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Go Math Grade 8 Answer Key Chapter 4 Nonproportional Relationships

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Lesson 1: Representing Linear Nonproportional Relationships

• Representing Linear Nonproportional Relationships – Page No. 98
• Representing Linear Nonproportional Relationships – Page No. 99
• Representing Linear Nonproportional Relationships – Page No. 100

Lesson 2: Determining Slope and y-intercept

• Determining Slope and y-intercept – Page No. 104
• Determining Slope and y-intercept – Page No. 105
• Determining Slope and y-intercept – Page No. 106

Lesson 3: Graphing Linear Nonproportional Relationships Using Slope and y-intercept

• Graphing Linear Nonproportional Relationships Using Slope and y-intercept – Page No. 110
• Graphing Linear Nonproportional Relationships Using Slope and y-intercept – Page No. 111
• Graphing Linear Nonproportional Relationships Using Slope and y-intercept – Page No. 112

Lesson 4: Proportional and Nonproportional Situations 

• Proportional and Nonproportional Situations – Page No. 117
• Proportional and Nonproportional Situations – Page No. 118
• Proportional and Nonproportional Situations – Page No. 119
• Proportional and Nonproportional Situations – Page No. 120

Lesson 5: Representing Linear Nonproportional Relationships – Model Quiz

• Representing Linear Nonproportional Relationships – Model Quiz – Page No. 121

Mixed Review 

• Mixed Review – Page No. 122

Guided Practice – Representing Linear Nonproportional Relationships – Page No. 98

Make a table of values for each equation.

Question 1.
y = 2x + 5

Type below:
____________

Answer:

Explanation:
y = 2x + 5
Choose several values for x and substitute in the equation to find y.
x = 2(-2) + 5 = 1
x = 2(-1) + 5 = 3
x = 2(0) + 5 = 5
x = 2(1) + 5 = 7
x = 2(2) + 5 = 9

Graphing Linear Nonproportional Relationships Worksheet Answers Question 2.
y = \(\frac{3}{8}\)x − 5

Type below:
____________

Answer:

Explanation:
y = \(\frac{3}{8}\)x − 5
Choose several values for x and substitute in the equation to find y.
x = 3/8(-8) – 5 = -8
x = 3/8(0) – 5 = -5
x = 3/8(8) – 5 = -2
x = 3/8(16) – 5 = 1
x = 3/8(24) – 5 = 4

Explain why each relationship is not proportional.

Question 3.

First calculate \(\frac{y}{x}\) for the values in the table.
____________

Answer:
The relationship is not proportional

Explanation:
Find y/x
3/0 = undefined
7/2 = 3.5
11/4 = 2.75
15/6 = 2.5
19/8 = 2.375
The ratio is not constant, hence relationship is not proportional.

Question 4.

__________________

Answer:
The graph is a straight line but does not pass through the origin. So, the relationship is not proportional.

Complete the table for the equation. Then use the table to graph the equation.

Question 5.
y = x − 1


Type below:
____________

Answer:

Explanation:
y = x – 1
Choose several values of x and substitute in the equation to find y to draw a table.
x = -2; y = -2 – 1 = -2
x = -1; y = -1 -1 = -2
x = 0; y = 0 -1 = -1
x = 1; y = 1 – 1 = 0
x = 2; y = 2 -1 = 1
Also, Plot the ordered pairs from the table. Then draw a line connecting the points to represent all the possible solutions

Essential Question Check-In

Question 6.
How can you choose values for x when making a table of values representing a real-world situation?
Type below:
____________

Answer:
When choosing values for x in a real-world situation, you choose positive values with an appropriate interval to represent the array of data.

Independent Practice – Representing Linear Nonproportional Relationships – Page No. 99

State whether the graph of each linear relationship is a solid line or a set of unconnected points. Explain your reasoning.

Question 7.
The relationship between the number of $4 lunches you buy with a $100 school lunch card and the money remaining on the card
____________

Answer:
Set of unconnected points.

Explanation:
You cannot buy a fractional part of a lunch.
Set of unconnected points.

