We included **H****MH Into Math Grade 8 Answer Key**** PDF** **Module 7 Lesson 3 Solve Systems by Substitution **to make students experts in learning maths.

## HMH Into Math Grade 8 Module 7 Lesson 3 Answer Key Solve Systems by Substitution

I Can solve systems of equations by substitution.

**Spark Your Learning**

A state fair offers two pricing plans. Each includes a flat fee for admission and a price per ride. The equations in the table show the total cost y, in dollars, to attend the fair and go on x rides. For how many rides do the two pricing plans cost the same? Solve this problem without graphing and explain your reasoning.

Answer:

Given that the equations are,

Super saver y = 4x + 7

Fun pack y = 2x + 17

The total cost y, in dollars, to attend the fair and go on x rides.

Let us consider number of rides = x

In super saver equation

If x = 1 then y = 4(1) + 7 = 11

If x = 2 then y = 4(2) + 7 = 15

If x = 4 then y = 4(4) + 7 = 23

If x = 5 then y = 4(5) + 7 = 27

In Fun pack

If x = 1 then y = 2(1) + 17 = 19

If x = 2 then y = 2(2) + 17 = 21

If x = 3 then y = 2(3) + 17 = 23

If x = 4 then y = 2(4) + 17 = 25

If x = 5 then y = 2(5) + 17 = 27

For 5 rides the cost of the Super saver and the Fun pack is the same.

**Turn and Talk** How can you check that you correctly found a solution to a system of linear equations?

**Build Understanding**

1.

Solve the system.

A. Graph the system to estimate a solution.

Answer:

B. Since y = -2x – 4 and y = 2x + 8, what do you know about -2x – 4 and 2x + 8 at the intersection point of the two lines?

C. Solve the equation -2x – 4 = 2x + 8.

Answer:

-2x – 4 = 2x + 8

-2x – 4 – 2x – 8 = 0

-4x – 12 = 0

-4x = 12

x = -3

D. What does the solution in Part C represent?

_____________________

_____________________

E. How can you find the value of the other variable? What is the value?

_____________________

_____________________

F. What does the y-value tell you?

_____________________

_____________________

G. Check your solution by substituting x and y back into both original equations. Show your work.

H. What is the solution to the system of equations? _____

**Step It Out**

2. Solve this system using substitution,

**Connect to Vocabulary**

To substitute is to replace a variable with a number or another expression in an algebraic expression.

A. How can you use x = 2y + 7 to substitute for x in the second equation?

Since the equation is solved for ___,

you can substitute ____ for x in the second equation.

Answer:

Since the equation is solved for y,

you can substitute y for x in the second equation.

B. Complete the following solution.

2x + 5y = 5

2__( )__ + 5y = 5

____ + 5y = 5

___ + 14 = 5

9y = ___

y = ____

Answer:

2x + 5y = 5

2(2y + 7) + 5y = 5

4y + 14 + 5y = 5

9y = 5 – 14

9y = -9

y = -9/9

y = -1

C. Since y = ___, x = 2 __( )__ + 7,

So, x = ____ + 7 or ___.

The solution is ____.

Answer:

Since y = -1, x = 2(-1) + 7

So, x = -2 + 7 or 5

The solution is 5.

3. For admission to a concert at the state fair, child tickets cost x dollars and adult tickets cost y dollars. Solve the system shown to find the price of each type of ticket.

A. Solve the system by first solving for x in the first equation. Then substitute the resulting expression in the second equation and solve for one of the variables.

Since x + y = 10, x = ___

Substitute into the other equation:

4x + 8y 64

4__( )__ + 8y = 64

____ + 8y = 64

___ + 4y = 64

4y = ___

y = ____

Answer:

Since x + y = 10, x = 10 – y

Substitute into the other equation:

4x + 8y = 64

4(10 – y) + 8y = 64

40 – 4y + 8y = 64

40 + 4y = 64

4y = 24

y = 6

B. Since y = ___, x = 10 – ___, and x = ___. The cost of each child ticket is ___, and the cost of each adult ticket is ____.

