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

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

I Can solve systems of equations by substitution.

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.

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.

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.
-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.
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 = ____
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 ____.
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 = ____
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 ____.
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 = ____
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 = ____
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 ___.
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.
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
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)

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.
_____________________
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?
_____________________
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.
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.
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.
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.

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?
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
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
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
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
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
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
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
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: ____
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?
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.
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.
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
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
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.
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?
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
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.
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
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.

∠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.
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).