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## Go Math Grade 8 Chapter 7 Solving Linear Equations Answer Key

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**Lesson 1: Equations with the Variable on Both Sides**

- · Equations with the Variable on Both Sides – Page No. 200
- · Equations with the Variable on Both Sides – Page No. 201
- · Equations with the Variable on Both Sides Lesson Check – Page No. 202

**Lesson 2: Equations with Rational Numbers**

- · Equations with Rational Numbers – Page No. 206
- · Equations with Rational Numbers – Page No. 207
- · Equations with Rational Numbers Lesson Check – Page No. 208

**Lesson 3: Equations with the Distributive Property**

- · Equations with the Distributive Property – Page No. 212
- · Equations with the Distributive Property – Page No. 213
- · Equations with the Distributive Property Lesson Check – Page No. 214

**Lesson 4: Equations with Many Solutions or No Solution**

- · Equations with Many Solutions or No Solution – Page No. 218
- · Equations with Many Solutions or No Solution – Page No. 219
- · Equations with Many Solutions or No Solution Lesson Check – Page No. 220

**Lesson 5: Equations with the Variable on Both Sides**

**Reviews**

### Guided Practice – Equations with the Variable on Both Sides – Page No. 200

**Use algebra tiles to model and solve each equation.**

Question 1.

x + 4 = -x – 4

x = ________

Answer:

x = -4

Explanation:

Model x + 4 on the left side of the mat and -x -4 on the right side.

Add one c-tile to both sides. This represents adding x to both sides of the equation. Remove zero pairs.

Place four -1-tiles on both sides. This represents subtracting -4 from both sides of the equation. Remove zero pairs.

Separate each side into 2 equal groups. One x-tile is equivalent to four -1-tiles.

x = -4

Question 2.

2 – 3x = -x – 8

x = ________

Answer:

Explanation:

Given 2 – 3x = -x – 8

Model 2-3x on the left side of the mat and -x-8 on the right side.

Place one x tile to both sides. This represents subtracting from both sides of the equation.

Remove 2 1 tiles from sides. This represents subtracting from both sides of the equation.

Separate each side into 2 equal groups. One -x tile is equivalent to 5 – 1 tile.

The solution is -x = -5 or x = 5

**Solving Linear Equations Grade 8 Question 3.**

At Silver Gym, membership is $25 per month, and personal training sessions are $30 each. At Fit Factor, membership is $65 per month, and personal training sessions are $20 each. In one month, how many personal training sessions would Sarah have to buy to make the total cost at the two gyms equal?

________ sessions

Answer:

4 sessions

Explanation:

At Silver Gym, membership is $25 per month, and personal training sessions are $30 each.

Membership + Personal training session = 25 + 30x

At Fit Factor, membership is $65 per month, and personal training sessions are $20 each.

Membership + Personal training session = 65 + 20x

Membership at Silver Gym = Membership at Fit Factor

25 + 30x = 65 + 20x

30x – 20x = 65 – 25

10x = 40

x = 4

Sarah would have to buy 4 sessions for the total cost at the two gyms to be equal.

Question 4.

Write a real-world situation that could be modeled by the equation 120 + 25x = 45x.

Type below:

_______________

Answer:

120 + 25x = 45x

Sarah offers a plan to tutor a student at $25 per her plus a one-time registration fee of $ 120.

Surah offers an alternative plan to tutor a student at $45 per hour and no registration fee.

120 + 25x = 45x

Question 5.

Write a real-world situation that could be modeled by the equation 100 – 6x = 160 – 10x.

Type below:

_______________

Answer:

100 – 6x = 160 – 10x

The initial water in Tank A is 100 gallons and leaks at 6 gallons per week.

The initial water in Tank B is 160 gallon and leaks at 10 gallons per week

100 – 6x = 160 – 10x

**Essential Question Check-In**

Question 6.

How can you solve an equation with the variable on both sides?

Type below:

_______________

Answer:

Isolate the variable on one side. Add/subtract the variable with a lower coefficient from both sides. Add/subtract the constant (with the variable) from both sides. Divide both sides by coefficient of the isolated variable.

### Independent Practice – Equations with the Variable on Both Sides – Page No. 201

Question 7.

Derrick’s Dog Sitting and Darlene’s Dog Sitting are competing for new business. The companies ran the ads shown.

a. Write and solve an equation to find the number of hours for which the total cost will be the same for the two services.

________ hours

Answer:

3 hours

Explanation:

Hourly rate + One time fee = 5x + 12

Hourly rate + One time fee = 3x + 18

5x + 12 = 3x + 18

5x – 3x = 18 – 12

2x = 6

x = 3

The cost of the two dog sitting would be the same for 3 hrs.

Question 7.

b. Analyze Relationships

Which dog-sitting service is more economical to use if you need 5 hours of service? Explain.

____________

Answer:

Darlene’s Dog Sitting would be cheaper

Explanation:

Let y be the cost of dog sitting after x hours for both companies

y = 5x +12

y = 3x +18

Substitute x = 5

y = 5(5) + 12 = 37

y = 3 (5) + 18 = 33

Compare the cost for both companies for x = 5hr.

$37 > $33

Darlene’s Dog Sitting would be cheaper

**Linear Equations 8th Grade Question 8.**

Country Carpets charges $22 per square yard for carpeting and an additional installation fee of $100. City Carpets charges $25 per square yard for the same carpeting and an additional installation fee of $70.

a. Write and solve an equation to find the number of square yards of carpeting for which the total cost charged by the two companies will be the same.

_______ square yards

Answer:

10 square yards

Explanation:

Unit square rate + One time installation fee = 22x + 100

Unit square rate + One time installation fee = 25x + 70

22x + 100 = 25x + 70

25x – 22x = 100 – 70

3x = 30

x = 10

The total cost charged by the two companies will be the same for 10 square yards of carpeting.

Question 8.

b. Justify Reasoning

Mr. Shu wants to hire one of the two carpet companies to install carpeting in his basement. Is he more likely to hire Country Carpets or City Carpets? Explain your reasoning.

