and the obtained slope value.
y = mx + b
-2 = \(\frac{3}{2}\) . 0 + b
-2 = 0 + b
b = -2

Lesson 4.2 Determining Slope and Y-Intercept Answer Key Pdf Question 4.

Slope m = ___ y-intercept b = ___
Answer:
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
First, read the points on the graph, the places where the graph intersects a certain value of the x and y axes
The first one is (0,9) and the second one is (3,0). Include them into the calculation:
m = \(\frac{0-9}{3-0}\) = \(\frac{-9}{3}\) = -3
m = -3

According to the equation y = mx + b we can calculate the y-intercept, that is b. Include the values of x and y from one point and the obtained slope value.
y = mx + b
9 = -3 . 0 + b
9 = 0 + b
b = 9

Find the slope and y-intercept of the line represented by each table. (Example 1)

Question 5.

slope m = ______y-intercept b = ______
Answer:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Include values of x and y from the table, for each and next point:

To get the y-intercept (“b”) of the line, we need to include values of x and y in the equation:
y = mx + b
Include values of x and y from any point
1 = 3 . 0 + b
1 = 0 + b
b = 1

Question 6.

Answer:
The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Include values of x and y from the table, for each and next point:

To get the y-intercept (“b”) of the line, we need to include values of x and y in the equation:
y = mx + b
Include values of x and y from any point
140 = -4 . 0 + b
140 = 0 + b
b = 140

Essential Question Check-In

Go Math Answer Key Grade 8 Lesson 4.2 Determining Slope and Y-Intercept Question 7.
How can you determine the slope and the y-intercept of a line from a graph?
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.

Texas Go Math Grade 8 Lesson 4.2 Independent Practice Answer Key 

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.

Answer:
It is already written in the task that the relationship is linear, so the slope (rate of change) can be found with only
two points.

The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

The first point is (1, 125) and the second point is (2, 175). Put those in the calculation:
m = \(\frac{175-125}{2-1}\) = \(\frac{50}{1}\) = 50
The standard form of our Linear function is
y = mx – 1- b
We need to get the initial value of this function.
Take a look at the equation, compare the data we already have, and find what we need to get. Thus, the y-intercept (”b”) is unknown.

Include the values of x, y, and the slope in the equation and calculate:
y = mx + b
125 = 50 . 1 + b
125 = 50 + b
b = 125 – 50
b = 75

Lesson 4.2 Slope of a Line Answer Key Go Math 8th Grade Pdf Question 9.
Make Predictions The total cost to pay for parking at a state park for the day and rent a paddleboat are shown.

a. Find the cost to park for a day and the hourly rate to rent a paddleboat.
Answer:
Slope = \(\frac{29-17}{2-1}\) = \(\frac{12}{1}\) = 12 Finding the slope using any two given points by Slope(m) = (y₂ – y₂) ÷ (x₂ – x₁)
where
(x₂, y₂) = (2, 29) and (x₁, y₁) = (1, 17)
The hourly rent is $\$12$ per hr

Work backward from x = 1 to x = 0 Find the initial value when the value of x is 0
\(\frac{29-17}{2-1}\) = \(\frac{12}{1}\)
x = 1 1 = 0 Subtract the difference of x and y from the first point.
y = 17 – 12 = 5
The cost to park for a day is $\$5$

b. What will Lin pay if she rents a paddleboat for 3.5 hours and splits the total cost with a friend? Explain.
Answer:
Total Cost = 3.5(12) + 5 = 47 When Lin paddles for 3.5 hr
Lin’s Cost = \(\frac{47}{2}\) = 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.

