We included **HMH Into Math Grade 8 Answer Key PDF** **Module 7 Review **to make students experts in learning maths.

## HMH Into Math Grade 8 Module 7 Review Answer Key

**Vocabulary**

elimination

substitution

system of

equations

**Choose the correct term from the Vocabulary box.**

Question 1.

A ____ is two or more equations that contain two or more variables.

Answer: A **system of equations** is two or more equations that contain two or more variables.

Question 2.

A method of solving a system of equations that involves adding system equations to remove a variable is called ____

Answer: A method of solving a system of equations that involves adding system equations to remove a variable is called **elimination**.

Question 3.

A method of solving a system of equations that involves replacing a variable with a number or another expression is called _____

Answer: A method of solving a system of equations that involves replacing a variable with a number or another expression is called **substitution**.

**Concepts and Skills**

Question 4.

North Shore Kayak and South Shore Kayak each charge a flat fee plus an hourly rate. The graph shows y, the cost in dollars, for renting a kayak from each company for x hours. Which statement about the companies is true?

A. The companies have the same hourly rate.

B. South Shore costs more when renting a kayak for 2 hours.

C. The flat fee is higher for North Shore than for South Shore.

D. The companies cost the same when renting a kayak for 1 hour.

Answer: D. The companies cost the same when renting a kayak for 1 hour.

Question 5.

A graph of a system of two equations is shown. What is the solution of the system? Write the ordered pair.

(_____, _____)

Answer:

A solution to a system of equations is a set of values for the variable that satisfy all the equations simultaneously.

In the above graph, we can see that two equations meet at the point (-3, -2)

So, the ordered pair is (-3, -2)

Question 6.

**Use Tools** Determine the solution of the system of equations shown. State what strategy and tool you will use to answer the question, explain your choice, and then find the answer.

y = x + 1

y = -2x + 4

Answer:

y = x + 1

y = -2x + 4

Substitute y = -2x + 4

-2x + 4 = x + 1

-2x – x = 1 – 4

-3x = -3

x = 1

Substitute x = 1 in the above equation

y = x + 1

y = 1 + 1

y = 2

The solution of system of equations are y = 2 and x = 1.

**Determine the solution of each system of equations.**

Question 7.

y = -6x + 4

y = 2 + 12 (___, ____)

Answer:

y = -6x + 4

y = 2 + 12

y = 14

Substitute y = 14 in the above equation.

14 = -6x + 4

14 – 4 = -6x

10 = -6x

x = -10/6

x = -5/3

Question 8.

3x + 2y = 9

4x – y = 34 (___, ____)

Answer:

3x + 2y = 9

4x – y = 34

3x + 2y = 9

3x = 9 – 2y

x = (9-2y)/3

4[(9-2y)/3] – y = 34

[(36-11y)/3 = 34]

36 – 11y = 34 × 3

36 – 11y = 102

-11y = 102 – 36

-11y = 66

y = -66/11

y = -6

3x + 2y = 9

3x + 2(-6) = 9

3x -12 = 9

3x = 9 + 12

3x = 21

x = 21/3

x = 7

Thus the solutions to the system of equations are x = 7 and y = -6.

Question 9.

A system of equations is shown. Which method could be used to eliminate a variable from the system?

5x + 6y = 28

4x + 2y = 14

A. Multiply the first equation by -4, and then add the equations.

B. Multiply the first equation by -2, and then add the equations.

C. Multiply the second equation by -3, and then add the equations.

D. Multiply the second equation by -2, and then add the equations.

Answer: C. Multiply the second equation by -3, and then add the equations.

5x + 6y = 28

4x + 2y = 14—- × (-3)

-12x – 6y = -42

5x + 6y – 28 -12x -6y + 42 = 0

-7x + 14 = 0

14 = 7x

x = 2

Question 10.

Select the number of solutions for each system of two linear equations.

Answer:

i. 3x + 5y = 10

2x + 5y = 10

3x + 5y – 10 = 2x + 5y -10

3x + 5y -10 – 2x – 5y + 10 = 0

x = 0

It has infinite number of solutions.

ii. x – 3y = 4 ⇒ x – 3y – 4 = 0

2x – 6y = 8 ⇒ 2x – 6y – 8 = 0

x – 3y -4 – 2x + 6y + 8 = 0

-x + 3y + 4 = 0

x = 3y + 4

2 (3y + 4) + 5y = 10

6y + 8 + 5y = 10

11y + 8 = 10

11y = 2

y = 2/11

x – 3(2/11) = 4

x – 6/11 – 4 = 0

x – 4.54 = 0

x = 4.54

It has one solution.

iii. 4x – 2y = 6

4x – 2y = 8

4x – 2y – 6 – 4x + 2y + 8 = 0

-6 + 8 = 0

2 = 0

It has no solution.

Question 11.

Tickets for a high school basketball game cost $4 for adults and $3 for students. The school sells 120 tickets and makes $412 in ticket sales. The system of equations shown can be used to determine the number of adult tickets a and the number of student tickets s the school sold. How many adult tickets and how many student tickets did the school sell?

a + s = 120

4a + 3s = 412

____ adult tickets ____ student tickets

Answer:

Given,

Tickets for a high school basketball game cost $4 for adults and $3 for students.

The school sells 120 tickets and makes $412 in ticket sales.

a + s = 120— × 3 = 3a + 3s = 360

4a + 3s = 412

3a + 3s – 360 – 4a – 3s – 412 = 0

-a – 772 = 0

a = -772

Substitute a = -772 in the given equation

-772 + s = 120

s = 120 + 772

s = 892

772 adult tickets and 892 student tickets.

Question 12.

Corinne’s pumpkin weighs 28 ounces and is growing at a rate of 5 ounces per week. Ron’s pumpkin weighs 10 ounces and is growing at a rate of 13 ounces per week. Let t represent time in weeks and w represent weight in ounces. Which system of equations can be used to determine when the weights of the two pumpkins will be equal?

A.

w = 5 + 28t

w = 13 + 10t

B.

w = 5(t + 28)

w = 13(t + 10)

C.

w = 5t – 28

w = 13t – 10

D.

w = 5t + 28

w = 13t + 10

Answer:

Given,

Corinne’s pumpkin weighs 28 ounces and is growing at a rate of 5 ounces per week.

Ron’s pumpkin weighs 10 ounces and is growing at a rate of 13 ounces per week.

Let t represent time in weeks and w represent weight in ounces.

Corrine’s: w = 28 + 5t

Ron’s: w = 10 + 13t

So, the system of equations is

w = 5t + 28

w = 3t + 10

Option D is the correct answer.