___________

Answer:

City Carpets are cheaper when x < 10

y = 25(9) + 70 = 295

y = 22(9) + 100 = 298

Country Carpets are cheaper when x > 10

y = 25(11) + 70 = 345

y = 25(11) + 100 = 342

If Mr.Shu needs the carpeting done for less than 10 square yards, he will hire City Carpets and if he needs carpeting for more than 10 square yards, he will hire Country Carpets.

**Write an equation to represent each relationship. Then solve the equation.**

Question 9.

Two less than 3 times a number is the same as the number plus 10.

________

Answer:

3x – 2 = x + 10

x = 6

Explanation:

Two less than 3 times a number is the same as the number plus 10.

Two less than 3 times x is the same as the x plus 10.

Two less than 3x is the same as the x + 10

3x – 2 is the same as x + 10

3x – 2 = x + 10

3x – x = 10 + 2

2x = 12

x = 6

Question 10.

A number increased by 4 is the same as 19 minus 2 times the number.

______

Answer:

x + 4 = 19 – 2x

x = 5

Explanation:

A number increased by 4 is the same as 19 minus 2 times the number.

x increased by 4 is the same as 19 minus 2x.

x + 4 is the same as 19 – 2x

x + 4 = 19 – 2x

x + 2x = 19 – 4

3x = 15

x = 15/3

x = 5

Question 11.

Twenty less than 8 times a number is the same as 15 more than the number.

Type below:

____________

Answer:

8x – 20 = x + 15

x = 5

Explanation:

Twenty less than 8 times a number is the same as 15 more than the number.

Twenty less than 8 times x is the same as 15 more than the x.

Twenty less than 8x is the same as 15 more than the x

8x – 20 is the same as x + 15

8x – 20 = x + 15

8x – x = 15 + 20

7x = 35

x = 35/7 = 5

x = 5

### Equations with the Variable on Both Sides – Page No. 202

Question 12.

The charges for an international call made using the calling card for two phone companies are shown in the table.

a. What is the length of a phone call that would cost the same no matter which company is used?

_______ minutes

Answer:

10 minutes

Explanation:

Cost of minutes + One time fee = 3x + 35

Cost of minutes + One time fee = 2x + 45

3x + 35 = 2x + 45

3x – 2x = 45 – 35

x = 10

The cost would be same for 10 minutes.

Question 12.

b. Analyze Relationships

When is it better to use the card from Company B?

Type below:

____________

Answer:

y = 3x + 35

y = 3(11) + 35 = $68

y = 2x + 45

y = 2(11) + 45 = $67

Since when x > 10, Company B is cheaper so it should be better to use when the length of the call is greater than 10 minutes.

**H.O.T.**

**Focus on Higher Order Thinking**

Question 13.

Draw Conclusions

Liam is setting up folding chairs for a meeting. If he arranges the chairs in 9 rows of the same length, he has 3 chairs left over. If he arranges the chairs in 7 rows of that same length, he has 19 left over. How many chairs does Liam have?

______ chairs

Answer:

75 chairs

Explanation:

Number of chairs in 9 row + leftover chairs = 9x + 3

Number of chairs in 7 row + leftover chairs = 7x + 19

9x + 3 = 7x + 19

9x – 7x = 19 – 3

2x = 16

x = 16/2

x = 8

Total number of chairs = 9(8) + 3 = 75

**Solving Linear Equations 8th Grade Question 14.**

Explain the Error

Rent-A-Tent rents party tents for a flat fee of $365 plus $125 a day. Capital Rentals rents party tents for a flat fee of $250 plus $175 a day. Delia wrote the following equation to find the number of days for which the total cost charged by the two companies would be the same:

365x + 125 = 250x + 175

Find and explain the error in Delia’s work. Then write the correct equation.

Type below:

____________

Answer:

Delia’s equation

365x + 125 = 250x + 175

The error is that she attached the variable with the flat fee (which is constant) and put the daily rent as a constant (which is variable).

Correct equation

125x + 365 = 175x + 250

Question 15.

Persevere in Problem-Solving

Lilliana is training for a marathon. She runs the same distance every day for a week. On Monday, Wednesday, and Friday, she runs 3 laps on a running trail and then runs 6 more miles. On Tuesday and Sunday, she runs 5 laps on the trail and then runs 2 more miles. On Saturday, she just runs laps. How many laps does Lilliana run on Saturday?

________ laps

Answer:

2 laps

Explanation:

Miles covered by lap + Addition number of miles = 3x + 6

Miles covered by lap + Addition number of miles = 5x + 2

3x + 6 = 5x + 2

5x – 3x = 6 – 2

2x = 4

x = 4/2

x = 2

### Guided Practice – Equations with Rational Numbers – Page No. 206

Question 1.

Sandy is upgrading her Internet service. Fast Internet charges $60 for installation and $50.45 per month. Quick Internet has free installation but charges $57.95 per month.

a. Write an equation that can be used to find the number of months at which the Internet service would cost the same.

Type below:

____________

Answer:

50.45x + 60 = 57.95x

Explanation:

Write an equation for Fast Internet, where x is the number of months.

Charge per Month × Number of Months + Installation Fee

50.45x + 60

Write an equation for Quick Internet, where x is the number of months.

Charge per Month × Number of Months + Installation Fee

57.95x

50.45x + 60 = 57.95x

Question 1.

b. Solve the equation.

_______ hours

Answer:

8

Explanation:

50.45x + 60 = 57.95x

57.95x – 50.45x = 60

7.5x = 60

x = 60/7.5

x = 8

The total cost will be the same for 8 months.

**Solve.**

Question 2.

\(\frac{3}{4}\) n – 18 = \(\frac{1}{4}\) n – 4

______

Answer:

n = 28

Explanation:

3/4 . n – 18 = 1/4 . n – 4

Determine the least common multiple of the denominators

LCM is 4

Multiply both sides of the equation by the LCM

4(3/4 . n – 18) = 4(1/4 . n – 4)

3n – 72 = n – 16

3n – n = -16 + 72

2n = 56

n = 56/2

n = 28

Question 3.