Question 8.
The relationship between time and the distance remaining on a 3-mile walk for someone walking at a steady rate of 2 miles per hour.
____________

Answer:
A solid line

Explanation:
The relationship between time and the distance remaining on a 3-mile walk for someone walking at a steady rate of 2 miles per hour. The distance remaining can be a fraction. The time can be in a fraction as well.
A solid line

Nonproportional Relationship Graph Question 9.
Analyze Relationships
Simone paid $12 for an initial year’s subscription to a magazine. The renewal rate is $8 per year. This situation can be represented by the equation y = 8x + 12, where x represents the number of years the subscription is renewed and y represents the total cost.
a. Make a table of values for this situation.

Type below:
____________

Answer:

Explanation:
y = 8x + 12
Choose several values for x and substitute in the equation to find y.

Question 9.
b. Draw a graph to represent the situation. Include a title and axis labels.
Type below:
____________

Answer:

Explanation:
Plot the ordered pairs from the table. Then draw a line connecting the points to represent all the possible solutions

Question 9.
c. Explain why this relationship is not proportional.

Type below:
____________

Answer:
It is not proportional as the graph does not pass through the origin

Explanation:
When a relationship is proportional, the graph of the equation passes through the origin.
It is not proportional as the graph does not pass through the origin

Question 9.
d. Does it make sense to connect the points on the graph with a solid line? Explain.
Type below:
____________

Answer:
No

Explanation:
No; The subscription is renewed for the entire year and cannot be done for a fraction of the year. The number of years must be a whole numb, so the total cost goes up in $8 increments.

Representing Linear Nonproportional Relationships – Page No. 100

Question 10.
Analyze Relationships
A proportional relationship is a linear relationship because the rate of change is constant (and equal to the constant of proportionality). What is required of a proportional relationship that is not required of a general linear relationship?
Type below:
____________

Answer:
The ratio between one quantity to the other quantity should be constant for a proportional linear relationship. The graph should be a straight line that passes through the origin.

Lesson 4.1 Representing Linear Nonproportional Relationships Answer Key Question 11.
Communicate Mathematical Ideas
Explain how you can identify a linear non-proportional relationship from a table, a graph, and an equation.
Type below:
____________

Answer:
In a table, the ratios y/x will not be equal. A graph will not pass through the origin. An equation will be in the form y = mx + b where b is not equal to 0.

Focus on Higher Order Thinking

Question 12.
Critique Reasoning
George observes that for every increase of 1 in the value of x, there is an increase of 60 in the corresponding value of y. He claims that the relationship represented by the table is proportional. Critique George’s reasoning.

Type below:
____________

Answer:
The ratio is not constant, hence the relationship cannot be proportional.

Explanation:
Find y/x
90/1 = 90
150/2 = 75
210/3 = 70
270/4 = 67.5
330/5 = 66
The ratio is not constant, hence the relationship cannot be proportional.

Question 13.
Make a Conjecture
Two parallel lines are graphed on a coordinate plane. How many of the lines could represent proportional relationships? Explain.
Type below:
____________

Answer:
Maximum one

Explanation:
When there are two parallel lines, only one can pass through the origin and a line representing a proportional relationship must pass through the origin.
Maximum one

Guided Practice – Determining Slope and y-intercept – Page No. 104

Find the slope and y-intercept of the line in each graph.

Question 1.

slope m = _____ y-intercept b = _____
m = ____________
b = ____________

Answer:
slope m = -2 y-intercept b = 1
m = -2
b = 1

Explanation:
Find the slope using two points from the grapgh by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (0, 1) and (x2, y2) = (2, -3)
Slope m = (y2 -y1)/(x2 – x1) = (-3 – 1)/(2 – 0) = -4/2 = -2
From the graph when x = 0
y-intercept (b) = 1

Lesson 4.2 Determining Slope and Y-Intercept Answer Key Pdf Question 2.

slope m = _____ y-intercept b = _____
m = ____________
b = ____________

Answer:
slope m = 5 y-intercept b = -15
m = 5
b = -15

Explanation:
Find the slope using two points from the graph by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (3, 0) and (x2, y2) = (0, -15)
Slope m = (y2 -y1)/(x2 – x1) = (-15 – 0)/(0 – 3) = 15/3 = 5
From the graph when x = 0
y-intercept (b) = -15

Question 3.

slope m = _____ y-intercept b = _____
Type below:
____________

Answer:
slope m = 3/2 y-intercept b = -2
m = 3/2
b = -2

Explanation:
Find the slope using two points from the graph by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (0, -2) and (x2, y2) = (2, 1)
Slope m = (y2 -y1)/(x2 – x1) = (1 – (-2))/(2 – 0) = 3/2
From the graph when x = 0
y-intercept (b) = -2

