# enVision Math Common Core Grade 8 Answer Key Topic 5 Analyze And Solve Systems Of Linear Equations

Practice with the help of enVision Math Common Core Grade 8 Answer Key Topic 5 Analyze and Solve Systems of Linear Equations regularly and improve your accuracy in solving questions.

## enVision Math Common Core 8th Grade Answers Key Topic 5 Analyze And Solve Systems Of Linear Equations

Topic Essential Question
What does it mean to solve a system of linear equations?
Answer:
A system of linear equations is just a set of two or more linear equations. In two variables (x and y), the graph of a system of two equations is a pair of lines in the plane

Topic 5 ënVision STEM Project

Did You Know?
After the Boston Tea Party of 1773, many Americans switched to drinking coffee rather than tea because drinking tea was considered unpatriotic.

Although Brazil is the largest coffee-producing nation in the world, Americans combine to drink 0.2% more coffee each year than Brazilians. The third-ranked nation for total coffee consumption, Germany, consumes approximately 44% as much coffee as either the United States or Brazil.

The United States consumes the most coffee by total weight, but Americans do not drink the most coffee per capita. People in northern European countries like Finland, Norway, and Holland drink more than twice as much coffee as their American counterparts each day.

In some coffee-producing nations, millions of acres of forest are cleared to make space for coffee farming. Sustainable farms grow coffee plants in natural growing conditions without chemicals and with minimal waste.

Coffee beans are actually seeds that are harvested from cherries that grow on coffee plants in tropical climates.

Your Task: Daily Grind

Coffee roasters create coffee blends by mixing specialty coffees with less expensive coffees in order to create unique coffees, reduce costs, and provide customers with consistent flavor. You and your classmates will explore coffee blends while considering the environmental and economic impact of the coffee trade.

### Topic 6 GET READY!

Review What You Know!

Vocabulary
Choose the best term from the box to complete each definition.
linear equation
parallel
slope
y-intercept

Question 1.
The value of m in the equation y = mx + b represents the __________ .
Answer:
We know that,
The value of m in the equation
y = mx + b
represents the “Slope”
Hence, from the above,
We can conclude that the best term from the box to complete the given definition is “Slope”

Question 2.
When lines are the same distance apart over their entire lengths, they are _________.
Answer:
We know that,
When lines are the same distance apart over their entire lengths, they are “Parallel”
Hence, from the above,
We can conclude that the best term from the box to complete the given definition is “Parallel”

Question 3.
The _________ is the value b in the equation y = mx + b.
Answer:
We know that,
The “y-intercept” is the value of b in the equation
y = mx + b
Hence, from the above,
We can conclude that the best term from the box to complete the given definition is “y-intercept”

Question 4.
A __________ is a relationship between two variables that gives a straight line when graphed.
Answer:
We know that,
A “Linear relationship” is a relationship between two variables that gives a straight line when graphed
Hence, from the above,
We can conclude that the best term from the box to complete the given definition is “Linear relationship”

Identifying Slope and y-Intercept

Identify the slope and the y-intercept of the equation.
Question 5.
y = 2x – 3
slope = _________
y-intercept = ________
Answer:
The given equation is:
y = 2x – 3
Compare the above equation with
y = mx + b
Where,
m is the slope
b is the y-intercept
Hence, from the above,
We can conclude that
The slope is: 2
The y-intercept is: -3

Question 6.
y =-0.5x + 2.5
slope = _________
y-intercept = ________
Answer:
The given equation is:
y = -0.5x + 2.5
Compare the above equation with
y = mx + b
Where,
m is the slope
b is the y-intercept
Hence, from the above,
We can conclude that
The slope is: -0.5
The y-intercept is: 2.5

Question 7.
y – 1 = -x
slope = _________
y-intercept = ________
Answer:
The given equation is:
y – 1 = -x
Add 1 on both sides
So,
y = -x + 1
Now,
Compare the above equation with
y = mx + b
Where,
m is the slope
b is the y-intercept
Hence, from the above,
We can conclude that
The slope is: -1
The y-intercept is: 1

Graphing Linear Equations

Graph the equation.
Question 8.
y = $$\frac{2}{3}$$x – 2

Answer:
The given equation is:
y = $$\frac{2}{3}$$x – 2
Hence,
The representation of the given equation in the coordinate plane is:

Question 9.
y = -2x + 1

Answer:
The given equation is:
y = -2x + 1
Hence,
The representation of the given equation in the coordinate plane is:

Solving Equations for Variables

Solve the equation for y.
Question 10.
y – x = 5
Answer:
The given equation is:
y – x = 5
Add x on both sides
So,
y – x + x = 5 + x
y = x + 5
Hence, from the above,
We can conclude that the value of y is:
y = x + 5

Question 11.
y + 0.2x = -4
Answer:
The given equation is:
y + 0.2x = -4
Subtract with 0.2x on both sides
So,
y + 0.2x – 0.2x = -4 – 0.2x
y = -0.2x – 4
Hence, from the above,
We can conclude that the value of y is:
y = -0.2x – 4

Question 12.
–$$\frac{2}{3}$$x + y = 8
Answer:
The given equation is:
–$$\frac{2}{3}$$x + y = 8
Add with $$\frac{2}{3}$$x on both sides
So,
–$$\frac{2}{3}$$x + $$\frac{2}{3}$$x + y = 8 + $$\frac{2}{3}$$x
y = $$\frac{2}{3}$$x + 8
Hence, from the above,
We can conclude that the value of y is:
y = $$\frac{2}{3}$$x + 8

Language Development

Complete the fishbone map by writing key terms or phrases related to systems of linear equations on each diagonal. Connect supporting ideas on the horizontal lines.

### Topic 5 PICK A PROJECT

PROJECT 5A
What is the most interesting discussion you’ve had with your classmates?
PROJECT: WRITE A SPEECH FOR A DEBATE

PROJECT 5B
What can you compare with a Venn diagram?
PROJECT: DRAW A VENN DIAGRAM

PROJECT 5C
What can you do with a smartphone?
PROJECT: CHOOSE A CELL PHONE PLAN

PROJECT 5D
If you made a stained-glass window, what colors and shapes would you use?
PROJECT: MAKE A MODEL OF A STAINED-GLASS WINDOW

### Lesson 5.1 Estimate Solutions by Inspection

Solve & Discuss It!
Draw three pairs of lines, each showing a different way that two lines can intersect or not intersect. How are these pairs of lines related?

I can… find the number of solutions of a system of equations by inspecting the equations.
Answer:
The representation of three pairs of lines, each showing a different way that two lines can intersect or not intersect is:

Now,
Parallel lines:
The lines that do not intersect each othse and have the same slope but different y-intercepts
Perpendicular lines:
The lines that intersect each other with the product of the slopes of lines -1 and different y-intercepts
Intersecting lines:
The lines that are neither parallel nor perpendicular are called “Intersecting lines”
Note:
All perpendicular lines should be intersecting lines but all intersecting lines should not be perpendicular lines

Focus on math practices
Look for Relationships Is it possible for any of the pairs of lines drawn to have exactly two points in common? Explain.
Answer:
We know that,
Any pair of lines should intersect at only 1 common point
Hence, from the above,
We can conclude that It is not possible for any of the pairs of lines drawn to have exactly two points in common

Essential Question
How are slopes and y-intercepts related to the number of solutions of a system of linear equations?
Answer:
When slopes and y-intercepts are different, the two lines intersect at one point,
There is only 1 solution.
When slopes are the same, the lines may be either parallel (different y-intercepts)
There are no solutions
If the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
So, there are infinitely many solutions

Try It!

How many solutions does this system of equations have? Explain.
y = x + 1
y = 2x + 2
The system of equations has ________ solution. The equations have _______ slopes, so the lines intersect at ________ point.
Answer:
The given system of equations are:
y = x + 1 —–(1)
y = 2x + 2 —-(2)
Compare the given system of equations with
y = mx + b
So,
For the first equation,
m = 1, b = 1
For the second equation,
m = 2 and b = 2
Now,
When we observe the slopes and y-intercepts, they are different
We know that,
When slopes and y-intercepts are different, the two lines intersect at one point,
There is only 1 solution.
Hence, from the above,
We can conclude that
The given system of equations have only 1 solution
The equation has 2 slopes
The lines intersect at (-1, 0)

Convince Me!
The equations of a system have the same slopes. What can you determine about the solution of the system of equations?
Answer:
It is given that the equations of a system have the same slopes
Hence,
When slopes are the same and the y-intercepts are different, the lines may be either parallel
So,
There are no solutions
If the lines have the same slopes and the same y-intercepts, they intersect at all points,
So,
There are infinitely many solutions

Try It!