6 + \(\frac{4}{5}\) b = \(\frac{9}{10}\) b

_______

Answer:

b = 60

Explanation:

6 + \(\frac{4}{5}\) b = \(\frac{9}{10}\) b

LCM is 10

10(6 + \(\frac{4}{5}\) b) = 10(\(\frac{9}{10}\) b)

60 + 8b = 9b

9b – 8b = 60

b = 60

**Solving Linear Equations Worksheets Grade 8 Pdf Question 4.**

\(\frac{2}{11}\) m + 16 = 4 + \(\frac{6}{11}\) m

_______

Answer:

m = 33

Explanation:

\(\frac{2}{11}\) m + 16 = 4 + \(\frac{6}{11}\) m

The LCM is 11

11(\(\frac{2}{11}\) m + 16) = 11(4 + \(\frac{6}{11}\) m)

2m + 176 = 44 + 6m

6m – 2m = 176 – 44

4m = 132

m = 132/4

m = 33

Question 5.

2.25t + 5 = 13.5t + 14

_______

Answer:

t = -0.8

Explanation:

2.25t + 5 = 13.5t + 14

13.5t – 2.25t = 5 – 14

11.25t = -9

t = -9/11.25

t = -0.8

Question 6.

3.6w = 1.6w + 24

_______

Answer:

w = 12

Explanation:

3.6w = 1.6w + 24

3.6w – 1.6w = 24

2w = 24

w = 24/2

w = 12

Question 7.

-0.75p – 2 = 0.25p

_______

Answer:

p = -2

Explanation:

-0.75p – 2 = 0.25p

-2 = 0.25p + 0.75p

-2 = p

p = -2

Question 8.

Write a real-world problem that can be modeled by the equation 1.25x = 0.75x + 50.

Type below:

______________

Answer:

1.25x = 0.75x + 50.

Cell offers Plan A for no base fee and $1.25 per minute.

Cell offers Plan B for a $50 base fee and $0.75 per minute.

The equation shows when the total cost of the plan would be equal.

**Essential Question Check-In**

Question 9.

How does the method for solving equations with fractional or decimal coefficients and constants compare with the method for solving equations with integer coefficients and constants?

Type below:

______________

Answer:

When solving equations with fractional or decimal coefficients, the equations need to be multiplied by the multiple of the denominator such that the equations have integer coefficients and constants.

### Independent Practice – Equations with Rational Numbers – Page No. 207

Question 10.

Members of the Wide Waters Club pay $105 per summer season, plus $9.50 each time they rent a boat. Nonmembers must pay $14.75 each time they rent a boat. How many times would a member and a non-member have to rent a boat in order to pay the same amount?

_______ times

Answer:

20 times

Explanation:

Members of the Wide Waters Club pay $105 per summer season, plus $9.50 each time they rent a boat.

9.5x + $105

Nonmembers must pay $14.75 each time they rent a boat.

9.5x + $105 = 14.75x

9.5x – 14.75x = $105

5.25x = 105

x = 105/5.25

x = 20

The cost for members and non-members will be the same for 8 visits.

Question 11.

Margo can purchase tile at a store for $0.79 per tile and rent a tile saw for $24. At another store, she can borrow the tile saw for free if she buys tiles there for $1.19 per tile. How many tiles must she buy for the cost to be the same at both stores?

_______ tiles

Answer:

60 tiles

Explanation:

Margo can purchase tile at a store for $0.79 per tile and rent a tile saw for $24.

0.79x + 24

At another store, she can borrow the tile saw for free if she buys tiles there for $1.19 per tile.

1.19x

0.79x + 24 = 1.19x

1.19x – 0.79x = 24

0.4x = 24

x = 24/0.4

x = 60

Margo should buy 60 tiles for the cost to be the same at both stores.

Question 12.

The charges for two shuttle services are shown in the table. Find the number of miles for which the cost of both shuttles is the same.

_______ miles

Answer:

40 miles

Explanation:

0.1x + 10

0.35x

0.1x + 10 = 0.35x

0.35x – 0.1x = 10

0.25x = 10

x = 10/0.25

x = 40

The cost of shuttles would be the same for 40 miles.

Question 13.

Multistep

Rapid Rental Car charges a $40 rental fee, $15 for gas, and $0.25 per mile driven. For the same car, Capital Cars charges $45 for rental and gas and $0.35 per mile.

a. For how many miles is the rental cost at both companies the same?

_______ miles

Answer:

100 miles

Explanation:

0.25x + 40 + 15 = 0.35x + 45

0.35x – 0.25x = 55 – 45

0.1x = 10

x = 10/0.1

x = 100

The cost of car rentals would be the same for 100 miles.

Question 13.

b. What is that cost?

$ _______

Answer:

$80

Explanation:

Let y be the total cost. Substitute 100 miles in any one of the two equations

y = 0.35x + 45

y = 0.35(100) + 45 = $80

Total cost would be $80.

Question 14.

Write an equation with the solution x = 20. The equation should have the variable on both sides, a fractional coefficient on the left side, and a fraction anywhere on the right side.

Type below:

______________

Answer:

4/3x + 10 = 50/3 + x

Explanation:

Write an equation with the solution x = 20. The equation should have the variable on both sides, a fractional coefficient on the left side, and a fraction anywhere on the right side.

1/3 . x = 1/3 . 20

1/3 . x + x = 1/3 . 20 + x

4/3x = 20/3 + x

4/3x + 10 = 20/3 + x + 10

4/3x + 10 = 50/3 + x

Question 15.

Write an equation with the solution x = 25. The equation should have the variable on both sides, a decimal coefficient on the left side, and a decimal anywhere on the right side. One of the decimals should be written in tenths, the other in hundredths.

Type below:

______________

Answer:

x=25

divide both sides by 25

x/25 = 1

convert 1/25 to decimal form 0.04

0.04x = 1

add x on both sides

1.04x = 1 + x

add 0.1 on both sides

1.04x + 0.1 = x + 1.1

Question 16.

Geometry

The perimeters of the rectangles shown are equal. What is the perimeter of each rectangle?