Answer:

Since y = 6, x = 10 – 6, and x = 4. The cost of each child ticket is 4, and the cost of each adult ticket is 6.

**Turn and Talk** In Part B, what would change if you substituted y = 6 into 4x + 8y = 64 instead? What would remain the same?

4. Solve this system of equations.

A. Solve for one of the variables in one of the equations.

4x – 3y = -5

4x = ___ – 5

x = ____

Answer:

4x – 3y = -5

4x = 3y – 5

x = 3y – 5/4

B. Make a substitution in the other equation and then solve for the variable that remains after the substitution.

Substitute x = ___ into -8x + 2y = 2. Then solve for y.

—8(____) + 2y = 2

____ + 2y = 2

___ + 10 = 2

-4y = ____

y = ____

Answer:

Substitute x = 3y – 5/4 into -8x + 2y = 2

Then solve for y

-8(3y – 5/4) + 2y = 2

-12y + 10 + 2y = 2

-10y + 10 = 2

-10y = 2 – 10

-10y = -8

y = 4/5

C. Solve for x.

Substitute y = ___ in 4x – 3y = -5. Then solve for x.

4x – 3(___) = -5

4x – ___ = -5

4x = ___

x = ____

The solution is ___.

Answer:

Substitute y = 4/5 in 4x – 3y = -5

Then solve for x

4x – 3(4/5) = -5

4x – 2.4 = -5

4x = -5 + 2.4

x = -2.6

The solution is (-2.6, 4/5).

**Check Understanding**

Question 1.

Describe the steps for solving a system of two equations in two variables by substitution. Explain how to check the solution.

Answer:

The steps for solving a system of two equations in two variables are

From one equation solve for x or y then.

Then the values of x or y are substituted in the second equation

Substitute the values in the equation involving both variables.

Question 2.

What is the solution to the system shown?

x – y = 3

2x – 0.5y = 0

Answer:

Given that,

The equations are

x – y = 3 is an equation 1

2x – 0.5y = 0 is an equation 2

In equation 1 x = 3 + y

Substitute x value in equation 2

2(3 + y) – 0.5y = 0

6 + 2y – 0.5y = 0

1.5y + 6 = 0

1.5y = 0 – 6

1.5y = -6

y = -4

Substitute x = -4 in equation 1

x – (-4) = 3

x + 4 = 3

x = 3 – 4

x = -1

The solution i s(-1, -4)

**On Your Own**

Question 3.

**STEM** Scientists use drones with digital cameras to help them identify plants, predict flooding, and construct 3-D maps of different landscapes. A team of scientists is using two drones to map a region. The heights of the drones are represented by the equations given, where x is the number of minutes since the drones were released by the scientists and y is the height in meters.

A. Solve the system of equations.

_____________________

Answer:

Given that the equations are

y = 8x + 5

y = 6x + 25

Rewrite the equations as

y – 8x = 5 is an equation 1

y – 6x = 25 is an equation 2

Substitute equation 2 from equation 1

-2x = -20

x = -20/-2

x = 10

Substitute x in equation 1

y – 8(10) = 5

y – 80 = 5

y = 5 + 80

y = 85

The solution is (10,85)

B. What does your solution tell you about the drones?

_____________________

Answer:

where x is the number of minutes since the drones were released by the scientists.

y is the height in meters.

For 10 minutes the drones were released by the scientists is 85 is meters.

Question 4.

Tickets for a school play have one price for students, x, and a different price for non-students, y. The system of equations shown is based on two different ticket orders in which the prices x and y are in dollars.

2x + 3y = 49

1 x + 2y = 30

A. What is the first step in solving the system by substitution? Justify your answer.

Answer:

Given that the equations are

2x + 3y = 49 is an equation 1

1x + 2y = 30 is an equation 2

The first step is solved for x

2x = 49 – 3y

x = 49 – 3y/2

B. Solve the system and explain what your solution represents.

Answer:

Given that the equations are

2x + 3y = 49 is an equation 1

1x + 2y = 30 is an equation 2

Multiply equation 2 with 2then we get

2x + 4y = 60 is an equation 3

Subtract equation 3 from equation 1

-y = -11

y = 11.