How many solutions does each system of equations have? Explain.
a. y = -3x + 5
y = -3x – 5
Answer:
The given system of equations are:
y = -3x + 5 ——(1)
y = -3x – 5 ——-(2)
Compare the given system of equations with
y = mx + b
So,
For the first equation,
m = -3, b = 5
For the second equation,
m = -3, b = -5
Now,
From the above,
We can observe that there are the same slopes but different y-intercepts
We know that,
When slopes are the same and the y-intercepts are different, the lines may be either parallel and there are no solutions
Hence, from the above,
We can conclude that there are no solutions for the given system of equations

b. y = 3x + 4
5y – 15x – 20 = 0
Answer:
The given system of equations are:
y = 3x + 4
5y – 15x – 20 = 0
Now,
5y = 15x + 20
Divide by 5 into both sides
So,
y = 3x + 4
So,
The given system of equations are:
y = 3x + 4 ——-(1)
y = 3x + 4 ——–(2)
Now,
Compare the given system of equations with
y = mx + b
So,
For the first equation,
m = 3, b = 4
For the second equation,
m = 3, b = 4
Now,
From the above,
We can observe that there are the same slopes and the same y-intercepts
We know that,
When slopes and the y-intercepts are the same,
The lines have infinitely many solutions
Hence, from the above,
We can conclude that there are infinitely many solutions for the given system of equations

KEY CONCEPT

You can inspect the slopes and y-intercepts of the equations in a system of linear equations in order to determine the number of solutions of the system.
One Solution
y = 2x + 4
y = 3x – 1
The slopes are different. The lines intersect at 1 point.

No Solution
y = 3x + 4
y = 3x + 5
The slopes are the same, and the y-intercepts are different. The lines are parallel.

Infinitely Many Solutions
y = 3x + 4
y = 4 + 3x
The slopes are the same, and the y-intercepts are the same. The lines are the same.

Do You Understand?
Question 1.
Essential Question How are slopes and y-intercepts related to the number of solutions of a system of linear equations?
Answer:
When slopes and y-intercepts are different, the two lines intersect at one point,
There is only 1 solution.
When slopes are the same, the lines may be either parallel (different y-intercepts)
There are no solutions
If the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
So, there are infinitely many solutions

Question 2.
Construct Arguments Macy says that any time the equations in a system have the same y-intercept, the system has infinitely many solutions. Is Macy correct? Explain.
Answer:
It is given that
Macy says that any time the equations in a system have the same y-intercept, the system has infinitely many solutions.
Now,
Consider two equations with the same y-intercept and different slopes.
y = 2x + 3
y = 5x + 3
Compare the above system of equation with
y = mx + b
Hence,
We can say that
This system has only one solution.
Now,
Consider another system of equations with the same y-intercepts and the same slopes
y = x + 7
y = x + 7
Compare the above system of equation with
y = mx + b
Hence,
We can say that
This system has infinitely many solutions.
Hence, from the above,
We can conclude that Macy is not correct

Question 3.
Use Structure How can you determine the number of solutions of a system of linear equations by inspecting its equations?
Answer:
By inspecting the equations, the number of solutions can be determined as mentioned below:
When slopes and y-intercepts are different, the two lines intersect at one point,
There is only 1 solution.
When slopes are the same, the lines may be either parallel (different y-intercepts)
There are no solutions
If the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
So, there are infinitely many solutions

Do You Know How?
Question 4.
Kyle has x 3-ounce blue marbles and a 5-ounce green marble. Lara has x 5-ounce green marbles and a 3-ounce blue marble. Is it possible for Kyle and Lara to have the same number of green marbles and the same total bag weight, y? Explain.

Answer:
It is given that
Kyle has x 3-ounce blue marbles and a 5-ounce green marble. Lara has x 5-ounce green marbles and a 3-ounce blue marble.
Now,
The total number of marbles = The total number of green marbles + The total number of blue marbles
So,
For Kyle,
The total number of marbles = x+ 1
For Lara,
The total number of marbles = x + 1
Now,
From the total number of marbles,
We can observe that Kyle and Lara have the same number of marbles
Since Kyla and Lara have the same number of marbles,
The total weight of the marbles will also be the same
Hence, from the above,
We can conclude that
Kyle and Lara have the same number of green marbles and the same total bag weight, y

Question 5.
How many solutions does this system of linear equations have? Explain.
$$\frac{1}{2}$$x = y
y = $$\frac{1}{2}$$x + 3
Answer:
The given system of equations are:
$$\frac{1}{2}$$x = y ——(1)
y = $$\frac{1}{2}$$x + 3 —–(2)
Now,
Comapre the given system of equations with
y = mx + b
So,
For the first equation,
m = $$\frac{1}{2}$$, b = 0
For the second equation,
m = $$\frac{1}{2}$$, b = 3
We know that,
When slopes are the same, the lines may be either parallel (different y-intercepts)
There are no solutions
Hence, from the above,
We can conclude that the given system of equations have no solutions

Question 6.
How many solutions does this system of linear equations have? Explain.
3y + 6x = 12
8x + 4y = 16
Answer:
The given system of equations are:
3y + 6x = 12
8x + 4y = 16
So,
3y = -6x + 12
Divide by 3 into both sides
So,
y = -2x + 4
So,
4y = -8x + 16
Divide by 4 into both sides
So,
y = -2x + 4
So,
The required system of equations are:
y = -2x + 4 —–(1)
y = -2x + 4 —–(2)
We know that,
If the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
So, there are infinitely many solutions
Hence, from the above,
We can conclude that the given system of equations have infinitely many solutions

Practice & Problem Solving

Question 7.
Leveled Practice Two rovers are exploring a planet. The system of equations below shows each rover’s elevation, y, at time x. What conclusion can you reach about the system of equations?

Rover A: y = 1.9x – 8
Rover B: 7y = 13.3x – 56
The slope for the Rover A equation is _________ the slope for the Rover B equation.
The y-intercepts of the equations are ___________.
The system of equations has __________ solution(s).
Answer:
It is given that
Two rovers are exploring a planet. The system of equations below shows each rover’s elevation, y, at time x.
Now,
The given system of equation are:
Rover A: y = 1.9x – 8 ——(1)
Rover B: 7y = 13.3x – 56
Now,
Divide the equation of Rover B with 7
So,
Rover B: y = 1.9x – 8 ——(2)
Now,
When we compare the given system of equations with
y = mx + b
For Rover A,
m = 1.9, b = -8
For Rover B,
m = 1.9, b = -8
We know that,
If the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
So, there are infinitely many solutions
Hence, from the above,
We can conclude that the given system of equations ahve infinitely many solutions

Question 8.
How many solutions does this system have?
y = x – 3
4x – 10y = 6
Answer:
The given system of equations are:
y = x – 3 ——(1)
4x – 10y = 6
So,
10y = 4x – 6
Divide by 10 into both sides
So,
y = 0.4x – 0.6 ——(2)
Now,
Compare the given system of equations with
y = mx + b
So,
For the first equation,
m = 1, b = -3
For the second equation,
m = 0.4, b = -0.6
We know that,
When slopes and y-intercepts are different, the two lines intersect at one point,
There is only 1 solution.
Hence, from the above,
We can conclude that the given system of equations have only 1 solution

Question 9.
How many solutions does this system have?
x + 3y = 0
12y = -4x
Answer:
The given system of equations are:
x + 3y = 0
12y = -4x
Now,
3y = -x + 0
y = –$$\frac{1}{3}$$x + 0 ——(1)
12y = -4x
y = –$$\frac{4}{12}$$x
y = –$$\frac{1}{3}$$x ——-(2)
Now,
Compare the given system of equations with
y = mx + b
So,
For the first equation,
m = –$$\frac{1}{3}$$, b = 0
For the second equation,
m = –$$\frac{1}{3}$$, b = 0
We know that,
If the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
So, there are infinitely many solutions
Hence, from the above,
We can conclude that the given system of equations ahve infinitely many solutions

Question 10.
What can you determine about the solution(s) of this system?
-64x + 96y = 176
56x – 84y = -147
Answer:
The given system of equations are:
-64x + 96y = 176
56x – 84y = -147
So,
96y = 64x + 176
84y = 56x + 147
Now,
Divide by 96 into both sides
y = $$\frac{2}{3}$$x + $$\frac{11}{6}$$ ——(1)
Now,
Divide by 84 into both sides
y = $$\frac{2}{3}$$x + $$\frac{7}{4}$$ ——-(2)
Now,
Compare the above equations with
y = mx + b
So,
For the first equation,
m = $$\frac{2}{3}$$, b = $$\frac{11}{6}$$
For the second equation,
m = $$\frac{2}{3}$$, b = $$\frac{7}{4}$$
We know that,
When slopes are the same, the lines may be either parallel (different y-intercepts)
There are no solutions
Hence, from the above,
We can conclude that the given system of equations have no solutions