Question 4.

slope m = _____ y-intercept b = _____
m = ____________
b = ____________

Answer:
slope m = -3 y-intercept b = 9
m = -3
b = 9

Explanation:
Find the slope using two points from the grapgh by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (3, 0) and (x2, y2) = (0, 9)
Slope m = (y2 -y1)/(x2 – x1) = (9 – 0))/(0 – 3) = -9/3 = -3
From the graph when x = 0
y-intercept (b) = 9

Find the slope and y-intercept of the line represented by each table.

Question 5.

slope m = _____ y-intercept b = _____
m = ____________
b = ____________

Answer:
slope m = 3 y-intercept b = 1
m = 3
b = 1

Explanation:
Find the slope using two points from the grapgh by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (8, 25) and (x2, y2) = (6, 19)
Slope m = (y2 -y1)/(x2 – x1) = (19 – 25)/(6 – 8) = 6/2 = 3
From the graph when x = 0
y-intercept (b) = 1

Question 6.

slope m = _____ y-intercept b = _____
m = ____________
b = ____________

Answer:
slope m = -4 y-intercept b = 140
m = -4
b = 140

Explanation:
Find the slope using two points from the grapgh by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (5, 120) and (x2, y2) = (15, 80)
Slope m = (y2 -y1)/(x2 – x1) = (80 – 120)/(15 – 5) = -40/10 = -4
From the graph when x = 0
y-intercept (b) = 140

Essential Question Check-In

Question 7.
How can you determine the slope and the y-intercept of a line from a graph?
Type below:
____________

Answer:
Choose any two points on the line from the graph and use it to find the slope. Determine the point where the line crosses the y-axis to find the y-intercept.

Independent Practice – Determining Slope and y-intercept – Page No. 105

Question 8.
Some carpet cleaning costs are shown in the table. The relationship is linear. Find and interpret the rate of change and the initial value for this situation.

Type below:
_____________

Answer:
Find the slope using two points
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (1, 125) and (x2, y2) = (3, 225)
Slope m = (y2 -y1)/(x2 – x1) = (225 – 125)/(3 – 1) = 100/2 = 50
Find the initial value when the value of x is 0
Work backward from x = 1 to x = 0
(175 – 125)/(2 – 1) = 50/1 = 50
Subtract the difference of x and y from the first point.
x = 1 – 1 = 0
y = 125 – 50 = 75
y-intercept (b) = 75
The slope/rate of change represents the increase in the cost of cleaning the rooms for a unit increase in the number of rooms. The y-intercept shows the initial cost of carpet cleaning.

Question 9.
Make Predictions
The total cost to pay for parking at a state park for the day and rent a paddleboat are shown.

a. Find the cost to park for a day and the hourly rate to rent a paddleboat.
Type below:
_____________

Answer:
$5

Explanation:
Find the slope using two points
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (1, 17) and (x2, y2) = (2, 29)
Slope m = (y2 -y1)/(x2 – x1) = (29 – 17)/(2 – 1) = 12/1 = 12
Find the initial value when the value of x is 0
Work backward from x = 1 to x = 0
(29 – 17)/(2 – 1) = 12/1 = 12
Subtract the difference of x and y from the first point.
x = 1 – 1 = 0
y = 17 – 12 = 5
The cost to park for a day is $5.

Question 9.
b. What will Lin pay if she rents a paddleboat for 3.5 hours and splits the total cost with a friend? Explain.
$ _____________

Answer:
$23.5

Explanation:
When Lin paddles for 3.5hr
Total Cost = 3.5(12) + 5 = 47
Lin’s cost = 47/2 = 23.5

Question 10.
Multi-Step
Raymond’s parents will pay for him to take sailboard lessons during the summer. He can take half-hour group lessons or half-hour private lessons. The relationship between cost and number of lessons is linear.

a. Find the rate of change and the initial value for the group lessons.
Type below:
____________

Answer:
$25

Explanation:
Find the slope using two points
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (1, 55) and (x2, y2) = (2, 85)
Slope m = (y2 -y1)/(x2 – x1) = (85 – 55)/(2 – 1) = 30/1 = 30
Rate of change is $30 for per lesson
Find the initial value when the value of x is 0
Work backward from x = 1 to x = 0
(85 – 55)/(2 – 1) = 30/1 = 30
Subtract the difference of x and y from the first point.
x = 1 – 1 = 0
y = 55 – 30 = 25
The initial value of the group lesson is $25.