Substitute y in equation 1

2x + 3(11) = 49

2x + 33 = 49

2x = 49 – 33

2x = 16

x = 16/2

x = 8

The solution is (8,11)

The prize for the students is $8

The prize for the non-students is $11.

Question 5.

Consider the system of equations

2x + 5y = 18

3x + 1.5y = 9

A. Graph to estimate the solution of the system. Estimated solution: _____

Given that the equations are

2x + 5y = 18 is an equation 1

3x + 1.5y = 9 is an equation 2

From equation 1

5y = 18 – 2x

y = 18 – 2x/5

if x = 1 then y = 18 – 2(1)/5 = 3.2

If x = 2 then y = 18 – 2(2) /5 = 2.8

If x = 4 then y = 18 – 2(4)/ 5 = 10/5 = 2

If x = 5 then y = 18 – 2(5)/5 = 1.6

From equation 2

1.5y = 9 – 3x

y = 9 – 3x/1.5

If x = 1 then y = 9 – 3(1)/1.5 = 4

If x = 2 then y = 9 – 3(2)/1.5 = 2

If x = 4 then y = 9 – 3(4)/1.5 = -2

If x = 5 then y = 9 – 3(5)/1.5 = -4

B. Solve the system by substitution.

Answer:

Given that the equations are

2x + 5y = 18 is an equation 1

3x + 1.5y = 9 is an equation 2

Multiply equation 1 with 3 and equation 2 with 2 Then we get

6x + 15y = 54 is an equation 3

6x + 3y = 18 is an equation 4

Subtract equation 4 from equation 3 then we get

12y = 36

y = 36/12

y = 3

substitute y in equation 1

2x + 5(3) = 18

2x + 15 = 18

2x = 18 – 15

2x = 3

x = 3/2

x = 1.5

The solution is (1.5, 3)

Question 6.

Complete the system of two linear equations so it has the solution (-1, 7). Check by using substitution to

solve the system.

Answer:

Question 7.

The map of a small city is placed on a coordinate plane. Two of the town’s straight roads can be represented by the equations in the system shown here.

-4x – 2y = -6

2x + 2 = 7

A. Without graphing, in what quadrant of the coordinate plane do the roads intersect? How do you know?

Answer:

Given that the equations are

-4x – 2y = -6 is an equation 1

2x + 2 = 7 is an equation 2

From equation 2

2x = 7 – 2

2x = 5

x = 5/2

x = 2.5

Substitute x in equation 1

-4(2.5) – 2y = -6

-10 – 2y = -6

-2y = -6 + 10

-2y = 4

y = 4/-2

y = -2

The solution is (2.5, -2).

The points (2.5, -2) lie in the 2nd quadrant.

The lines intersect at the points (2.5, -2)

B. **Attend to Precision** Check your answer by graphing and labeling the equations.

**For Problems 8-13, solve the system of equations.**

Question 8.

3x – y = 15

x + y = 1

Answer:

Given that the equations are

x – y = 4 is an equation 1

x + 2y = 4 is an equation 2

From equation 1

x = 4 + y

Substitute x in equation 2

4 + y + 2y = 4

4 + 3y = 4

3y = 4 – 4

3y = 0

y = 0/3

y = 0

Substitute y = 0 in equation 1

x – 0 = 4

x = 4

The solution is (4,0)

Question 9.

-2x + y = 8

y = 6

Answer:

Given that the equations are

-2x + y = 8 is an equation 1

y = 6 is an equation 2

Substitute equation 2 in equation 1

-2x + 6 = 8

-2x = 8 – 6

-2x = 2

x = 2/-2

x = -1

Substitute x in equation 1

-2(-1) + y = 8

2 + y = 8

y = 8 – 2

y = 6

The solution is (-1, 6)

Question 10.