Question 11.
Determine whether this system of equations has one solution, no solution, or infinitely many solutions.
y = 8x + 2
y = -8x + 2
Answer:
The given system of equations are:
y = 8x + 2 —–(1)
y = -8x + 2 —–(2)
Now,
Compare the given system of equations with
y = mx + b
So,
For the first equation,
m = 8, b = 2
For the second equation,
m = -8, b = 2
We know that,
When slopes and y-intercepts are different or y-intercepts are the same, the two lines intersect at one point,
There is only 1 solution.
Hence, from the above,
We can conclude that the given system of equations have only 1 solution

Question 12.
Construct Arguments Maia says that the two lines in this system of linear equations are parallel. Is she correct? Explain.
2x + y = 14
2y + 4x = 14
Answer:
It is given that
Maia said the below system of linear equations to be paralle
Now,
The given system of equations are:
2x + y = 14
2y + 4x = 14
Now,
y = -2x + 14 —-(1)
2y = -4x + 14
Divide by 2 into both sides
So,
y = -2x + 7 —–(2)
Now,
Compare the given system of equations with
y = mx + b
So,
For the first equation,
m = -2, b = 14
For the second equation,
m = -2, b = 7
We know that,
For the lines to be parallel, the slopes have to be the same and the y-intercepts be different
Hence, from the above,
We can conclude that Maia is correct

Question 13.
Reasoning Describe a situation that can be represented by using this system of equations. Inspect the system to determine the number of solutions and interpret the solution within the context of your situation.
y = 2x + 10
y = x + 15
Answer:
The given system of equations are:
y = 2x + 10 — (1)
y = x + 15 —–(2)
Now,
Compare the above system of equations with
y = mx + b
So,
For the first equation,
m = 2, b = 10
For the second equation,
m = 1, b = 15
We know that,
When the slopes and y-intercepts are different for a system of linear equations,
There is only 1 solution
Hence, from the above,
We can conclude that the given system of equations has only 1 solution

Question 14.
Look for Relationships Does this system have one solution, no solutions, or infinitely many solutions? Write another system of equations with the same number of solutions that uses the first equation only.
12x + 51y = 156
-8x – 34y = -104
Answer:
The given system of equations are:
12x + 51y = 156
-8x – 34y = -104
So,
51y = -12x + 156
Divide by 51 into both sides
So,
y = –$$\frac{4}{17}$$x + $$\frac{52}{17}$$ —-(1)
So,
8x + 34y = 104
34y = -8x + 104
Divide by 34 into both sides
So,
y = –$$\frac{4}{17}$$x + $$\frac{52}{17}$$ ——-(2)
Now,
Compare the above equations with
y = mx + b
So,
For the first equation,
m = –$$\frac{4}{17}$$, b = $$\frac{52}{17}$$
For the second equation,
m = –$$\frac{4}{17}$$, b = $$\frac{52}{17}$$
We know that,
When we have the same slopes and the same y-intercepts, the solutions for a system of linear equations are infinite
Now,
Another system of equations with the first equation as one of the equations is:
12x + 51y = 156
48x + 204y = 624
Hence, from the above,
We can conclude that there are infinitely many solutions for the given system of equations

Question 15.
The equations represent the heights, y, of the flowers, in inches, after x days. What does the y-intercept of each equation represent? Will the flowers ever be the same height? Explain.

Answer:
It is given that
The equations represent the heights, y, of the flowers, in inches, after x days.
Now,
The given system of equations are:
y = 0.7x + 2 —–(1)
y = 0.4x + 2 —–(2)
Now,
Compare the above system of equations with
y = mx + b
So,
For the first equation,
m = 0.7, b = 2
For the second equation,
m = 0.4, b = 2
Now,
We know that
b represents the y-intercept or the initial value
Now,
When we compare the slopes of the equations,
We know that,
The slopes are different
Hence, from the above,
We can conclude that
The y-intercept of each equation represents the initial height of the flowers
The heights of the flowers are not the same

Question 16.
Does this system have one solution, no solution, or an infinite number of solutions?
4x + 3y = 8
8x + y = 2
Answer:
The given system of equations are:
4x + 3y = 8
8x + y = 2
Now,
3y = -4x + 8
Divide by 3 into both sides
So,
y = –$$\frac{4}{3}$$x + $$\frac{8}{3}$$ —-(1)
So,
y = -8x + 2 —–(2)
Now,
Compare the above equations with
y = mx + b
So,
For the equation 1,
m = –$$\frac{4}{3}$$, b = $$\frac{8}{3}$$
For the equation 2,
m = -8, b = 2
We know that,
When the slopes and y-intercepts of a system of equations are different, the system has only a solution
Hence, from the above,
We can conclude that the given system of equations has only 1 solution

Question 17.
Higher-Order Thinking Under what circumstances does the system of equations Qx + Ry = S and y = Tx + S have infinitely many solutions?
Answer:
The given system of equations are:
Qx + Ry = S
y = Tx + S
So,
Ry = -Qx + S
Divide by R into both sides
So,
y = –$$\frac{Q}{R}$$x + $$\frac{S}{R}$$ —–(1)
y = Tx + S —–(2)
Now,
For the given system of equations to have infinitely many solutions,
The slopes must be the same
The y-intercepts must be the same
So,
–$$\frac{Q}{R}$$ = T
$$\frac{S}{R}$$ = S
Hence, from the above,
We can conclude that the given system of equations will have infinitely many solutions when
–$$\frac{Q}{R}$$ = T
$$\frac{S}{R}$$ = S

Assessment Practice
Question 18.
By inspecting the equations, what can you determine about the solution(s) of this system?
12y = 9x + 33
20y = 15x + 55
Answer:
The given system of equations are:
12y = 9x + 33 —-(1)
20y = 15x + 55 —-(2)
So,
Divide eq (1) by 12 into both sides
So,
y = $$\frac{3}{4}$$x + $$\frac{11}{4}$$ —-(3)
So,
Divide eq (2) by 20 into both sides
So,
y = $$\frac{3}{4}$$x + $$\frac{11}{4}$$ —–(4)
Now,
Compare the above equations with
y = mx + b
So,
For the 3rd equation,
m = $$\frac{3}{4}$$, b = $$\frac{11}{4}$$
For the 4th equation,
m = $$\frac{3}{4}$$, b = $$\frac{11}{4}$$
We know that,
When the slopes and the y-intercepts of a system of equations are different, the system has only 1 solution
Hence, from the above,
We can conclude that the given system of equations has infinitely many solutions

Question 19.
Choose the statement that correctly describes how many solutions there are for this system of equations.
y = $$\frac{2}{3}$$x + 3
y = $$\frac{5}{4}$$x + 3
A. Infinitely many solutions because the slopes are equal and the y-intercepts are equal
B. Exactly one solution because the slopes are equal but the y-intercepts are NOT equal
C. No solution because the slopes are equal and the y-intercepts are NOT equal
D. Exactly one solution because the slopes are NOT equal
Answer:
The given system of equations are:
y = $$\frac{2}{3}$$x + 3 —-(1)
y = $$\frac{5}{4}$$x + 3 —–(2)
Now,
Compare the abovee quations with
y = mx + b
So,
For the first equation,
m = $$\frac{2}{3}$$, b = 3
Fpr the second equation,
m = $$\frac{5}{4}$$, b = 3
We know that
When the slopes are different and the y-intercepts are the same or different, the system of equations has only 1 solution
Hence, from the above,
We can conclude that option D matches with the given system of equations

### Lesson 5.2 Solve Systems by Graphing

Explore It!
Beth and Dante pass by the library as they walk home using separate straight paths.

I can… find the solution to a system of equations using graphs.

A. Model with Math The point on the graph represents the location of the library. Draw and label lines on the graph to show each possible path to the library.