Perimeter = _______

Answer:

Perimeter = 3.2

Explanation:

Perimeter of the first rectangle

P = 2(n + n + 0.6) = 2(2n + 0.6) = 4n + 1.2

Perimeter of the second rectangle

P = 2(n + 0.1 + 2n) = 2(3n + 0.1) = 6n + 0.2

the perimeter is equal

4n + 1.2 = 6n + 0.2

6n – 4n = 1.2 – 0.2

2n = 1

n = 1/2

n = 0.5

P = 4n + 1.2 = 4(0.5) + 1.2 = 3.2

Question 17.

Analyze Relationships

The formula F = 1.8C + 32 gives the temperature in degrees Fahrenheit (F) for a given temperature in degrees Celsius (C). There is one temperature for which the number of degrees Fahrenheit is equal to the number of degrees Celsius. Write an equation you can solve to find that temperature and then use it to find the temperature

Type below:

______________

Answer:

x = 1.8x + 32

Explanation:

F = 1.8C +32

let x be the temperature such that it is the same in both Celsius and in Fahrenheit

Then the required equation is

x = 1.8x + 32

subtract 1.8x from both sides

-0.8x = 32

divide by -0.8 on both sides

x = -40

So -40 degree celsius

### Equations with Rational Numbers – Page No. 208

Question 18.

Explain the Error

Agustin solved an equation as shown. What error did Agustin make? What is the correct answer?

x = _______

Answer:

x = -12

Explanation:

Agustin did not multiply by 12 on both sides in step 2. He only partially multiplied the variable and left the constants as such, which doesn’t make any sense.

The correct solution is

12(x/3 – 4) = 12(3x/4 + 1)

4x – 48 = 9x + 12

subtract 12 on both sides

4x – 60 = 9x

subtract 4x on both sides

-60 = 5x

x = -12

**H.O.T.**

**Focus on Higher Order Thinking**

Question 19.

Draw Conclusions

Solve the equation \(\frac{1}{2} x-5+\frac{2}{3} x=\frac{7}{6} x+4\). Explain your results.

Type below:

_____________

Answer:

\(\frac{1}{2} x-5+\frac{2}{3} x=\frac{7}{6} x+4\)

The least common multiple of the denominators: LCM(2, 3, 6) = 6

6(\(\frac{1}{2} x-5+\frac{2}{3} x=\frac{7}{6} x+4\))

6.1/2x – 6.5 + 6.2/3x = 6.7/6x +6.4

3x – 30 + 4x = 7x + 24

7x – 30 = 7x + 24

-30 = 24

This is not true. The equation has no solution.

Question 20.

Look for a Pattern

Describe the pattern in the equation. Then solve the equation.

0.3x + 0.03x + 0.003x + 0.0003x + .. = 3

x = ______

Answer:

x = 9

Explanation:

0.3x + 0.03x + 0.003x + 0.0003x + .. = 3

0.3x = 3

0.9x = 9

x = 9

Question 21.

Critique Reasoning

Jared wanted to find three consecutive even integers whose sum was 4 times the first of those integers. He let k represent the first integer, then wrote and solved this equation : k + (k + 1) + (k + 2) = 4k. Did he get the correct answer? Explain.

__________

Answer:

No, it is wrong on two accounts.

First, he has not specified if k is even or not. An easy way of doing so would assume x to be any integer and k=2a

This ensures that k is an even integer.

Nest the question asks for 3 consecutive even integers, Jared just took 3 consecutive integers, and thus at least 1 of them is odd.

So correct representation would be

k + (k+2) + (k + 4) = 4k

which upon solving yields k=6

### Guided Practice – Equations with the Distributive Property – Page No. 212

**Solve each equation.**

Question 1.

4(x + 8) – 4 = 34 – 2x

________

Answer:

x = 1

Explanation:

4(x + 8) – 4 = 34 – 2x

4x + 32 – 4 = 34 – 2x

4x + 2x = 34 – 28

6x = 6

x = 6/6

x = 1

**7.3 Equations with the Distributive Property Question 2.**

\(\frac{2}{3}\)(9 + x) = -5(4 – x)

________

Answer:

x = 6

Explanation:

\(\frac{2}{3}\)(9 + x) = -5(4 – x)

2/3(9 + x) = -5(4 – x)

3 (2/3(9 + x)) = 3(-5(4 – x))

2(9 + x ) = -15 (4 – x)

18 + 2x = -60 + 15x

15x – 2x = 18 + 60

13x = 78

x = 78/13

x = 6

Question 3.

-3(x + 4) + 15 = 6 – 4x

________

Answer:

x = 3

Explanation:

-3(x + 4) + 15 = 6 – 4x

-3x – 12 + 15 = 6 – 4x

-3x + 3 = 6 – 4x

-3x + 4x = 6 – 3

x = 3

Question 4.

10 + 4x = 5(x – 6) + 33

________

Answer:

x = 7

Explanation:

10 + 4x = 5(x – 6) + 33

10 + 4x = 5x – 30 + 33

10 + 4x = 5x + 3

5x – 4x = 10 – 3

x = 7

Question 5.

x – 9 = 8(2x + 3) – 18

________

Answer:

x = -1

Explanation:

x – 9 = 8(2x + 3) – 18

x – 9 = 16x + 24 – 18

x – 9 = 16x + 6

16x – x = -9 – 6

15x = – 15

x = -15/15

x = -1

Question 6.

-6(x – 1) – 7 = -7x + 2

________

Answer:

x = 3

Explanation:

-6(x – 1) – 7 = -7x + 2

-6x + 6 – 7 = -7x + 2

-6x – 1 = -7x + 2

-7x + 6x = -1 -2

-x = -3

x = 3

Question 7.

\(\frac{1}{10}\)(x + 11) = -2(8 – x)

________

Answer:

x = 9

Explanation:

\(\frac{1}{10}\)(x + 11) = -2(8 – x)

10(\(\frac{1}{10}\)(x + 11)) = 10 (-2(8 – x))

x + 11 = -20(8 – x)

x + 11 = -160 + 20x

20x – x = 11 + 160

19x = 171

x = 171/19 = 9

Question 8.

-(4 – x) = \(\frac{3}{4}\)(x – 6)

________

Answer:

x = -2

Explanation:

-(4 – x) = \(\frac{3}{4}\)(x – 6)

4(-(4 – x)) = 4 (3/4(x – 6))

-16 + 4x = 3x – 18

4x – 3x = -18 + 16

x = -2

Question 9.