4y = 20

x – y = 7

Answer:

Given that the equations are

4y = 20 is an equation 1

x – y = 7 is an equation 2

From equation 1

y = 20/4

y = 5

Substitute y in equation 2

x – 5 = 7

x = 7 – 5

x = 2

The solution is (2,5)

Question 11.

3x – 6y = 5

2x + y = 0

Answer:

Given that the equations are

3x – 6y = 5 is an equation 1

2x + y = 0 is an equation 2

From equation 2

2x = 0 – y

x = -y/2

Substitute x in equation 1

3(-0.5y) – 6y = 5

-1.5y – 6y = 5

-7.5y = 5

y = 5/-7.5

y = -0.6

Substitute y in equation 2

2x -0.6 = 0

2x = 0 – 0.6

2x = -0.6

x = -0.6/2

x = -0.3

The solution is (-0.3, -0.6)

Question 12.

-5x + 2y = -8

2x – 3y = 12

Answer:

Given that the equations are

-5x + 2y = -8 is an equation 1

2x – 3y = 12 is an equation 2

Multiply equation 1 with 2 then we get

2(-5x + 2y) = 2(-8)

-10x + 4y = -16 is an equation 3

Multiply equation 2 with -5

-5(2x – 3y) = -5(12)

-10x + 15y = -60 is an equation 4

Subtract equation 4 from equation 3 then we get

-11y = 44

y = 44/-11

y = -4

Substitute y in equation 1

-5x + 2(-4) = -8

-5x – 8 = -8

-5x = -8 + 8

-5x = 0

x = 0/-5

x = 0

The solution is (0,-4)

Question 13.

4x + 2y = 18

-2x + 3y = 23

Answer:

Given that the equations are

4x + 2y = 18 is an equation 1

-2x + 3y = 23 is an equation 2

Multiply equation 1 with -2 and equation 2 with 4.

-8x -4y = -36 is an equation 3

-8x + 12y = 92 is an equation 4

Substitute equation 4 from equation 3

-16y = -128

y = -128/-16

y = 8

Substitute y in equation 1

4x + 2(8) = 18

4x + 16 = 18

4x = 18 – 16

4x = 2

x = 2/4

x = 1/2

x = 0.5

The solution is (0.5, 8)

**I’m in a Learning Mindset!**

How can I modify my process for solving systems by substitution to maintain an appropriate level of challenge?

**Lesson 7.3 More Practice/Homework**

Question 1.

There are x trumpet players and y saxophone players in a school’s jazz band. The equations in the system shown here relate x and y. Solve the system by substitution. What does the solution mean?

2x + 3y = 23

y = 3x – 7

Answer:

Given that the equations are

2x + 3y = 23 is an equation 1

y = 3x – 7 is an equation 2

Substitute equation 2 in equation 1

2x + 3(3x – 7) = 23

2x + 9x – 21 = 23

2x + 9x = 23 + 21

11x = 44

x = 44/11

x = 4

Substitute x in equation 2

y = 3(4) – 7

y = 12 – 7

y = 5

The solution is (4,5).

Question 2.

Students in Ms. Chu’s science class are building model rockets. Jars of baking soda cost x dollars each, and bottles of vinegar cost y dollars each. The system shown relates the prices of these items.

5x + 3y = 17

x + y = 4

A. Graph to estimate the solution of the system.

Estimated solution: ____

Answer:

Given that the equations are

5x + 3y = 17

x + y = 4

Rewrite the above equations

3y = 17 – 5x

y = 17 – 5x/3

If x = 1 then y = 17 – 5/1 = 12

If x = 2 then y = 17 – 5/2 = 6

If x = 3 then y = 17 – 5/3 = 4

x + y = 4

y = 4 – x

If x = 1 then y= 4 – 1 = 3

If x = 2 then y = 4 – 2 = 2

If x = 3 then y = 4 – 3 = 1

From the graph the two equations are parallel.