a. Find the rate of change and the initial value for the group lessons.
Answer:
Slope = \(\frac{85-55}{2-1}\) = \(\frac{30}{1}\) = 30 Finding the slope using any two given points by
slope (m) = (y₂ – y₁) ÷ (x₂ – x₁)
Rate of change is $\$30$ per lesson
where (x₂, y₂) = (2, 85) and (x₁, y₁) = (1, 55)
Work backward from x = 1 to x = 0 Find the initial value when the value of x is 0
\(\frac{85-55}{2-1}\) = \(\frac{30}{1}\)
x = 1 – 1 = 0 Subtract the difference of x and y from the first point
y = 55 – 30 = 25
The initial value of group lessons is $\$25$

b. Find the rate of change and the initial value for the private lessons.
Answer:
Slope = \(\frac{125-75}{2-1}\) = \(\frac{50}{1}\) = 50 Finding the slope using any two given points by
slope (m) = (y₂ – y₁) ÷ (x₂ – x₁)
Rate of change is $\$50$ per lesson
where (x₂, y₂) = (2, 125) and (x₁, y₁) = (1, 75)
Work backward from x = 1 to x = 0 Find the initial value when the value of x is 0
\(\frac{125-75}{2-1}\) = \(\frac{50}{1}\)
x = 1 – 1 = 0 Subtract the difference of x and y from the first point
y = 75 – 50 = 25
The initial value of group lessons is $\$25$

c. Compare and contrast the rates of change and the initial values.
Answer:
The initial value for both types of lessons is the same. Comparing results from parts a and b.
The rate of change is higher for private lessons than group lessons.

Vocabulary Explain why each relationship is not linear.

Lesson 4.2 Determining Slope and Y-Intercept Reteach Answer Key Question 11.

Answer:
\(\frac{6.5-4.5}{2-1}\) Finding the rate of change using given points by
the slope (m) = (y₂ – y₁) ÷ (x₂ – x₁)
\(\frac{8.5-6.5}{3-2}\) = 2
\(\frac{11.5-8.5}{4-3}\) = 3
The rate of change is not constant, hence the relationship is not linear

Question 12.

Answer:
Calculate the slope for each and next point.

Question 13.
Communicate Mathematical Ideas Describe the procedure you performed to derive the slope-intercept form of a linear equation. (Explore Activity 2)
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.

Texas Go Math Grade 8 Lesson 4.2 H.O.T. Focus On Higher Order Thinking Answer Key 

Question 14.
Critique Reasoning Your teacher asked your class to describe a real-world situation in which the 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?
Answer:
a. If the brother loses 5 blocks every month, the slope would be -5 and not 5.
When the initial value is decreasing, the slope is negative.

b. Describe a real-world situation that does match the situation.
Answer:
I bought a 1oo card pack and bought 5 additional cards every month. Real-world situation

8th Grade Go Math Answer Key Practice and Homework Lesson 4.2 Question 15.
Justify Reasoning John has a job parking cars. 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 from his fixed salary? Justify your answer.

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.
\(\frac{300}{5}\) = 60
He earns the same ¡n fees as his fixed salary for parking.

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Module 5 Answer Key Writing Linear Equations.

Texas Go Math Grade 8 Module 5 Answer Key Writing Linear Equations

Essential Question
How can you use linear equations to solve real world problems?

Texas Go Math Grade 8 Module 5 Are You Ready? Answer Key

Complete these exercises to review skills you will need for this chapter.

Write each fraction as a decimal.

Question 1.
\(\frac{1}{8}\) ________
Answer:
Given,
\(\frac{1}{8}\)
Convert from fraction to the decimal.
= 1 ÷ 8
= 0.125

Texas Go Math Grade 8 Answer Key Module 5 Answer Key Question 2.
\(\frac{0.3}{0.5}\) ________
Answer:
Given,
\(\frac{0.3}{0.5}\)
Convert from fraction to the decimal.
\(\frac{3}{10}\) ÷ \(\frac{5}{10}\) = \(\frac{3}{5}\) = 0.6

Question 3.
\(\frac{0.13}{0.8}\) ________
Answer:
Given,
\(\frac{0.13}{0.8}\)
Convert from fraction to decimal.
\(\frac{13}{10}\) ÷ \(\frac{8}{10}\) = \(\frac{13}{8}\) = 1.625

Question 4.
\(\frac{0.39}{0.52}\) ________
Answer:
Given,
\(\frac{0.39}{0.52}\)
Convert from fraction to the decimal.
\(\frac{39}{100}\) ÷ \(\frac{52}{100}\) = \(\frac{39}{52}\) = 0.75

Solve each equation using the inverse operation.