Answer:
It is given that
Beth and Dante pass by the library as they walk home using separate straight paths.
Now,
In the graph,
the location of the graph is given
Hence,
The representation of one of the paths of Beth and Dante that are passing through the library in a straight line is:

B. Write a system of equations that represents the paths taken by Beth and Dante.
Answer:
From part (a),
The representation of one of the paths of Beth and Dante that are passing through the library in a straight line is:

So,
From the graph,
The points that are passing through the path of Beth is: (-4, 0), (0, 2)
Now,
Compare the given points with (x1, y1), (x2, y2)
We know that,
Slope (m) = y2 – y1 / x2 – x1
So,
Slope (m) = $$\frac{2 – 0}{0 + 4}$$
= $$\frac{2}{4}$$
= $$\frac{1}{2}$$
We know that,
The form of the linear equation in the slope-intercept form is:
y = mx + b
So,
y = $$\frac{1}{2}$$x + b
2y = x + 2b
Substitute (-4, 0) (or) (0, 2) in the above equation
So,
2b = 4
b = 2
So,
The required equation is:
2y = x + 4
Now,
The points that are passing through the path of Dante is: (-1, -2), (0, 0)
Now,
Compare the given points with (x1, y1), (x2, y2)
We know that,
Slope (m) = y2 – y1 / x2 – x1
So,
Slope (m) = $$\frac{0 + 2}{0 + 1}$$
= $$\frac{2}{1}$$
= 2
We know that,
The form of the linear equation in the slope-intercept form is:
y = mx + b
So,
y = 2x + b
Substitute (-1, -2) (or) (0, 0) in the above equation
So,
b = 0
So,
The required equation is:
y = 2x
Hence, from the above,
We can conclude that
The required system of linear equations are:
2y = x + 4
y = 2x

Focus on math practices
Reasoning What does the point of intersection of the lines represent in the situation?
Answer:
Point of intersection means the point at which two lines intersect. These two lines are represented by the equation
a1x + b1y + c1= 0 and
a2x + b2y + c2 = 0, respectively.
By solving the two equations, we can find the solution for the point of intersection of two lines.

Essential Question
How does the graph of a system of linear equations represent its solution?
Answer:
Each shows two lines that make up a system of equations. If the graphs of the equations intersect, then there is one solution that is true for both equations. If the graphs of the equations do not intersect (for example, if they are parallel), then there are no solutions that are true for both equations.

Try It!

Solve the system by graphing.

y = 3x + 5
y = 2x + 4
The solution is the point of intersection (______, _______)
Answer:
The given system of equations are:
y = 3x + 5
y = 2x + 4
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the solution or the intersection point of the given system of equations is: (-1, 2)

Convince Me!
How does the point of intersection of the graphs represent the solution of a system of linear equations?
Answer:
When you graph an equation, each point (x,y) on the line satisfies the equation. Therefore, when 2 lines intersect, the coordinates of the intersection point satisfy both equations, i.e. the intersection point represents the solution of the set

Try It!

Solve each system by graphing. Describe the solutions.

a. 5x + y = -3
10x + 2y = -6
Answer:
The given system of equations are:
5x + y = -3
10x + 2y = -6
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the given system of equations has infinitely many solutions since both equations are on the same line

b. x + y = 7
2x + 6y = 12
Answer:
The given system of equations are:
x + y = 7
2x + 6y = 12
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the given system of equations has only one solution i.e., (7.5, -0.5)

KEY CONCEPT

The solution of a system of linear equations is the point of intersection of the lines defined by the equations.

Do You Understand?
Question 1.
Essential Question How does the graph of a system of linear equations represent its solution?
Answer:
Each shows two lines that make up a system of equations. If the graphs of the equations intersect, then there is one solution that is true for both equations. If the graphs of the equations do not intersect (for example, if they are parallel), then there are no solutions that are true for both equations.

Question 2.
Reasoning If a system has no solution, what do you know about the lines being graphed?
Answer:
If a system has no solution, it is said to be inconsistent. The graphs of the lines do not intersect, so the graphs are parallel and there is no solution.

Question 3.
Construct Arguments in a system of linear equations, the lines described by each equation have the same slopes. What are the possible solutions to the system? Explain.
Answer:
If the given system of equations has the same slope, then
For the same slope and the same y-intercept – There ate infinitely many solutions
For the same slope and different outputs – There are no solutions

Do You Know How?
In 4-6, graph each system of equations and find the solution.
Question 4.
y = -3x – 5
y = 9x + 7

Answer:
The given system of equations are:
y = -3x – 5
y = 9x + 7
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the solution for the given system of equations is: (-1, -2)

Question 5.
y = 2x – 5
6x + 3y = -15

Answer:
The given system of equations are:
y = 2x – 5
6x + 3y = -15
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the solution for the given system of equations is: (0, -5)

Question 6.
y = -4x + 3
8x + 2y = 8

Answer:
The given system of equations are:
y = -4x + 3
8x + 2y = 8
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that there are no solutions for the given system of equations

Practice & Problem Solving

In 7 and 8, graph each system of equations to determine the solution.
Question 7.
x + 4y = 8
3x + 4y = 0

Answer:
The given system of equations are:
x + 4y = 8
3x + 4y = 0
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the solution for the given system of equations is: (-4, 3)

Question 8.
2x – 3y = 6
4x – y = 12

Answer:
The given system of equations are:
2x – 3y = 6
4x – y = 12
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the solution for the given system of equations is: (3, 0)

Question 9.
The total cost, c, of renting a canoe for n hours can be represented by a system of equations.

a. Write the system of equations that could be used to find the total cost, c, of renting a canoe for n hours.
Answer:
It is given that
The total cost, c, of renting a canoe for n hours can be represented by a system of equations.
Now,
The total cost for renting a canoe (y) = The cost of renting a canoe per hour × The number of hours (n) + Deposit
So,
For River Y,
y = 33n
For River Z,
y = 5n + 13
Hence, from the above,
We can conclude that the system of equations for the total cost of a canoe is:
y = 33n
y = 5n + 13

b. Graph the system of equations.

Answer:
From part (a),
The system of equations is:
y = 33n
y = 5n + 13
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the solution of the given system of equations is: (0.5, 16)

c. When would the total cost for renting a canoe be the same on both rivers? Explain.
Answer:
From the graph that is in part (b),
We can observe that the y-axis passes through 12
Hence,from the above,
We can conclude that the total cost for renting a canoe can be the same on both rivers after 12 hours

Question 10.
Graph the system of equations and determine the solution.
x + 2y = 4
4x + 8y = 64

Answer:
The given system of equations are:
x + 2y = 4
4x + 8y = 64
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that there are no solutions for the given system of equations since they are parallel

Question 11.
Graph the system of equations, then estimate the solution.
y = 1.5x + 1
y = -1.5x + 5.5

Answer:
The given system of equations are:
y = 1.5x + 1
y = -1.5x + 5.5
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the estimated solution for the given system of equations is: (1, 3)

In 12 and 13, graph and determine the solution of the system of equations.
Question 12.
-3y = -9x + 3
-6y = -18x – 12

Answer:
The given system of equations are:
-3y = -9x + 3
-6y = -18x – 12
So,
The representation of the given syetem of equations in the coordinate plane is:

Hence, from the above,
We can conclude that there are no solutions for the given system of equations since they are parallel lines

Question 13.
x + 5y = 0
25y = -5x

Answer:
The given system of equations are:
x + 5y = 0
25y = -5x
So,
The representation of the given system of equations i the coordinate plane is:

Hence, from the above,
We can conclude that there are infinitely many solutions for the given system of equations since both equations are in the same line

Question 14.
Higher Order Thinking The total cost, c, of making n copies can be represented by a system of equations.

a. Estimate how many copies you need to make for the total cost to be the same at both stores.
Answer:
It is given that
The total cost, c, of making n copies can be represented by a system of equations.
We know that,
1 dollar = 0.1 cent
So,
The total cost of making n copies (y) = The number of copies per hour × The number of copies + the cost of machine use
So,
For Store W,
y = 5n —— (1)
For Store Z,
y = 0.20n + 2 —- (2)
Now,
From the given graph,
We can observe that the given system of equations pass through (6, 5)
Hence, from the above,
We can conclude that we have to make 6 copies for the total cost to be the same at both stores

b. If you have to make a small number of copies, which store should you go to? Explain.
Answer:
To make a small number of copies,
We will see which store gives us the less cost for printing the number of copies
So,
From the given information,
We can observe that Store Z gives us the less cost for printing the more number of copies
Hence, from the above,
We can conclude that we will goto Store Z if we have to make small number of copies

Assessment Practice
Question 15.
Consider the following system of equations.
y =-3x + 6
y = 3x – 12
Which statement is true about the system?
A. The graph of the system is a pair of lines that do not intersect.
B. The graph of the system is a pair of lines that intersect at exactly one point.
C. The graph of the system is a pair of lines that intersect at every point.
D. The system has infinitely many solutions.
Answer:
The given system of equations are:
y = -3x + 6
y = 3x – 12
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that option B matches with the description of the above graph

Question 16.
What is the solution of the system of equations?

Answer:
The given graph is:

So,
From the given graph,
We can observe that
The intersection point is: (2.5, -3)
Hence, from the above,
We can conclude that the solution for the given system of equations are: (2.5, 3)

### Topic 5 MID-TOPIC CHECKPOINT

Question 1.
Vocabulary How can you determine the number of solutions of a system by looking at the equations? Lesson 5-1
Answer:
A system of two equations can be classified as follows: If the slopes are the same but the y-intercepts are different, the system has no solution. If the slopes are different, the system has one solution. If the slopes are the same and the y-intercepts are the same, the system has infinitely many solutions.