-8(8 – x) = \(\frac{4}{5}\)(x + 10)

________

Answer:

x = 10

Explanation:

-8(8 – x) = \(\frac{4}{5}\)(x + 10)

5(-8(8 – x)) = 5(\(\frac{4}{5}\)(x + 10))

-40(8 – x) = 4(x + 10)

-320 + 40x = 4x + 40

40x – 4x = 40 + 320

36x = 360

x = 360/36

x = 10

Question 10.

\(\frac{1}{2}\)(16 – x) = -12(x + 7)

________

Answer:

x = 8

Explanation:

\(\frac{1}{2}\)(16 – x) = -12(x + 7)

2 (\(\frac{1}{2}\)(16 – x)) = 2 (-12(x + 7))

16 – x = -24 (x + 7)

16 – x = -24x – 168

24x – x = -168 – 16

23x = 184

x = 184/23

x = 8

**Lesson 7.3 Equations with the Distributive Property Answer Key Question 11.**

Sandra saves 12% of her salary for retirement. This year her salary was $3,000 more than in the previous year, and she saved $4,200.What was her salary in the previous year?

Write an equation _____

Sandra’s salary in the previous year was _____

Salary = $ _____

Answer:

Write an equation 0.12x + 360 = 4200

Sandra’s salary in the previous year was $32000

Salary = $3000

Explanation:

0.12(x + 3000) = 4200

0.12x + 360 = 4200

0.12x = 4200 – 360

0.12x = 3840

x = 3840/0.12

x = 32000

Sandra’s salary in the previous year was $32000

**Essential Question Check-In**

Question 12.

When solving an equation using the Distributive Property, if the numbers being distributed are fractions, what is your first step? Why?

Type below:

___________

Answer:

Multiply both sides by the denominator of the fraction

### Independent Practice – Equations with the Distributive Property – Page No. 213

Question 13.

Multistep

Martina is currently 14 years older than her cousin Joey. In 5 years she will be 3 times as old as Joey. Use this information to answer the following questions.

a. If you let x represent Joey’s current age, what expression can you use to represent Martina’s current age?

Type below:

___________

Answer:

y = x + 14

Explanation:

y = x + 14

where x is Joey’s current age and t is Martna’s current age.

Question 13.

b. Based on your answer to part a, what expression represents Joey’s age in 5 years? What expression represents Martina’s age in 5 years?

Type below:

___________

Answer:

Ages in 5 years

Joey’s age = x + 5

Martina’s age = x + 14 + 5 = x + 19

Question 13.

c. What equation can you write based on the information given?

Type below:

___________

Answer:

3(x + 5) = x + 19

Explanation:

In 5 years, Martina will be three times as old as Joey

3(x + 5) = x + 19

Question 13.

d. What is Joey’s current age? What is Martina’s current age?

Joey’s current age ___________

Martina’s current age ___________

Answer:

Joey’s current age 2

Martina’s current age 16

Explanation:

3(x + 5) = x + 19

3x + 15 = x + 19

3x – x = 19 – 15

2x = 4

x = 2

Question 14.

As part of a school contest, Sarah and Luis are playing a math game. Sarah must pick a number between 1 and 50 and give Luis clues so he can write an equation to find her number. Sarah says, “If I subtract 5 from my number, multiply that quantity by 4, and then add 7 to the result, I get 35.” What equation can Luis write based on Sarah’s clues and what is Sarah’s number?

Type below:

___________

Answer:

x = 12

Explanation:

As part of a school contest, Sarah and Luis are playing a math game. Sarah must pick a number between 1 and 50 and give Luis clues so he can write an equation to find her number. Sarah says, “If I subtract 5 from my number, multiply that quantity by 4, and then add 7 to the result, I get 35.”

4 (x – 5) + 7 = 35

4x – 20 + 7 = 35

4x – 13 = 35

4x = 35 + 13

4x = 48

x = 48/4

x = 12

Question 15.

Critical Thinking

When solving an equation using the Distributive Property that involves distributing fractions, usually the first step is to multiply by the LCD to eliminate the fractions in order to simplify computation. Is it necessary to do this to solve \(\frac{1}{2}\)(4x + 6) = 13(9x – 24)? Why or why not?

___________

Answer:

It is not necessary. In this case, distributing the fractions directly results in whole-number coefficients and constants, however, if the results are not in whole-number coefficients and constants it is harder to solve fractions.

Question 16.

Solve the equation given in Exercise 15 with and without using the LCD of the fractions. Are your answers the same?

___________

Answer:

x = 11

Explanation:

\(\frac{1}{2}\)(4x + 6) = 13(9x – 24)

6(\(\frac{1}{2}\)(4x + 6)) = 6(13(9x – 24))

3(4x + 6) = 2(9x – 24)

12x + 18 = 18x – 48

18x – 12x = 18 + 48

6x = 66

x = 66/6

x = 11

### Equations with the Distributive Property – Page No. 214

Question 17.

Represent Real-World Problems

A chemist mixed x milliliters of 25% acid solution with some 15% acid solution to produce 100 milliliters of a 19% acid solution. Use this information to fill in the missing information in the table and answer the questions that follow.

a. What is the relationship between the milliliters of acid in the 25% solution, the milliliters of acid in the 15% solution, and the milliliters of acid in the mixture?

Type below:

_____________

Answer:

The milliliters of acid in the 25% solution plus the milliliters of acid in the 15% solution equals the milliliters of acid in the mixture

Explanation:

Question 17.

b. What equation can you use to solve for x based on your answer to part a?

Type below:

_____________

Answer:

0.25x + 0.15(100 – x) = 19

Question 17.

c. How many milliliters of the 25% solution and the 15% solution did the chemist use in the mixture?

Type below:

_____________

Answer:

0.25x + 0.15(100 – x) = 19

0.25x + 15 – 0.15x = 19

0.1x + 15 = 19

0.1x = 4

x = 4/0.1

x = 40

The chemist used 40ml of the 25% solution and 100 – 40 = 60ml of the 15% solution.

**H.O.T.**

**Focus on Higher Order Thinking**

Question 18.