Question 5.
7p = 28 __________________
Answer:
7p = 28 p is multiplied by 7
\(\frac{7 p}{7}\) = \(\frac{28}{7}\) To solve the equation, use the inverse operation, division.
p = 4

Question 6.
h – 13 = 5 ________
Answer:
h – 3 = 13 is subtracted from h
h – 13 + 13 = 5 + 13 To solve the equation, use the inverse operation, addition.
h = 18

Go Math Grade 8 Module 5 Answer Key Question 7.
\(\frac{y}{3}\) = -6 ______
Answer:
\(\frac{y}{3}\) = -6 y is divided by 3
\(\frac{y}{3}\) * 3 = -6 * 3 To solve the equation, use the inverse operation, multiplication.
y = -18

Question 8.
b + 9 = 21 _________
Answer:
b + 9 = 21 9 is added to b
b + 9 – 9 = To solve the equation, use the inverse operation, subtraction.
b = 12

Question 9.
c – 8 = -8 ________________
Answer:
c – 8 = 8 8 is subtracted from c
c – 8 + 8 = -8 + 8 To solve the equation, use the inverse operation, addition
c = 0

Question 10.
3n = -12 _______
Answer:
3n = -12
n is multiplied by 3
\(\frac{3 n}{3}\) = \(\frac{-12}{3}\) To solve the equation, use the inverse operation, division.
n = -4

Question 11.
-16 = m + 7 _______
Answer:
-16 = m + 7 7 is added to m

-16 – 7 = m + 7 To solve the equation, use the inverse operation, subtraction.
m = -23

Grade 8 Math Module 5 Answer Key Question 12.
\(\frac{t}{-5}\) = -5 ____
Answer:
\(\frac{t}{-5}\) = -5
As we can see t is divided by (-5), so to solve the equation, we use the inverse operation: muLtiplication.
\(\frac{t}{-5}\) . (-5) = (-5) . (-5)
t = 25

Texas Go Math Grade 8 Module 5 Reading Start-Up Answer Key

Visualize Vocabulary

Use the ✓ words to complete the diagram. You can put more than one word in each bubble.

Understand Vocabulary

Complete the sentences using the preview words.

Question 1.
A set of data that is made up of two paired variables is ____
Answer:
A set of data that is made up of two paired variables is Bivariate data
Bivariate data is defined as a set of data that is made up of two o paired variables. Each value of one of the variables is paired with a value of the other variable.
By definition, a set of data that is made up of two variables is bivariate data.
Thus, the blank part of the statement is bivariate data.

Texas Go Math Grade 8 Answer Key Pdf Module 5 Question 2.
When the rate of change varies from point to point, the relationship is a ____.
Answer:
When the rate of change varies from point to point, the relationship is a nonlinear
A nonlinear relationship is a type of relationship between two points in which change in one point does not correspond with change in another point. It is shown when the rate of change varies between pairs of points.
By description, when the rate of change varies from point to point the relationship is nonlinear.
Thus, the blank part of the statement is nonlinear.

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Module 5 Quiz Answer Key.

Texas Go Math Grade 8 Module 5 Quiz Answer Key

Texas Go Math Grade 8 Module 5 Ready to Go On? Answer Key

5.1 Writing Linear Equations from Situations and Graphs

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

Question 1.

Answer:
The standard form of an equation for the linear function is y = mx + b. To write an equation in slope-intercept form, we need to get values for the slope and y-intercept.

We can read the starting point from the graph, which is the intersection on the y-axis when x = 0. So we can read what our value is for the y-intercept. → starting point is (0, b)
Thus, b = 20.