Question 2.
How many solutions does the system of equations have? Explain. Lesson 5-1
2x – 9y = -5
4x – y = 2
Answer:
The given system of equations are:
2x – 9y = -5
4x – y = 2
So,
9y = 2x + 5
y = $$\frac{2}{9}$$x + $$\frac{5}{9}$$ —— (1)
So,
y = 4x – 2 —— (2)
Now,
Compare the above equations with
y = mx + b
So,
For the first equation,
m = $$\frac{2}{9}$$, b = $$\frac{5}{9}$$
For the second equation,
m = 4, b = -2
We know that,
When the slopes and y-intercepts of a given system of equations are different, the system of equations has only one solution
Hence, from the above,
We can conclude that the given system of equations has only one solution

Question 3.
Graph the system of equations and find the solution. Lesson 5-2
y = 2x – 1
y = $$\frac{1}{2}$$x + 2

Answer:
The given system of equations are:
y = 2x – 1
y = $$\frac{1}{2}$$x + 2
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the solution of the given system of equations is: (2, 3)

Question 4.
One equation in a system is y =-3x + 7. Which equation gives the system no solution? Lesson 5-1
A. y = -3x + 7
B. y = 3x + 5
C. y = -3x + 5
D. y= $$\frac{1}{3}$$x – 7
Answer:
It is given that
One of the equations in a system of equations is:
y = -3x + 7
Now,
For a system of equations to have no solution,
The slopes must be equal but the y-intercepts must be different
So,
With the above description, we have 2 options matched but option A has the same slope and y-intercept as the first equation
Hence, from the above,
We can conclude that option C must be the other equation in the given system of equations

Question 5.
Finn bought 12 movie tickets. Student tickets cost $4, and adult tickets cost$8. Finn spent a total of $60. Write and graph a system of equations to find the number of student and adult tickets Finn bought. Lesson 5-2 Answer: It is given that Finn bought 12 movie tickets. Student tickets cost$4, and adult tickets cost $8. Finn spent a total of$60.
Now,
Let x be the number of student tickets
Let y be the number of adult tickets
So,
x + y = 12 —– (1) [The total number of tickets]
So,
4x + 8y = 60 —– (2) [The total cost of the tickets]
So,
The representation of the above equations in the coordinate plane is:

So,
From the above graph,
The intersection point of the graph is: (9, 3)
Hence, from the above,
We can conclude that
The number of student tickets is: 9
The number of adult tickets is: 3

Question 6.
What value of m gives the system infinitely many solutions? Lesson 5-1
-x + 4y = 32
y = mx + 8
Answer:
The given system of equations are:
-x + 4y = 32
y = mx + 8
So,
4y = x + 32
y = $$\frac{1}{4}$$x + 8 —– (1)
y = mx + 8 ——- (2)
Now,
For the given system of equations to have infinitely many solutions,
The slopes and the y-intercepts of the 2 equations must be equal
So,
m = $$\frac{1}{4}$$
Hence, from the above,
We can conclude that the value of m is: $$\frac{1}{4}$$

### Topic 5 MID-TOPIC PERFORMANCE TASK

Perpendicular lines intersect to form right angles. The system of equations below shows perpendicular lines.

PART A
How many solutions does the system have? Explain.
Answer:
The given graph is:

From the given graph,
We can observe that there is only 1 intersection point
Hence, from the above,
We can conclude that the given system of equations has only 1 solution

PART B
Identify the slope and y-intercept of each line. What do you notice about the slopes of the lines?
Answer:
From the given graph,
The system of equations is:
y = $$\frac{3}{4}$$x + 1 —— (1)
y = –$$\frac{4}{3}$$x – 2 —— (2)
Now,
Compare the above equations with
y = mx + b
So,
For the first equation,
m = $$\frac{3}{4}$$, b = 1
For the second equation,
m = –$$\frac{4}{3}$$, b = -2
Now,
From the slopes of the 2 equations,
We can observe that the product of the 2 slopes are equal to -1

PART C
What value of m makes the system show perpendicular lines? Explain.
y = $$\frac{1}{2}$$x + 8
y = mx – 6
Answer:
The given system of equations are:
y = $$\frac{1}{2}$$x + 8 —- (1)
y = mx – 6 —— (2)
Now,
Compare the above equations with
y = mx + b
So,
For the first equation,
m = $$\frac{1}{2}$$, b = 8
For the second equation,
m = m, b = -6
We know that,
For the system of equations to be perpendicular,
The product of the slopes must be equal to -1
So,
m1 . m2 = -1
So,
$$\frac{1}{2}$$m = -1
m = -2
Hence, from the above,
We ca conclude that the value of m is: -1

### Lesson 5.3 Solve Systems by Substitution

Explain It!
Jackson needs a taxi to take him to a destination that is a little over 4 miles away. He has a graph that shows the rates for two companies. Jackson says that because the slope of the line that represents the rates for On-Time Cabs is less than the slope of the line that represents Speedy Cab Co., the cab ride from On-Time Cabs will cost less.

I can… solve systems of equations using substitution.

A. Do you agree with Jackson? Explain.
Answer:
It is given that
Jackson needs a taxi to take him to a destination that is a little over 4 miles away. He has a graph that shows the rates for two companies. Jackson says that because the slope of the line that represents the rates for On-Time Cabs is less than the slope of the line that represents Speedy Cab Co., the cab ride from On-Time Cabs will cost less.
Now,
We know that,
For a linear graph,
Quantity 1 (The component of the x-axis) ∝ Quantity 2 (The component of the y-axis)
So,
From the given information,
Slope ∝ Cost
Hence, from the above information,
We can agree with Jackson

B. Which taxi service company should Jackson call? Explain your reasoning.
Answer:
We know that,
The taxi service company must be chosen according to the cost
Hence, from the above,
We can conclude that Jackson should call for an On-Time cabs taxi service since the cost is less

Focus on math practices
Be Precise Can you use the graph to determine the exact number of miles for which the cost of the taxi ride will be the same? Explain.
Answer:
From the given graph,
We can observe that the intersection point is the point where the cost of the taxi ride will be the same
So,
Corresponding to that intersection point, the distance will be calculated for the same cost of the taxi rides
Hence, from the above,
We can conclude that you can use the graph to determine the exact number of miles for which the cost of the taxi ride will be the same

Essential Question When is substitution a useful method for solving systems of equations?
Answer:
The substitution method is most useful for systems of 2 equations in 2 unknowns. The main idea here is that we solve one of the equations for one of the unknowns, and then substitute the result into the other equation.

Try It!

Brandon took a 50-question exam worth a total of 160 points. There were x two-point questions and y five-point questions. How many of each type of question were on the exam?
x + y = 50
2x + 5y = 160
y = _____ – ______
Substitute for y: 2x + 5(_____ – _____) = 160
2x + _____ – ______x = 160
x = ______ two-point questions
Substitute for x: _____ + y = 50
y = _____ five-point questions
Answer:
It is given that
Brandon took a 50-question exam worth a total of 160 points. There were x two-point questions and y five-point questions.
Now,
The given system of equations are:
x + y = 50 —– (1)
2x + 5y = 160 —–(2)
So,
From eq (1),
y = 50 – x
Now,
Substitute y in eq (2)
So,
2x + 5 (50 – x) = 160
2x + 5 (50) – 5x = 160
2x + 250 – 5x = 160
-3x = 160 – 250
-3x = -90
3x = 90
x = $$\frac{90}{3}$$
x = 30
So,
y = 50 – x
y = 50 – 30
y = 20
Hence from the above,
We can conclude that
The number of 2-point questions is: 30
The number of 5-point questions is: 20

Convince Me!
How do you know which equation to choose to solve for one of the variables?
Answer:
The idea of substitution is that if one variable lets you express one variable in terms of the other, you can substitute that expression for the variable in the other equation. That way the second equation only has one variable, and you can solve that

Try It!

Use substitution to solve each system of equations. Explain.
a. y + $$\frac{1}{2}$$x = 3
4y + 2x = -6
Answer:
The given system of equations are:
y + $$\frac{1}{2}$$x = 3 —- (1)
4y + 2x = -6 ——- (2)
So,
From eq (1),
y = 3 – $$\frac{1}{2}$$x
Now,
Substitute y in eq (2)
So,
4 (3 – $$\frac{1}{2}$$x) + 2x = -6
4 (3) – 2x + 2x = -6
12 = -6
Hence, from the above,
We can conclude that there is no solution for the given system of equations

b. y = $$\frac{1}{4}$$x – 2
8y – 2x = -16
Answer:
The given system of equations are:
y = $$\frac{1}{4}$$x – 2 —– (1)
8y – 2x = -16
Now,
Substitute eq (1) in eq (2)
So,
8 ($$\frac{1}{4}$$x – 2) – 2x = -16
2x – 8 (2) – 2x = -16
-16 = -16
16 = 16
Hence, from the above,
We can conclude that there are infinitely many solutions for the given system of equations

KEY CONCEPT

Systems of linear equations can be solved algebraically. When one of the equations can be easily solved for one of the variables, you can use substitution to solve the system efficiently.
STEP 1 Solve one of the equations for one of the variables. Then substitute the expression into the other equation and solve.
STEP 2 Solve for the other variable using either equation.