Explain the Error

Anne solved 5(2x) – 3 = 20x + 15 for x by first distributing 5 on the left side of the equation. She got the answer x = -3. However, when she substituted -3 into the original equation for x, she saw that her answer was wrong. What did Anne do wrong, and what is the correct answer?

x = ________

Answer:

x = -1.8

Explanation:

Dado que 5 solo se multiplica por 2x, no tiene sentido usar la distribución aquí. Básicamente, distribuir 5 fue el problema

Solución correcta:

5 (2x) – 3 = 20x + 15

10x -3 = 20x + 15

restar 15 en ambos lados

10x – 18 = 20x

restar 10x de ambos lados

-18 = 10x

x = -1.8

Question 19.

Communicate Mathematical Ideas

Explain a procedure that can be used to solve 5[3(x + 4) – 2(1 – x)] – x – 15 = 14x + 45. Then solve the equation.

x = ________

Answer:

x = 1

Explanation:

5[3(x + 4) – 2(1 – x)] – x – 15 = 14x + 45

5[3x + 12 – 2 + 2x] – x – 15 = 14x + 45

5[5x + 10] – x – 15 = 14x + 45

25x + 50 – x – 15 = 14x + 45

24x + 35 = 14x + 45

24x – 14x = 45 – 35

10x = 10

x = 1

### Guided Practice – Equations with Many Solutions or No Solution – Page No. 218

**Use the properties of equality to simplify each equation. Tell whether the final equation is a true statement.**

Question 1.

The statement is: _______

Answer:

The statement is: true

Explanation:

3x – 2 = 25 – 6x

3x + 6x -2 = 25 -6x + 6x

9x – 2 = 25

9x -2 + 2 = 25 + 2

9x = 27

x = 27/9

x = 3

The statement is true.

**Solving Equations by Clearing Fractions Worksheet Question 2.**

____________

Answer:

The statement is false.

Explanation:

2x – 4 = 2(x – 1) + 3

2x – 4 = 2x – 2 + 3

2x – 4 = 2x + 1

2x – 4 – 2x = 2x + 1 – 2x

-4 not equal to 1

The statement is false.

Question 3.

How many solutions are there to the equation in Exercise 2?

____________

Answer:

There is no solution to exercise 2.

Question 4.

After simplifying an equation, Juana gets 6 = 6. Explain what this means.

____________

Answer:

When 6 = 6, there are infinite solutions.

**Write a linear equation in one variable that has infinitely many solutions.**

Question 5.

Start with a _____ statement.

Add the _____ to both sides.

Add the _____ to both sides.

Combine _____ terms.

Type below:

____________

Answer:

Start with a “true” statement

Add the “same variable” to both sides

Add the “same constant” to both sides

Combine “like” terms

Explanation:

Start with a “true” statement

10 = 10

Add the “same variable” to both sides

10 + x = 10 + x

Add the “same constant” to both sides

10 + x + 5 = 10 + x + 5

Combine “like” terms

15 + x = 15 + x

**Essential Question Check-In**

Question 6.

Give an example of an equation with an infinite number of solutions. Then make one change to the equation so that it has no solution.

Type below:

____________

Answer:

An equation with infinitely many solutions

x – 2x + 3 = 3 – x

-x + 3 = 3 – x

+x/3 = +x/3

An equation for no solution

x – 2x + 3 = 3 – x + 4

-x + 3 = 7 – x

-x/3 = -x/7

### Independent Practice – Equations with Many Solutions or No Solution – Page No. 219

**Tell whether each equation has one, zero, or infinitely many solutions.**

Question 7.

-(2x + 2) – 1 = -x – (x + 3)

____________

Answer:

The statement is true

Explanation:

-(2x + 2) – 1 = -x – (x + 3)

-2x – 2 – 1 = -x – x + 3

-2x – 3 = -2x + 3

-3 = -3

The statement is true

Question 8.

-2(z + 3) – z = -z – 4(z + 2)

____________

Answer:

The statement is false.

Explanation:

-2(z + 3) – z = -z – 4(z + 2)

-3z – 6 = -3z -8

-3z -6 + 3z = -3z – 8 + 3z

-6 not equal to -8

The statement is false.

**Create an equation with the indicated number of solutions.**

Question 9.

No solution:

3(x – \(\frac{4}{3}\)) = 3x + _____

Type below:

______________

Answer:

3(x – \(\frac{4}{3}\)) = 3x + ?

3x – 4 = 3x + ?

3x – 4 = 3x + 2

When there is no solution, the statement should be false. Any number except -4 would make the equation have no solutions.

Question 10.

Infinitely many solutions:

2(x – 1) + 6x = 4( _____ – 1) + 2

Type below:

______________

Answer:

2(x – 1) + 6x = 4( _____ – 1) + 2

2(x – 1) + 6x = 4( ? – 1) + 2

2x – 2 + 6x = 4(? – 1) + 2

8x – 2 = 4(? – 1) + 2

8x – 2 = 4(2x – 1) + 2

8x – 2 = 8x – 4 + 2

8x – 2 = 8x – 2

When there are infinitely many solutions, the statement should be true

Question 11.

One solution of x = -1:

5x – (x – 2) = 2x – ( _____ )

Type below:

______________

Answer:

Put x = -1 in the equation

-5 – (-1 – 2) = -2 – blank

simplifying

-2 = -2 – blank

add 2 on both sides

0 = blank

Question 12.

Infinitely many solutions:

-(x – 8) + 4x = 2( _____ ) + x

Type below:

______________

Answer:

-(x – 8) + 4x = 2( ?) + x

-x + 8 + 4x = 2(?) + x

3x + 8 = 2(?) + x

3x + 8 = 2 (x + 4) + x

3x + 8 = 2x + 8x + x

3x + 8 = 3x + 8

When there are infinitely many solutions, the statement should be true.

Question 13.

Persevere in Problem Solving

The Dig It Project is designing two gardens that have the same perimeter. One garden is a trapezoid whose nonparallel sides are equal. The other is a quadrilateral. Two possible designs are shown at the right.

a. Based on these designs, is there more than one value for x? Explain how you know this.