The slope is calculated by:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Choose two points from the graph and put them in thecalculation I’ll take (0, 20) for (x₁, y₁) and (2, 80) for (x₂, y₂)
m = \(\frac{80-20}{2-0}\)
m = \(\frac{60}{2}\)
m = 30
The equation in slope-intercept form is: y = 30x + 20

Module 5 Test Answers Texas Go Math Grade 8 Pdf Question 2.

Answer:
The standard form of an equation for the linear function is y = mx + b. To write an equation in slope-intercept form, we need to get values for the slope and y-intercept.

We can read the starting point from the graph, which is the intersection on the y-axis when x = 0. So we can read what our value is for the y-intercept. → starting point is (0, b)
Thus, b = 60

The slope is calculated by:
\(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

Choose two points from the graph and put them in the calculation (0, 60) for (x₁, y₁) and (6, 0) for (x₂, y₂)
m = \(\frac{0-60}{6-0}\)
m = \(\frac{-60}{6}\)
m = -10
The equation in slope-intercept form is: y = -10x + 60

5.2 Writing Linear Equations from a Table

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

Question 3.

Answer:
The standard form of an equation for the linear function is y = mx + b. To write an equation in slope-intercept form, we need to get values for the slope and y-intercept.

We can read the starting point from the graph, which is the intersection on the y-axis when x = 0. So we can read what our value is for the y-intercept. → starting point is (0, b)
Thus, b = 1.5

The slope is calculated by:
\(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

Choose two points from the graph and put them in the calculation (0, 1.5) for (x₁, y₁) and (100, 36.5) for (x₂, y₂)
m = \(\frac{36.5-1.5}{100-0}\)
m = \(\frac{35}{100}\)
m = 0.35
The equation in slope-intercept form is: y = 0.35x + 1.5

Grade 8 Texas Go Math Module 5 Answer Key Question 4.

Answer:
The standard form of an equation for the linear function is y = mx + b. To write an equation in slope-intercept form, we need to get values for the slope and y-intercept.

The slope is calculated by:
\(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

Choose two points from the graph and put them in the calculation (25, 94) for (x₁, y₁) and (35, 88) for (x₂, y₂)
m = \(\frac{88-94}{35-25}\)
m = \(\frac{-6}{10}\)
m = -0.6
To find the y-intercept (b), we can include one point from the table and the value of the slope in the standard form of the equation. So:
y = mx + b
94 = -0.6 . 25 + b
94 = -15 + b
b = 94 + 15
b = 109

The equation in slope-intercept form is: 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.

Answer:
The standard form of an equation for the linear function is y = mx + b. To write an equation in slope-intercept form, we need to get values for the slope and y-intercept.

The slope is calculated by:
\(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

Choose two points from the graph and put them in the calculation (20, 40) for (x₁, y₁) and (50, 60) for (x₂, y₂)
m = \(\frac{60-40}{50-20}\)
m = \(\frac{20}{30}\)
m = \(\frac{2}{3}\)
m ≈ 0.67

To find the y-intercept (b), we can include one point from the table and the value of the slope in the standard form of the equation. So:
y = mx + b
40 = 0.67 . 20 + b
40 = 13.4 + b
b = 40 – 13.4
b ≈ 26.6
The equation in slope-intercept form is: y = 0.67x + 26.6

Question 6.

Answer:

Confirm the points represent a linear relationship

Slope = \(\frac{50-65}{30-25}\) = \(\frac{-15}{5}\) = -3
Find the slope using given points from the table by Slope(m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where
(x₂, y₂) = (30, 50) and (x₁, y₁) = (25, 65)
50 = -3(30) + b Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b):
50 = -90 + b y = mx + b
b = 50 + 90 = 140
y = -3x + 140 Substitute the value of slope (m) and y-intercept (b) in slope-intercept form: y = mx + b

Essential Question

Go Math 8th Grade Pdf Module 5 Linear Functions Answer Key Question 7.
Write a real-world situation that can be represented by a linear relationship.
Answer:
A linear equation uses one or more variables where one variable is independent of the other. The degree of the linear equation is one.
Imagine that you are taking a cab while on vacation. You know that the cab service charges $10 to pick your family up from your hotel and another $0.20 per mile for the trip. Without knowing how many miles it will be to each destination, you can set up a linear equation that can be used to find the cost of any taxi trip you take on your trip. By using “x” to represent the number of miles to your destination and “y” to represent the cost of that taxi ride, the linear equation would be y = 0.20x + 10.