Do You Understand?
Question 1.
Essential Question when is substitution a useful method for solving systems of equations?
Answer:
The substitution method is most useful for systems of 2 equations in 2 unknowns. The main idea here is that we solve one of the equations for one of the unknowns, and then substitute the result into the other equation.

Question 2.
Generalize when using substitution to solve a system of equations, how can you tell when a system has no solution?
Answer:
When a system has no solution or an infinite number of solutions and we attempt to find a single, unique solution using an algebraic method, such as substitution, the variables will cancel out and we will have an equation consisting of only constants. If the equation is untrue then the system has no solution.

Question 3.
Construct Arguments Kavi solved the system of equations as shown. What mistake did Kavi make? What is the correct solution?
3x + 4y = 33
2x + y = 17
y = 17 – 2x
2x + (17 – 2x) = 17
2x + 17 – 2x = 17
2x – 2x + 17 = 17
17 = 17
Infinitely many solutions
Answer:
The given system of equations are:
3x + 4y = 33 —– (1)
2x + y = 17 —– (2)
From eq (2),
y = 17 – 2x
Now,
Substitute y in eq (1)
So,
3x + 4 (17 – 2x) = 33
3x + 4 (17) – 4 (2x) = 33
3x + 68 – 8x = 33
-5x = 33 – 68
-5x = -35
5x = 35
x = $$\frac{35}{5}$$
x = 7
Hence, from the above,
We can conclude that the mistake did by Kavi is the miswriting of eq (1)

Do You Know How?
In 4-6, solve each system using substitution.
Question 4.
y = $$\frac{1}{2}$$x + 4
x – y = 8
Answer:
The given system of equations are:
y = $$\frac{1}{2}$$x + 4 —– (1)
x – y = 8 —– (2)
Now,
From eq (2),
y = x – 8
Substitute y in eq (1)
So,
x – 8 = $$\frac{1}{2}$$x + 4
x – $$\frac{1}{2}$$x = 4 + 8
$$\frac{1}{2}$$x = 12
x = 12 (2)
x = 24
So,
y = x – 8
y = 24 – 8
y = 16
Hence, from the above,
We can conclude that the solution for the given system of equations is: (24, 16)

Question 5.
3.25x – 1.5y = 1.25
13x – y = 10
Answer:
The given system of equations are:
3.25x – 1.5y = 1.25 —- (1)
13x – y = 10 —-(2)
Now,
From eq (2),
y = 13x – 10
Now,
Substitute y in eq (1)
So,
3.25x – 1.5 (13x – 10) = 1.25
3.25x – 1.5 (13x) + 1.5(10) = 1.25
3.25x – 19.5x + 15 = 1.25
-16.25x = -13.75
16.25x = 13.75
x = 0.84
So,
y = 13x – 10
y = 13 (0.84) – 10
y = 0.92
Hence, from the above,
We can conclude that the solution for the given system of equations is: (0.84, 0.92)

Question 6.
y – 0.8x = 0.5
5y – 2.5 = 4x
Answer:
The given system of equations are:
y – 0.8x = 0.5 —- (1)
5y – 2.5 = 4x —— (2)
Now,
From eq (1),
y = 0.8x + 0.5
Now,
Substitute y in eq (2)
So,
5 (0.8x + 0.5) – 2.5 = 4x
5 (0.8x) + 5 (0.5) – 2.5 = 4x
4x + 2.5 – 2.5 = 4x
4x – 4x + 2.5 = 2.5
2.5 = 2.5
Hence, from the above,
We can conclude that the given system of equations has infinitely many solutions

Practice & Problem Solving

Leveled Practice In 7-9, solve the systems of equations.
Question 7.
Pedro has 276 more hits than Ricky. Use substitution to solve the system of equations to find how many hits Pedro, p, and Ricky, r, have each recorded.
p + r = 2,666
p = r + 276

It is given that Pedro has 276 more hits than Ricky
Now,
The given system of equations are:
p + r = 2,666
p = r + 276
Now,
STEP 1 Substitute for p to solve for r.
p + r = 2,666
r + 276 + r = 2,666
2r + 276 = 2,666
2r = 2,390
r = 1,195
STEP 2 Substitute for r to solve for p.
p = r + 276
p = 1,195 + 276
p = 1,471
Hence, from the above,
We can conclude that
Pedro has 1,471 hits, and Ricky has 1,195 hits.

Question 8.
2y + 4.4x = -5
y = -2.2x + 4.5
2 (-2.2x + 4.5) + 4.4x = -5
-4.4x + 9 + 4.4x = -5
9 = -5
Hence, from the above,
We can conclude that
The statement is not true. So, there is no solution.

Question 9.
x + 5y = 0
25y = -5x
x = -5y
25y = -5 (-5y)
25y = 25y
Hence, from the above,
We can conclude that
The statement is true. So, there are infinitely many solutions

Question 10.
On a certain hot summer day, 481 people used the public swimming pool. The daily prices are $1.25 for children and$2.25 for adults. The receipts for admission totaled $865.25. How many children and how many adults swam at the public pool that day? Answer: It is given that On a certain hot summer day, 481 people used the public swimming pool. The daily prices are$1.25 for children and $2.25 for adults. The receipts for admission totaled$865.25.
Now,
Let x be the number of children
Let y be the number of adults
So,
x + y = 481 —– (1) [The number of people that used the public swimming pool]
1.25x + 2.25y = 865.25 —– (2) [The receipts for admission]
Now,
From eq (1),
y = 481 – x
Substitute y in eq (2)
So,
1.25x + 2.25 (481 – x) = 865.25
1.25x + 2.25 (481) – 2.25 (x) = 865.25
1.25x + 1,082.25 – 2.25x = 865.25
-x = -217
x = 217
Now,
y = 481 – x
y = 481 – 217
y = 264
Hence,from the above,
We can conclude that
The number of children that swam at the pool is: 217
The number of adults that swam at the pool is: 264

Question 11.
Construct Arguments Tim incorrectly says that the solution of the system of equations is x = -9, y = -4.
6x – 2y = -6
11 = y – 5x
a. What is the correct solution?
Answer:
The given system of equations are:
6x – 2y = -6 —– (1)
11 = y – 5x —— (2)
Now,
From eq (2),
y = 5x + 11
Substitute y in eq (1)
So,
6x – 2 (5x + 11) = -6
6x – 2 (5x) – 2 (11) = -6
6x – 10x – 22 = -6
-4x = 16
x = –$$\frac{16}{4}$$
x = -4
Now,
y = 5x + 11
y = 5 (-4) + 11
y = -20 + 11
y = -9
Hence, from the above,
We can conclude that the correct solution for the given system of equations is: (-4, -9)

b. What error might Tim have made?
Answer:
It is given that
Tim incorrectly says that the solution of the system of equations is x = -9, y = -4.
But, from part (a),
We can observe that x = -4, y = -9
Hence, from the above,
We can conclude that the error that Tim has made is the reversal of the values of x and y in the solution

Question 12.
The number of water bottles, y, filled in x minutes by each of two machines is given by the equations below. Use substitution to determine if there is a point at which the machines will have filled the same number of bottles.
160x + 2y = 50
y + 80x = 50
Answer:
The given system of equations are:
160x + 2y = 50 —— (1)
y + 80x = 50 ——- (2)
Now,
From eq (2),
y = 50 – 80x
Now,
Substitute y in eq (1)
So,
160x + 2 (50 – 80x) = 50
160x + 2 (50) – 2 (80x) = 50
160x + 100 – 160x = 50
100 = 50
Hence, from the above,
We can conclude that at any point, the machines will not have filled the same number of bottles

Question 13.
a. Use substitution to solve the system below.
x = 8y – 4
x + 8y = 6
Answer:
The given system of equations are:
x = 8y – 4 —– (1)
x + 8y = 6 —– (2)
Now,
Substitute eq (1) in eq (2)
So,
8y – 4 + 8y = 6
16y – 4 = 6
16y = 10
y = $$\frac{16}{10}$$
y = 1.6
Now,
x = 8y – 4
x = 8 (1.6) – 4
x = 12.8 – 4
x = 8.8
Hence, from the above,
We can conclude that

b. Reasoning Which expression would be easier to substitute into the other equation in order to solve the problem? Explain.
Answer:
From part (a),
The given system of equations are:
x = 8y – 4 —– (1)
x + 8y = 6 —– (2)
Hence,
From the given equations,
In order to reduce the number of steps,
Eq (1) would be easier to substitute into the other equation in order to solve the problem

Question 14.
The perimeter of a frame is 36 inches. The length is 2 inches greater than the width. What are the dimensions of the frame?