B. **Attend to Precision** Solve the system. What does the solution represent?

Answer:

Given that the equations are

5x + 3y = 17 is an equation 1

x + y = 4 is an equation 2

From equation 2

x = 4 – y

Substitute x in equation 1

5(4 – y) + 3y = 17

20 – 5y + 3y = 17

20 – 2y = 17

-2y = 17 – 20

-2y = -3

y = -3/-2

y = 1.5

Substitute y in equation 1

5x + 3(1.5) = 17

5x + 4.5 = 17

5x = 17 – 4.5

5x = 12.5

x = 12.5/5

x = 2.5

The solution is (2.5, 1.5)

The cost of backing soda is 2.5 dollars

The cost of vinegar is 1.5 dollars.

Question 3.

Use the system of equations shown for Parts A and B.

3x – 2y = -10

-4x + 3y = 13

A. Solve one of the equations for either variable.

Answer:

Given that the equations are

3x – 2y = -10 is an equation 1

-4x + 3y = 13 is an equation 2

From equation 1

3x – 2y = -10

3x = -10 + 2y

x = -10 + 2y/3

x = -3.3 + 0.6y

B. Use substitution to find the solution of the system.

Answer:

Given that the equations are

3x – 2y = -10 is an equation 1

-4x + 3y = 13 is an equation 2

Substitute x = -3.3 + 0.6y in equation 2

-4(-3.3 + 0.6y) + 3y = 13

13.2 – 2.4y + 3y = 13

0.6y + 13.2 = 13

0.6y = -0.2

y = -0.2/0.6

y = -0.3

Substitute y = -0.3 in equation 1

3x – 2(-0.3) = -10

3x + 0.6 = -10

3x = -10 – 0.6

3x = -10.6

x = -10.6/3

x = -3.53

The solution is (-3.53, -0.3)

Question 4.

**Math on the Spot** Solve each system by substitution.

A.

y = 3x

x + y = 3

Answer:

Given that the equations are

y = 3x is an equation 1

x + y = 3 is an equation 2

Substitute equation 1 in equation 2

x + 3x = 3

4x = 3

x = 3/4

x = 0.75

Substitute x in equation 1

y = 3(0.75)

y = 2.25

The solution is (0.85, 2.25)

B.

x – y = 4

x + 2y = 4

Answer:

Given that the equations are

x – y = 4 is an equation 1

x + 2y = 4 is an equation 2

From equation 1

x = 4 + y

Substitute x in equation 2

4 + y + 2y = 4

4 + 3y = 4

3y = 4 – 4

3y = 0

y = 0/3

y = 0

Substitute y in equation 1

x – 0 = 4

x = 4

The solution is (4,0)

**Test Prep**

Question 5.

Which is a correct step in solving this system of equations by substitution?

x + y = 3

3x – 4y = -5

A. Substitute x + 3 for x in the equation 3x – 4y = -5.

B. Substitute x + 3 for y in the equation 3x – 4y = -5.

C. Substitute -x + 3 for x in the equation 3x – 4y = -5.

D. Substitute -x + 3 for y in the equation 3x – 4y = -5.

Answer:

Given that the equation is

x + y = 3

3x – 4y = -5

The next step is

Substitute -x + 3 for y in the equation 3x – 4y = -5.

Option D is the correct answer.

Question 6.

Brodie is using a coordinate plane to design two straight paths in a community garden. The paths are represented by the lines 2x + 3y = 6 and -3x – 2y = 1. At what point, if any, do the two paths intersect?

Answer:

Given that the equations are

2x + 3y = 6 is an equation 1

-3x – 2y = 1. Is an equation 2

Multiply equation 1 with -3 and equation 2 with 2. Then we get

-6x – 9y = -18 is an equation 3

-6x – 4y = 2 is an equation 4

Subtract equation 4 from equation 3

-5y = -20

y = 4

Substitute y in equation 1

2x + 3(4) = 6

2x + 12 = 6

2x = 6 – 12

2x = -6

x = -6/2

x = -3

The solution is (4, -3)

The lines are intersected at the point (4, -3)

Question 7.

Celia used substitution correctly to solve one of the systems of equations shown here. As part of her solution process, she solved the equation -2x + 3(-2x + 4) = -3. Which system did Celia solve?