Texas Go Math Grade 8 Module 5 Mixed Review Texas Test Prep Answer Key

Selected Response

Question 1.
An hourglass is turned over with the top part filled with sand. After 3 minutes, there are 855 mm of sand in the top half. After 10 minutes, there are 750 mm of sand in the top half. Which equation represents this situation?
(A) y = 285x
(B) y = -10.5x + 900
(C) y = -15x + 900
(D) y = 75x
Answer:
(C) y = -15x + 900

Explanation:
(3, 855) Write the given information as ordered pair

(10, 750)

Slope = \(\frac{855-750}{3-10}\) = \(\frac{105}{-7}\) = -15 Find the slope using given points by Slope (m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

750 = -15(10) + b Substituting the value of slope (m) and (x, y) in the
750 = -15(10) + b slope intercept form to find y-intercept (b):
750= -150 + b
b = 750 + 150 = 900y = mx + b
y = -15x + 900 Substitute the value of slope (m) and y-intercept (b) in slope-intercept form:
y = mx + b

Question 2.
Which graph shows a linear relationship?

Answer:

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

8th Grade Math Module 5 Writing Linear Equations Module Quiz Answer Key Question 3.
What are the slope and y-intercept of the relationship shown in the table?

(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:
slope = \(\frac{3000-2500}{20000-10000}\) = \(\frac{500}{10000}\) = 0.05
Find the slope using given points from the table by Slope(m) = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) where (x₂, y₂) = (20000, 3000) and (x₁, y₁) = (10000, 2500)

3000 = 0.05(2000) + b Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b):
3000 = 1000 + b y = mx + b
b = 3000 – 1000 = 2000

Gridded Response

Question 4.
Franklin’s faucet was leaking water, so he put a bucket underneath to catch the water. After a while, Franklin started keeping track of how much water was leaking. His data is in the table below.

Predict how many gallons of water will have leaked if Franklin hasn’t stopped the leak after 14 hours.

10
Answer:
9.5 – 8 = 1.5
So add 1.5 and it will show you what you get in 14 hours.
In 14 hours if Franklin doesn’t stop the leak it will be 23 quarts

Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 5.1 Answer Key Writing Linear Equations from Situations and Graphs.

Texas Go Math Grade 8 Lesson 5.1 Answer Key Writing Linear Equations from Situations and Graphs

Essential Question
How do you write an equation to model a linear relationship given a graph or a description?

Texas Go Math Grade 8 Lesson 5.1 Explore Activity Answer Key 

Explore Activity
Writing an Equation in Slope-intercept Form

Greta makes clay mugs and bowls as gifts at the Crafty Studio. She pays a membership fee of $15 a month and an equipment fee of $3.00 an hour to use the potter’s wheel, table, and kiln.

Write an equation in the form y = mx + b that Greta can use to calculate her monthly costs.

A. What is the input variable, x, for this situation?
What is the output variable, y for this situation?

B. During April, Greta does not use the equipment at all.
What will be her number of hours (x) for April? ______
What will be her cost (y) for April? ______
What will be the y-intercept, b, in the equation? _____

C. Greta spends 8 hours in May for a cost of $15 + 8($3) = ____
In June, she spends 11 hours for a cost of _________________
From May to June, the change in x-values is _______________
From May to June, the change in y-values is _______________
What will be the slope, m, in the equation? ______

D. Use the values for m and b to write an equation for Greta’s costs in the form y = mx + b: __________________

Reflect

Question 1.
What does the value of the slope represent in this context?
Answer:
The slope represents the rental for each DVD borrowed.