______________

Answer:

There are more than one value of x

Explanation:

Perimeter of the trapezoid

P = 2x – 2 + x + 1 + x + x + 1 = 5x

Perimeter of the quadrilateral

P = 2x – 9 + x + x + 8 + x + 1 = 5x

5x = 5x

There are more than one value of x

Question 13.

b. Why does your answer to part a make sense in this context?

Type below:

______________

Answer:

The condition was that the two perimeters are to be equal. However, a specific number was not given, so there are an infinite number of possible perimeters

Explanation:

Interpretation of part a in this context

The condition was that the two perimeters are to be equal. However, a specific number was not given, so there are an infinite number of possible perimeters

Question 13.

c. Suppose the Dig It Project wants the perimeter of each garden to be 60 meters. What is the value of x in this case? How did you find this?

______ meters

Answer:

12 meters

Explanation:

2x – 2 + x + 1 + x + x + 1 = 60

5x = 60

x = 60/5

x = 12

### Equations with Many Solutions or No Solution – Page No. 220

Question 14.

Critique Reasoning

Lisa says that the indicated angles cannot have the same measure. Marita disagrees and says she can prove that they can have the same measure. Who do you agree with? Justify your answer.

I agree with: ______________

Answer:

I agree with: Marita

Explanation:

9x – 25 + x = x + 50 + 2x – 12

10x – 25 = 3x + 38

10x – 3x = 38 + 25

7x = 63

x = 63/7

x = 9

When x = 9 the angles will be same and for any other value of x, the angles will not be the same.

Question 15.

Represent Real-World Problems

Adele opens an account with $100 and deposits $35 a month. Kent opens an account with $50 and also deposits $35 a month. Will they have the same amount in their accounts at any point? If so, in how many months and how much will be in each account? Explain.

______________

Answer:

Adele’s amount after x months

A = 100 + 35x

Kent’s amount after x months

A = 50 + 35x

100 + 35x = 50 + 35x

100 is not equal to 50

The statement is false, the amounts in two accounts would never be equal.

**H.O.T.**

**Focus on Higher Order Thinking**

Question 16.

Communicate Mathematical Ideas

Frank solved an equation and got the result x = x. Sarah solved the same equation and got 12 = 12. Frank says that one of them is incorrect because you cannot get different results for the same equation. What would you say to Frank? If both results are indeed correct, explain how this happened.

Frank is: ____________

Answer:

Both of them can be correct as both equations give the same result i.e. there are infinitely many solutions. Frank eliminated the constant from both sides while Sarah eliminated the variable from both sides.

Question 17.

Critique Reasoning

Matt said 2x – 7 = 2(x – 7) has infinitely many solutions. Is he correct? Justify Matt’s answer or show how he is incorrect.

Matt is: ____________

Answer:

Explanation:

2x – 7 = 2(x – 7)

2x – 7 = 2x – 14

-7 not equal to -14

The statement is false, there is no solution. Matt is incorrect.

### 7.1 Equations with the Variable on Both Sides – Model Quiz – Page No. 221

**Solve.**

Question 1.

4a – 4 = 8 + a

_______

Answer:

a = 4

Explanation:

4a – 4 = 8 + a

4a – a = 8 + 4

3a = 12

a = 12/3

a = 4

Question 2.

4x + 5 = x + 8

_______

Answer:

x = 1

Explanation:

4x + 5 = x + 8

4x – x = 8 – 5

3x = 3

x = 3/3

x = 1

**Homework and Practice Solving Linear Equations 7.1 Answer Key Question 3.**

Hue is arranging chairs. She can form 6 rows of a given length with 3 chairs left over, or 8 rows of that same length if she gets 11 more chairs. Write and solve an equation to find how many chairs are in that row length.

_______ chairs

Answer:

7 chairs

Explanation:

Hue is arranging chairs. She can form 6 rows of a given length with 3 chairs left over, or 8 rows of that same length if she gets 11 more chairs.

6x + 3 = 8x – 11

8x – 6x = 3 + 11

2x = 14

x = 14/2

x = 7

There are 7 chairs in each row.

**7.2 Equations with Rational Numbers**

**Solve.**

Question 4.

\(\frac{2}{3} n-\frac{2}{3}=\frac{n}{6}+\frac{4}{3}\)

_______

Answer:

n = 4

Explanation:

\(\frac{2}{3} n-\frac{2}{3}=\frac{n}{6}+\frac{4}{3}\)

The LCM is 6.

6(2/3n – 2/3) = 6(n/6 + 4/3)

6(2/3n) -6(2/3) = 6(n/6) + 6(4/3)

4n – 4 = n + 8

4n – n = 8 + 4

3n = 12

n = 12/3

n = 4

Question 5.

1.5d + 3.25 = 1 + 2.25d

_______

Answer:

d = 3

Explanation:

1.5d + 3.25 = 1 + 2.25d

2.25d – 1.5d = 3.25 – 1

0.75d = 2.25

d = 2.25/0.75

d = 3

Question 6.

Happy Paws charges $19.00 plus $1.50 per hour to keep a dog during the day. Woof Watchers charges $14.00 plus $2.75 per hour. Write and solve an equation to find for how many hours the total cost of the services is equal.

_______ hours

Answer:

3.2 hours

Explanation:

Happy Paws charges $19.00 plus $1.50 per hour to keep a dog during the day.

1.5x + 19

Woof Watchers charges $14.00 plus $2.75 per hour.

2.75x + 15

1.5x + 19 = 2.75x + 15

2.75x – 1.5x = 19 – 15

1.25x = 4

x = 4/1.25

x = 3.2

The total cost of the services is equal after 3.2 hrs.

**7.3 Equations with the Distributive Property**

**Solve.**

Question 7.

14 + 5x = 3(-x + 3) – 11

_______

Answer:

x = -2

Explanation:

14 + 5x = 3(-x + 3) – 11

14 + 5x = -3x + 9 – 11

14 + 5x = -3x – 2

5x + 3x = -2 – 14

8x = – 16

x = -16/8

x = -2

Question 8.