A. -2x + y = 4

-2x + 3y = -3

B. 2x – y = 4

-2x + 3y = -3

C. 2x + y = 4

-2x + 3y = -3

D. -2x – y = 4

-2x + 3y = -3

Answer:

Given that the solved equation is -2x + 3(-2x + 4) = -3.

A)

Given that the equations are

-2x + y = 4 is an equation 1

-2x + 3y = -3 is an equation 2

From equation 1

y = 4 + 2x

Substitute in equation 2 then we get

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

B)

Given that the equations are

2x – y = 4 is an equation 1

-2x + 3y = -3 is an equation 2

From equation 1

y = -4 + 2x

Substitute y in equation 2 then we get

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

C)

Given that the equations are

2x + y = 4 in equation 1

-2x + 3y = -3 in equation 2

From equation 1

y = -2x + 4

Substitute y in equation 2 then we get

-2x + 3(4 – 2x) = -3

D)

Given that the equations are

-2x – y = 4 is an equation 1

-2x + 3y = -3 is an equation 2

From equation 1

y = -4 – 2x

Substitute y in equation 2 then we get

-2x + 3(-4 – 2x) = -3

Therefore option C is the correct answer.

-2x + 3(4 – 2x) = -3 is equal to the solved equation

Question 8.

Which is a true statement about the solution of this system of equations?

6x – 2y = -3

4x + 6y = 9

A. The values of both x and y are integers.

B. The values of x and y are equal.

C. The solution lies in Quadrant III of the coordinate plane.

D. The solution lies on one of the axes of the coordinate plane.

Answer:

Given that the equations are

6x – 2y = -3 is an equation 1

4x + 6y = 9 is an equation 2

Multiply equation 1 with 4 then we get

30x – 8y = -12 is an equation 1

Multiply equation 2 with 6 then we get

30x + 36y = 54 is an equation 2

Subtract equation 2 from equation 1 then we get

-44y = -66

y = -66/-44

y = 1.5

Substitute y = 1.5 in equation 1

6x – 2(1.5) = -3

6x – 3 = -3

6x = -3 + 3

6x = 0

x = 0

The solution is (0, 1.5)

The solution lies on one of the axes of the coordinate plane.

Option D is the correct answer.

Question 9.

Solve the system

2x + 3y = -9

—x + 4y = 10

Answer:

Given that the equations are

2x + 3y = -9 is an equation 1

-x + 4y = 10 is an equation 2

In equation 2

-x = 10 – 4y

x = -10 + 4y

Substitute x in equation 1

2(-10 + 4y) + 3y = -9

-20 + 8y + 3y = -9

-20 + 11y = -9

11y = -9 + 20

11y = 11

y = 1

Substitute y = 1 in equation 2

-x + 4(1) = 10

-x = 10 – 4

-x = 6

x = -6

The solution is (-6, 1).

**Spiral Review**

**In the figure, Line m is parallel to Une n. Use the figure to solve Problems 10-12.**

Question 10.

Name all of the pairs of corresponding angles in the figure.

Answer:

∠2 and ∠6 are the corresponding angles

∠4 and ∠8 are the corresponding angles

∠1 and ∠5 are the corresponding angles

∠3 and ∠7 are the corresponding angles

Question 11.

The measure of ∠3 is (4x + 1)° and the measure of ∠6 is (6x – 29)°. Find the value of x and the measures of ∠3 and ∠6.

Answer:

Given,

The measure of ∠3 is (4x + 1)° and the measure of ∠6 is (6x – 29)°.

(4x + 1)° = (6x – 29)°

4x + 1 = 6x – 29

4x – 6x = -29 – 1

-2x = -30

x = 15°

Question 12.

Solve the equation 4(x + 3) + 3 = 5(x + 4).

Answer:

4(x + 3) + 3 = 5(x + 4)

4x + 12 + 3 = 5x + 20

4x + 15 – 5x – 20 = 0

-x – 5 = 0

x + 5 = 0

x = -5