Go Math Grade 8 Lesson 5.1 Answer Key Question 2.
Describe the meaning of the y-intercept.
Answer:
The y-intercept represents the one-time membership fee.

Your Turn

Question 3.
The cash register subtracts $2.50 from a $25 Coffee Café gift card for every medium coffee the customer buys. Use the graph to write an equation in slope-intercept form to represent this situation.

Answer:
slope = \(\frac{0-25}{10-0}\) = \(\frac{-25}{10}\) = -2.5 Finding the slope using given points by Slope(m) = (y₂ – y₁) ÷ (x₂ – x₁) where (x₂, y₂) = (0, 25) and (x₁, y₁) = (10, 0)
y-intercept (b) = 25 From the graph
y = -2.5x + 25 Substituting the value of slope (m) and y-intercept in slope-intercept form:
y = mx + b
y = -2.5x + 25

Reflect

Question 4.
Without graphing, tell whether the graph of this equation rises or falls from left to right. What does the sign of the slope mean in this context?
Answer:
Whether the graph is ascending or descending depends on the sign of the slope. If the slope is positive then the graph is increasing, because we get the second point by counting upwards from the starting point (y-intercept, b) for the number written in the slope numerator. If the slope is negative then the graph is descending, because we get the second point by counting down from the starting point (y-intercept, b) for the number written in the slope numerator.

Your Turn

Lesson 5.1 Writing Linear Equations from Situations and Graphs Question 5.
Hari’s weekly allowance varies depending on the number of chores he does. He received $16 in allowance the week he did 12 chores, and $14 in allowance the week he did 8 chores. Write an equation for his allowance in slope-intercept form. _______
Answer:
Input variables: Number of chores
Output variables Weekly allowance Identify the input and output variables.
(12, 16) Write the information given in the problem as ordered pairs.
(8, 14)
Slope = \(\frac{16-14}{12-8}\) = \(\frac{2}{4}\) = \(\frac{1}{2}\) Finding the slope using given points by slope (m) = (y₂ – y₁) ÷ (x₂ – x₁)

14 = \(\frac{1}{2}\)(8) + b Substituting the value of slope (m) and (x, y) in the slope-intercept form to find y-intercept (b):
14 = 4 + b y = mx + b
b = 14 – 4 = 10
y = \(\frac{1}{2}\)x + 10 Substituting the value of slope (m) and y-intercept in slope-intercept form.

Texas Go Math Grade 8 Lesson 5.1 Guided Practice Answer Key 

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. (Explore Activity)
a. input variable: _____________________________________________
Answer:
Input variable: The length of the necklace in inches.

b. output variable: _____________________________________
Answer:
Output variable: The total number of beads in the necklace.

c. equation: ______________
Answer:
Equation:
y = 5x + 27

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. (Example 1)

Choose two points on the graph to find the slope.
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) = _______
Read the y-intercept from the graph: b = ______________
Use your slope and y-intercept values to write an equation in slope-intercept form. _________________________
Answer:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) = \(\frac{0-300}{5-0}\) = \(\frac{-300}{5}\) = -60
Finding the slope using given points (x₂, y₂) = (0, 300) and (x₁, y₁) = (5, 0)
y-intercept (b) = 300 Read from the graph (when x = 0)
y = -60x + 300 Substituting the value of slope (m) and y-intercept in slope-intercept form:
y = mx + b

Texas Go Math Grade 8 Writing Linear Equations from Graphs 5.1 Answer Key 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. (Example 2)
Input variable: _____________________
Output variable: __________________________
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) = __________
Substitute in y = mx + b: _______________ + b; __________ = b
Write an equation in slope-intercept form. ___________________
Answer:
To determine the equation in slope intercept form y = mx + b that represents the situation At 59° F crickets chirp at a rate of 76 times per minute, and at 65° F, they chirp 100 times per minute, let us follow the following steps.

First, let us identify the input and output variables. The input variable x is the temperature while the output variable y is the number of chirps per minute.