Answer:
The given frame is:

From the given frame,
We can observe that it is in the form of a rectangle
Now,
We know that,
The perimeter of a rectangle = 2 (Length + Width)
So,
36 = 2 (W + 2 + W)
36 = 2 (2W + 2)
2W + 2 = $$\frac{36}{2}$$
2W + 2 = 18
2W = 18 – 2
2W = 16
W = $$\frac{16}{2}$$
W = 8
So,
L = W + 2
L = 8 + 2
L = 10
Hence, from the above,
We can conclude that the dimensions of the frame are:
The length of the frame is: 10 inches
The width of the frame is: 8 inches

Question 15.
Higher-Order Thinking The members of the city cultural center have decided to put on a play once a night for a week. Their auditorium holds 500 people. By selling tickets, the members would like to raise $2,050 every night to cover all expenses. Let d represent the number of adult tickets sold at$6.50. Let s represent the number of student tickets sold at $3.50 each. a. If all 500 seats are filled for a performance, how many of each type of ticket must have been sold for the members to raise exactly$2,050?
Answer:
It is given that
The members of the city cultural center have decided to put on a play once a night for a week. Their auditorium holds 500 people. By selling tickets, the members would like to raise $2,050 every night to cover all expenses. Let d represent the number of adult tickets sold at$6.50. Let s represent the number of student tickets sold at $3.50 each. Now, d + s = 500 —– (1) [The total number of people] 6.50d + 3.50s = 2,050 —– (2) [The total expenses] So, From eq (1), d = 500 – s Now, Substitute d in eq (1) So, 6.50 (500 – s) + 3.50s = 2,050 6.50 (500) – 6.50s + 3.50s = 2,050 3,250 – 3s = 2,050 -3s =-1,200 3s = 1,200 s = $$\frac{1,200}{3}$$ s = 400 So, d = 500 – s d = 500 – 400 d = 100 Hence, from the above, We can conclude that The number of student tickets is: 400 The number of adult tickets is: 100 b. At one performance there were three times as many student tickets sold as adult tickets. If there were 480 tickets sold at that performance, how much below the goal of$2,050 did ticket sales fall?
Answer:
It is given that there were three times as many student tickets as adult tickets
So,
s = 3d
Now,
d + s = 480 —– (1)
So,
d + 3d = 480
4d = 480
d = $$\frac{480}{4}$$
d = 120
So,
s = 3 (120)
s = 360
Now,
The equation for the total expenses is:
6.50d + 3.50s = 6.50 (120) + 3.50 (360)
= 780 + 1,260
= 2,040
Now,
The fall in the ticket sales = $2,050 –$2,040
The cost of each turkey beef sandwich is: $7 Question 14. Consider the system of equations. x – 3.1y = 11.5 -x + 3.5y = -13.5 a. Solve the system by elimination. Answer: The given system of equations are: x – 3.1y = 11.5 —- (1) -x + 3.5y = -13.5 —— (2) Now, Eq (1) – Eq (2) So, Hence, from the above, We can conclude that the solution for the given system of equations is: (-4, -5) b. If you solved this equation by substitution instead, what would the solution be? Explain. Answer: By using the substitution method, The solution for the given system of equations is: Hence, from the above, We can conclude that the solution for the substitution method is the same as the solution for the elimination method Question 15. Higher-Order Thinking Determine the number of solutions for this system of equations by inspection only. Explain. 3x + 4y = 17 21x + 28y = 109 Answer: The given system of equations are: 3x + 4y = 17 —– (1) 21x + 28y = 109 —— (2) So, Hence, from the above, We can conclude that there is no solution for the given system of equations Assessment Practice Question 16. Four times a number r plus half a number s equals 12. Twice the number r plus one fourth of the number s equals 8. What are the two numbers? Answer: It is given that Four times a number r plus half a number s equals 12. Twice the number r plus one-fourth of the number s equals 8. So, 4r + $$\frac{1}{2}$$s = 12 —- (1) 2r + $$\frac{1}{4}$$s = 8 —- (2) So, Hence, from the above, We can conclude that the values for the 2 numbers are not possible Question 17. Solve the system of equations. 3m + 3n = 36 8m – 5n = 31 Answer: The given system of equations are: 3m + 3n = 36 —- (1) 8m – 5n = 31 —– (2) So, Hence, from the above, We can conclude that the solution for the given system of equations is: (7, 5) 3-ACT MATH 3-Act Mathematical Modeling: Ups and Downs ACT 1 Question 1. After watching the video, what is the first question that comes to mind? Answer: Question 2. Write the Main Question you will answer. Answer: Question 3. Make a prediction to answer this Main Question. The person who wins took the ______. Answer: Question 4. Construct Arguments Explain how you arrived at your prediction. Answer: ACT 2 Question 5. What information in this situation would be helpful to know? How would you use that information? Answer: Question 6. Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically. Answer: Question 7. Model with Math Represent the situation using mathematics. Use your representation to answer the Main Question. Answer: Question 8. What is your answer to the Main Question? Does it differ from your prediction? Explain. Answer: ACT 3 Question 9. Write the answer you saw in the video. Answer: Question 10. Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference? Answer: Question 11. Make Sense and Persevere Would you change your model now that you know the answer? Explain. Answer: ACT 3 Extension Reflect Question 12. Model with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question? Answer: Question 13. Reason Abstractly A classmate solved the problem using equations with independent variable a and dependent variable b. What do these variables represent in the situation? Answer: SEQUEL Question 14. Generalize Write an equation or inequality to represent all numbers of flights for which the elevator is faster. Answer: ### Topic 5 REVIEW Topic Essential Question What does it mean to solve a system of linear equations? Answer: The solution to a system of linear equations is the point at which the lines representing the linear equations intersect. Two lines in the XY -plane can intersect once, never intersect, or completely overlap. Vocabulary Review Complete each definition and then provide an example of each vocabulary word. Vocabulary solution of a system of linear equations system of linear equations Answer: Use Vocabulary in Writing Describe how you can find the number of solutions of two or more equations by using the slope and the y-intercept. Use vocabulary terms in your description. Answer: When slopes and y-intercepts are different, the two lines intersect at one point, There is only 1 solution. When slopes are the same, the lines may be either parallel (different y-intercepts) There are no solutions If the lines are the same line (Same slopes and same y-intercepts), they intersect at all points, So, there are infinitely many solutions Concepts and Skills Review Lesson 5.1 Estimate Solutions by Inspection Quick Review The slopes and y-intercepts of the linear equations in a system determine the relationship between the lines and the number of solutions. Example How many solutions does the system of equations have? Explain. Answer: y + 2x = 6 y – 8 = -2x Write each equation in slope-intercept form. y = -2x + 6 y = -2x + 8 Identify the slope and y-intercept of each equation. For the equation, y = -2x + 6, the slope is –2 and the y-intercept is 6. For the equation, y = -2x + 8, the slope is -2 and the y-intercept is 8. The equations have the same slope but different y-intercepts, so the system has no solution. Practice Determine whether the system of equations has one solution, no solution, or infinitely many solutions. Question 1. y – 13 = 5x y – 5x = 12 Answer: The given system of equations are: y – 13 = 5x y – 5x = 12 So, The required system of equations are: y = 5x + 13 —– (1) y = 5x + 12 —– (2) Now, Compare the above equations with y = mx + b So, For the first equation, m = 5, b = 13 For the second equation, m = 5, b = 12 We know that, When slopes are the same, the lines may be either parallel (different y-intercepts) There are no solutions Hence, from the above, We can conclude that there is no solution for the given system of equations Question 2. y = 2x + 10 3y – 6x = 30 Answer: The given system of equations are: y = 2x + 10 —– (1) 3y – 6x = 30 —– (2) So, Divide eq (2) by 3 y – 2x = 10 So, y = 2x + 10 —– (3) Now, Compare the above equations with y = mx + b So, For the first equation, m = 2, b = 10 For the third equation, m = 2, b = 10 We know that, If the lines are the same line (Same slopes and same y-intercepts), they intersect at all points, So, there are infinitely many solutions Hence, from the above, We can conclude that there are infinitely many solutions for the given system of equations Question 3. -3x + $$\frac{1}{3}$$y = 12 2y = 18x + 72 Answer: The given system of equations are: -3x + $$\frac{1}{3}$$y = 12 —– (1) 2y = 18x + 72 —- (2) So, Multiply eq (1) with 3 -9x + y = 36 y = 9x + 36 —– (3) Now, Divide eq (2) with 2 So, y = 9x + 36 —– (4) Now, Compare the above equations with y = mx + b So, From the third equation, m = 9, b = 36 From the fourth equation, m = 9, b = 36 We know that, If the lines are the same line (Same slopes and same y-intercepts), they intersect at all points, So, there are infinitely many solutions Hence, from the above, We can conclude that there are infinitely many solutions for the given system of equations Question 4. y – $$\frac{1}{4}$$x = -1 y – 2 = 4x Answer: The given system of equations are: y – $$\frac{1}{4}$$x = -1 —- (1) y – 2 = 4x —— (2) Now, y = $$\frac{1}{4}$$x – 1 —- (3) y = 4x + 2 —- (4) Compare the above equations with y = mx + b So, From the third equation, m = $$\frac{1}{4}$$, b = -1 From the fourth equation, m = 4, b = 2 We know that, When slopes and y-intercepts are different, the two lines intersect at one point, There is only 1 solution. Hence, from the above, We can conclude that there is only 1 solution for the given system of equations Question 5. Michael and Ashley each buy x pounds of turkey and y pounds of ham. Turkey costs$3 per pound at Store A and $4.50 per pound at Store B. Ham costs$4 per pound at Store A and $6 per pound at Store B. Michael spends$18 at Store A, and Ashley spends $27 at Store B. Could Michael and Ashley have bought the same amount of turkey and ham? Explain. Answer: It is given that Michael and Ashley each buy x pounds of turkey and y pounds of ham. Turkey costs$3 per pound at Store A and $4.50 per pound at Store B. Ham costs$4 per pound at Store A and $6 per pound at Store B. Michael spends$18 at Store A, and Ashley spends $27 at Store B. Now, Take the number of pounds for the turkey to be x and that for the ham to be y For store A where Michael spent$18,
Turkey cost $3 per pound —- 3x Ham cost$4 per pound——4y
So,
The equation for cost will be;
3x + 4y = 18
Now,
For store B where Ashley spent $27 Turkey cost$4.5 per pound
Ham cost $6 per pound So, The equation for cost is: 4.5x + 6y = 27 So, The two equations are; 3x + 4y = 18 —— (1) 4.5x + 6y = 27 —– (2) Now, Divide eq (2) with 3 So, 1.5x + 2y = 9 —- (3) Multiply the above equation with 2 So, 3x + 4y = 18 — (4) Now, Compare eq (1) and eq (4) with y = mx + b So, From the eq (1), m = 3, b = 4 From the eq (2), m = 3, b = 4 From the above, We can observe that the slopes and the y-intercepts are equal So, Both the equations are in the same line Hence, from the above, We can conclude that Michael and Ashley bought the same amount of Turkey and Ham Lesson 5.2 Solve Systems by Graphing Quick Review Systems of equations can be solved by looking at their graphs. A system with one solution has one point of intersection. A system with infinitely many solutions has infinite points of intersection. A system with no solution has no points of intersection. Example Graph the system and determine its solution. y = x + 4 y = -2x + 1 Answer: Graph each equation in the system on the same coordinate plane. The point of intersection is (-1, 3). This means the solution to the system is (-1, 3). Practice Graph each system and find the solution(s). Question 1. y = $$\frac{1}{2}$$x + 1 -2x + 4y = 4 Answer: The given system of equations are: y = $$\frac{1}{2}$$x + 1 -2x + 4y = 4 So, The representation of the given system of equations in a coordinate plane is: Hence, from the above, We can conclude that the given system of equations has infinitely many solutions Question 2. y = -x – 3 y + x = 2 Answer: The given system of equations are: y = -x – 3 y + x = 2 So, The representation of the given system of equations in the coordinate plane is: Hence, from the above, We can conclude that there is no solution for the given system of equations Question 3. 2y = 6x + 4 y = -2x + 2 Answer: The given system of equations are: 2y = 6x + 4 y = -2x + 2 So, The representation of the given system of equations in the coordinate plane is: Hence, from the above, We can conclude that the solution for the given system of equations is: (0, 2) Lesson 5.3 Solve Systems by Substitution Quick Review To solve a system by substitution, write one equation for a variable in terms of the other. Substitute the expression into the other equation and solve. If the result is false, the system has no solution. If true, it has infinitely many solutions. If the result is a value, substitute to solve for the other variable. Example Use substitution to solve the system. y = x + 1 y = 5x – 3 Answer: Substitute x + 1 for y in the second equation. (x + 1) = 5x – 3 4 = 4x 1 = x Substitute 1 for x in the first equation. y = (1) + 1 = 2 The solution is x = 1, y = 2. Practice Use substitution to solve each system. Question 1. -3y = -2x – 1 y = x – 1 Answer: The given system of equations are: -3y = -2x – 1 —– (1) y = x – 1 —- (2) Now, Substitute eq (2) in eq (1) So, -3 (x – 1) = -2x – 1 -3 (x) + 3 (1) = -2x – 1 -3x + 3 = -2x – 1 -3x + 2x = -1 – 3 -x = -4 x = 4 So, y = x – 1 y = 4 – 1 y = 3 Hence, from the above, We can conclude that the solution for the given system of equations is: (4, 3) Question 2. y = 5x + 2 2y – 4 = 10x Answer: The given system of equations are: y = 5x + 2 —- (1) 2y – 4 = 10 —– (2) Now, Substitute eq (1) in eq (2) So, 2 (5x + 2) – 4 = 10 2 (5x) + 2 (2) – 4 = 10 10x = 10 x = 1 So, y = 5x + 2 y = 5 + 2 y = 7 Hence, from the above, We can conclude that the solution for the given system of equations is: (1, 7) Question 3. 2y – 8 = 6x y = 3x + 2 Answer: The given system of equations are: 2y – 8 = 6x —- (1) y = 3x + 2 —– (2) Now, Substitute eq (2) in eq (1) So, 2 (3x + 2) – 8 = 6x 2 (3x) + 2 (2) – 8 = 6x 6x + 4 – 8 = 6x 4 = 8 Hence, from the above, We can conclude that there is no solution for the given system of equations Question 4. 2y – 2 = 4x y = -x + 4 Answer: The given system of equations are: 2y – 2 = 4x —– (1) y = -x + 4 —– (2) Now, Substitute eq (2) in eq (1) So, 2 (-x + 4) – 2 = 4x 2 (-x) + 2 (4) – 2 = 4x -2x + 8 – 2 = 4x 4x + 2x = 6 6x = 6 x = 1 So, y = -x + 4 y = -1 + 4 y = 3 Hence, from the above, We can conclude that the solution for the given system of equations is: (1, 3) Lesson 5.4 Solve Systems by Elimination Quick Review To solve a system by elimination, multiply one or both equations to make opposite terms. Add (or subtract) the equations to eliminate one variable. Substitute to solve for the other variable. Example Use elimination to solve the system. 2x – 9y = -5 4x – 6y = 2 Multiply the first equation by -2.Then add. y = 1 Substitute 1 for y in the first equation. 2x – 9(1) = -5 2x – 9 = -5 2x = 4 x = 2 The solution is x = 2, y = 1. Practice Use elimination to solve each system. Question 1. -2x + 2y = 2 4x – 4y = 4 Answer: The given system of equations are: -2x + 2y = 2 —– (1) 4x – 4y = 4 ——- (2) So, Hence, from the above, We can conclude that the given system of equations has no solution Question 2. 4x + 6y = 40 -2x + y = 4 Answer: The given system of equations are: 4x + 6y = 40 -2x + y = 4 So, Hence, from the above, We can conclude that the solution for the given system of equations is: (1, 6) Question 3. A customer at a concession stand bought 2 boxes of popcorn and 3 drinks for$12. Another customer bought 3 boxes of popcorn and 5 drinks for $19. How much does a box of popcorn cost? How much does a drink cost? Answer: It is given that A customer at a concession stands bought 2 boxes of popcorn and 3 drinks for$12. Another customer bought 3 boxes of popcorn and 5 drinks for $19 Now, Let each box of popcorn be p Let each drink be d So, For Customer A, 2p + 3d =$12 —– (1)
For Customer B,
3p + 5d = $19 —– (2) Now, Hence, from the above, We can conclude that The cost of each popcorn box is:$3
The cost of each drink is: \$2

### Topic 6 Fluency Practice

Pathfinder
Shade a path from START to FINISH. Follow the solutions to the equations from least to greatest. You can only move up, down, right, or left.
I can… solve multistep equations using the Distributive Property.

Scroll to Top