\(\frac{1}{4}\)(x – 7) = 1 + 3x

_______

Answer:

x = -1

Explanation:

\(\frac{1}{4}\)(x – 7) = 1 + 3x

4(\(\frac{1}{4}\)(x – 7)) = 4(1 + 3x)

(x – 7) = 4 + 12x

12x – x = -7 – 4

11x = -11

x = -11/11

x = -1

Question 9.

-5(2x – 9) = 2(x – 8) – 11

_______

Answer:

x = 6

Explanation:

-5(2x – 9) = 2(x – 8) – 11

-10x + 45 = 2x – 16 – 11

-10x + 45 = 2x – 27

2x + 10x = 45 + 27

12x = 72

x = 72/12

x = 6

Question 10.

3(x + 5) = 2(3x + 12)

_______

Answer:

x = -3

Explanation:

3(x + 5) = 2(3x + 12)

3x + 15 = 6x + 24

6x – 3x = 15 – 24

3x = -9

x = -9/3

x = -3

**7.4 Equations with Many Solutions or No Solution**

**Tell whether each equation has one, zero, or infinitely many solutions.**

Question 11.

5(x – 3) + 6 = 5x – 9

____________

Answer:

There are infinitely many solutions

Explanation:

5(x – 3) + 6 = 5x – 9

5x – 15 + 6 = 5x – 9

5x – 9 = 5x – 9

The statement is true. There are infinitely many solutions.

Question 12.

5(x – 3) + 6 = 5x – 10

____________

Answer:

There are no solutions

Explanation:

5(x – 3) + 6 = 5x – 10

5x – 15 + 6 = 5x – 10

5x – 9 = 5x – 10

-9 not equal to -10

The statement is false. There are no solutions.

Question 13.

5(x – 3) + 6 = 4x + 3

____________

Answer:

There is one solution

Explanation:

5(x – 3) + 6 = 4x + 3

5x – 15 + 6 = 4x + 3

5x – 9 = 4x + 3

5x – 4x = 3 + 9

x = 12

There is one solution

### Selected Response – Mixed Review – Page No. 222

Question 1.

Two cars are traveling in the same direction. The first car is going 40 mi/h, and the second car is going 55 mi/h. The first car left 3 hours before the second car. Which equation could you solve to find how many hours it will take for the second car to catch up to the first car?

Options:

a. 55t + 3 = 40t

b. 55t + 165 = 40t

c. 40t + 3 = 55t

d. 40t + 120 = 55t

Answer:

d. 40t + 120 = 55t

Explanation:

Two cars are traveling in the same direction. The first car is going 40 mi/h, and the second car is going 55 mi/h. The first car left 3 hours before the second car.

3 × 40 + 40t = 120 + 40t

55t

40t + 120 = 55t

Question 2.

Which linear equation is represented by the table?

Options:

a. y = -x + 5

b. y = 2x – 1

c. y = x + 3

d. y = -3x + 11

Answer:

a. y = -x + 5

Explanation:

Find the slope using

m = (y2 – y1)/(x2 – x1)

where (x1, y1) = (3, 2), (x2, y2) = (1, 4)

Slope = (4 – 2)/(1 – 3) = -2/2 = -1

Question 3.

Shawn’s Rentals charges $27.50 per hour to rent a surfboard and a wetsuit. Darla’s Surf Shop charges $23.25 per hour to rent a surfboard plus $17 extra for a wetsuit. For what total number of hours are the charges for Shawn’s Rentals the same as the charges for Darla’s Surf Shop?

Options:

a. 3

b. 4

c. 5

d. 6

Answer:

b. 4

Explanation:

Shawn’s Rentals charges $27.50 per hour to rent a surfboard and a wetsuit.

27.5x

Darla’s Surf Shop charges $23.25 per hour to rent a surfboard plus $17 extra for a wetsuit.

23.25x + 17

23.25x + 17 = 27.5x

27.5x – 23.25x = 17

4.25x = 17

x = 17/4.25

x = 4

The charge would be equal after 4 hrs

Question 4.

Which of the following is irrational?

Options:

a. -8

b. 4.63

c. \(\sqrt { x } \)

d. \(\frac{1}{3}\)

Answer:

c. \(\sqrt { x } \)

Explanation:

\(\sqrt { x } \) is irrational

Question 5.

Greg and Jane left a 15% tip after dinner. The amount of the tip was $9. Greg’s dinner cost $24. Which equation can you use to find x, the cost of Jane’s dinner?

Options:

a. 0.15x + 24 = 9

b. 0.15(x + 24) = 9

c. 15(x + 24) = 9

d. 0.15x = 24 + 9

Answer:

b. 0.15(x + 24) = 9

Explanation:

Let x be the cost of Jane’s dinner. The amount of tip is the 15% of the total cost of dinner.

0.15(x + 24) = 9

Question 6.

For the equation 3(2x − 5) = 6x + k, which value of k will create an equation with infinitely many solutions?

Options:

a. 15

b. -5

c. 5

d. -15

Answer:

d. -15

Explanation:

3(2x – 5) = 6x + k

6x – 15 = 6x + k

6x – 15 = 6x – 15

The statement is true. k = -15

Question 7.

Which of the following is equivalent to 2^{−4}?

Options:

a. \(\frac{1}{16}\)

b. \(\frac{1}{8}\)

c. -2

d. -16

Answer:

a. \(\frac{1}{16}\)

Explanation:

2^{−4}

1/2^{4}

1/16

**Mini-Task**

Question 8.

Use the figures below for parts a and b.

a. Both figures have the same perimeter. Solve for x.

_______

Answer:

x=12

Explanation:

4x+10=3x+22

4x – 3x = 22 – 10

x = 12

Answer: x=12

Question 8.

b. What is the perimeter of each figure?

_______

Answer:

Both are 58

Explanation:

x + x + 5 + x + x + 5

12 + 12 + 5 + 12 + 12 + 5

58

x + 7 + x + 4 + x + 11

12 + 7 + 12 + 4 + 12 + 11

58

### Conclusion:

Go Math Grade 8 Answer Key Chapter 7 Solving Linear Equations are available along with the all practice questions here. Get all the questions, answers and also Go Math Grade 8 text books for free online. Refer to Go Math Grade 8 Chapter 7 Solving Linear Equations Answer Key to learn the quick maths in an easy way.