Second, let us write the ordered pairs.
at 59 °F, crickets chirp at a rate of 76 times per minute:(59, 76)
at 65 °F, crickets chirp at a rate of 100 times per minute:(65, 100)
x₁ = 59°
y₁ = 76
x₂ = 65°
y₂ = 100

Third, let us determine the slope in using the formula
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Substituting the values of x₁, y₁, x₂ and y₂.
m = \(\frac{100-76}{65-59}\)
m = \(\frac{24}{6}\)
m = 4

Fourth, let us find the value of the y-intercept b. Using the variables and the slope 4, we can have two equations
for us to find b, these are
76 = 4 • 59 + b
and
100 = 4 • 65 + b
We can use any of these two to determine the value of b, that is
100 = 4 • 65
100 = 260 + b
b = 100 – 260
b = -160

Now, let us form the equation in slope-intercept form, having the slope m = 4 and y-intercept b = -160, that is
y = mx + b
y = 4x – 160
see the explanation

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.
Answer:
m indicates the steepness of the line and whether the graph is rising or falling (left from right). b indicates the point where the line crosses the y-axis.

Texas Go Math Grade 8 Lesson 5.1 Independent Practice Answer Key 

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.
Answer:
Input variable: Flying time in seconds Identify the input and output variables.
Output variable: Number of times the dragonfly beats its wing
y = 30x Substituting the value of slope (m) and y-intercept in slope-intercept form.
where m = 30 times per second and b = 0

Go Math 8th Grade Pdf Lesson 5.1 Equations and Their Graphs 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.
Answer:
The problem is asking for the equation that tells the height of the balloon above the ground after a given number of seconds in slope-intercept form.

The statement that the balloon is released from the top of a platform that is 50 meters tall and rises at the rate of 4 meters per second
Let
y be the height of the balloon above the ground, x be the number of seconds, and 4s be the rate of the balloon as it rises per second.

Using the representation and the given, we should equate the height of the balloon above the ground y to the rate of the balloon as it rises per second 4x added to 50 which is the height of the platform, which gives us the equation,
y = 4x + 50

The graph shows a scuba diver’s ascent over time.

Question 7.
Use the graph to find the slope of the line. Tell what the slope means in this context.
______________________
Answer:
First, we choose two points on the graph to find the slope. We substitute (0, -10) for (x₁, y₁) and (80, 0) for (x₂, y₂)

Therefore, the slope equals to 0.125 means that the scuba driver ascends 0.125 meters each second.

Question 8.
Identify the y-intercept. Tell what the y-intercept means in this context.
Answer:
The starting point on the graph is (0, -10), i.e. (0, b). So the value of the y-intercept is -10. Reading from the graph, the y-intercept is the value for the depth (in meters) from which the scuba diver started to swim to the surface.

Question 9.
Write an equation in slope-intercept form that represents the diver’s depth over time.
Answer:
From the given graph, find two points that will help you calculate the slope.
I’ll take (0, -10) for (x₁, y₁) and (80, 0) for (x₂, y₂).
Thus:

The starting point on the graph is (0, -10), ie. (0, b). So the value of the y-intercept is -10. → b = -10

The standard form of the linear equation is y = mx + b, so include the obtained values of the slope and y-intercept.
y = \(\frac{1}{8}\)x – 10

Texas Go Math Grade 8 Pdf Lesson 5.1 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.
Answer:
To determine the equation in slope intercept form y = mx + b that converts degrees Celsius into degrees. Fahrenheit, let us follow the following steps by finding the slope and y-intercept form given that the water freezes at 0°C or 32°F and it boils at 100°C or 212°F

First, let us identify the input and output variables. The input variable x is the temperature in degrees Celsius while the output variable y is the temperature in degrees Fahrenheit

Second, let us write the ordered pairs.
water freezes at 0°C or 32°F:(0, 32)
water boils at 100°C or 212°F:(100, 212)
x₁ = 0
y₁ = 32
x₂ = 100
y₂ = 212

Third, let us determine the slope ni using the formula
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Substituting the values of x₁, y₁, x₂, and y₂
m = \(\frac{212-32}{100-0}\)
m = \(\frac{180}{100}\)
m = 1.8

Fourth, let us find the value of the y-intercept b. Using the variables and the slope 4, we can have two equations for us to find b, these are
32 = 1.8 . 0 + b
and
212 = 1.8 . 100 + b
We can use any of these two to determine the value of b, that is
212 = 1.8 . 100 + b
212 = 180 + b
b = 212 – 180
b = 32
Now, let us form the equation in slope-intercept form, having the slope m = 1.8 and y-intercept b = 32, that is
y = mx + b
y = 1.8x + 32

See the explanation.

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.

Answer:
To determine the equation in slope-intercept form to calculate the total amount you would pay for using a sailboat given that the cost of renting a sailboat at a lake is $20 per hour plus $12 for life jackets, let us represent the different variables in the problem.

Let y be the total cost you would pay for using a sailboat, x be the number of hours you would use the sailboat, $20 which is the cost of renting a sailboat per hour be the slope, and $12 is the cost of lifejackets be the y-intercept.

Now using the slope-intercept form, we can have the equation by substituting the value of where m = 20 and b = 12, that is
y = mx + b
y = 20x + 12

The graph shows the activity in a savings account.

Question 12.
What was the amount of the initial deposit that started this savings account?
Answer:
Reading from the graph, the y-intercept is the starting point: (0, 1000), so the value for the y-intercept is 1000. Thus,
b = 1000$.
So that value is the amount of the initial deposit that he started with his savings account.

Lesson 5.1 Answer Key 8th Grade Writing Linear Equations Question 13.
Find the slope and y-intercept of the graphed line.
Answer:
From the given graph, find two points that will help you calculate the slope.
I’ll take (0, 1000) for (x₁, y₁) and (6, 4000) for (x₂, y₂).
Thus:
m = \(\frac{4000-1000}{6-0}\) = \(\frac{3000}{6}\) = 500
The starting point on the graph is (0, 1000), i.e. (0, b). So value of the y-intercept is 1000. → b = 1000

Question 14.
Write an equation in slope-intercept form for the activity in this savings account.
Answer:
The standard form of the linear equation is y = mx + b. In previous exercise we got values for the slope and y intercept → m = 500, b = 1000. Include the obtained values of the slope and y-intercept into the equation.
y = 500x + 1000

Let us describe the equation: y represents the total amount of saved money, x represents the time of collecting money (in months), the slope represents the money saved during x months and y-intercept represents the amount with which he started to save.

Question 15.
Explain the meaning of the slope in this graph.
Answer:
The slope represents the amount of money saved in dollars per month in plan.

Texas Go Math Grade 8 Lesson 5.1 H.O.T. Higher Order Thinking Answer Key 

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.
Answer:
Explanation A:
You need to analyze the problem to find out which part depends on another. That dependent part will be represented by variable y, and the independent part by x.
You need to analyze the problem to find out which part depends on another.

Explanation B:
Variable x is the input, which represents the variable that affects the other variable when changed and is independent.
Variable y is the output, which represents the resulting variable or the variable that is dependent on the other.

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

Lesson 5.1 Writing Linear Equations from Situations and Graphs Answer Key Question 18.
Draw Conclusions Must m, in the equation y = mx + b, always be a positive number? Explain.
Answer:
In the equation y = mx + b, the variable m represents the slope of a line which is defined as the number that describes the steepness and direction of the line.

The slope of a line can be positive, negative, zero, or undefined.

• If the slope is upward to the right, then it is a positive slope.
• If the slope is downward to the right, then it is a negative slope.
• If the slope is horizontal, then the slope is zero.
• If the slope is vertical, then the slope is undefined.

We can conclude that m which is the slope of a line is not always a positive number as stated by the definition of the slope it can be a positive, negative, zero, or undefined.