Question 10.
b. Find the rate of change and the initial value for the private lessons.
Type below:
_____________

Answer:
$25

Explanation:
Find the slope using two points
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (1, 75) and (x2, y2) = (2, 125)
Slope m = (y2 -y1)/(x2 – x1) = (125 – 75)/(2 – 1) = 50/1 = 50
The rate of change is $50 per lesson
Find the initial value when the value of x is 0
Work backward from x = 1 to x = 0
(125 – 75)/(2 – 1) = 50/1 = 50
Subtract the difference of x and y from the first point.
x = 1 – 1 = 0
y = 75 – 50 = 25
The initial value of the private lesson is $25.

Question 10.
c. Compare and contrast the rates of change and the initial values.
Type below:
_____________

Answer:
The initial value for both types of lessons is the same. The rate of change is higher for private lessons than group lesson

Explanation:
Compare the results of a and b
The initial value for both types of lessons is the same. The rate of change is higher for private lessons than group lesson

Vocabulary – Determining Slope and y-intercept – Page No. 106

Explain why each relationship is not linear.

Question 11.

Type below:
_____________

Answer:
The rate of change is not constant, hence the relationship is not linear

Explanation:
Find the rate of change using two points Slope m = (y2 -y1)/(x2 – x1)
(6.5 – 4.5)/(2 – 1) = 2
(8.5 – 6.5)/(3 – 2) = 2
(11.5 – 8.5)/(4 – 3) = 3
The rate of change is not constant, hence the relationship is not linear

Question 12.

Type below:
_____________

Answer:
The rate of change is not constant, hence the relationship is not linear

Explanation:
Find the rate of change using two points Slope m = (y2 -y1)/(x2 – x1)
(126 – 140)/(5 – 3) = -7
(110 – 126)/(7 – 5) = -8
(92 – 110)/(9 – 7) = -9
The rate of change is not constant, hence the relationship is not linear

Question 13.
Communicate Mathematical Ideas
Describe the procedure you performed to derive the slope-intercept form of a linear equation.
Type below:
_____________

Answer:
Express the slope m between a random point (x, y) on the line and the point (0, b) where the line crosses the y-axis. Then solve the equation for y.

H.O.T.

Focus on Higher Order Thinking

Question 14.
Critique Reasoning
Your teacher asked your class to describe a real-world situation in which a y-intercept is 100 and the slope is 5. Your partner gave the following description: My younger brother originally had 100 small building blocks, but he has lost 5 of them every month since.
a. What mistake did your partner make?
Type below:
_____________

Answer:
If the brother loses 5 blocks every month, the slope would be -5 and not 5.

Explanation:
When the initial value is decreasing, the slope is negative.
If the brother loses 5 blocks every month, the slope would be -5 and not 5.

Question 14.
b. Describe a real-world situation that does match the situation.
Type below:
_____________

Answer:
I bought a 100-card pack and buy 5 additional cards every month.

Explanation:
Real-world situation
I bought a 100-card pack and buy 5 additional cards every month.

Question 15.
Justify Reasoning
John has a job parking car. He earns a fixed weekly salary of $300 plus a fee of $5 for each car he parks. His potential earnings for a week are shown in the graph. At what point does John begin to earn more from fees than his fixed salary? Justify your answer.

Type below:
_____________

Answer:
After parking 60 cars, John’s earnings become $600 double his initial base salary of $300.
Hence, after parking 61 cars, his earning from the fee becomes more than his fixed salary.

Explanation:
He earns the same in fees as his fixed salary for parking 300/5 = 60
After parking 60 cars, John’s earnings became $600 double his initial base salary of $300.
Hence, after parking 61 cars, his earning from the fee becomes more than his fixed salary.

Guided Practice – Graphing Linear Nonproportional Relationships Using Slope and Y-intercept – Page No. 110

Graph each equation using the slope and the y-intercept.

Question 1.
y = \(\frac{1}{2}\)x − 3
slope = _____ y-intercept = _____

Type below:
_____________

Answer:
slope = 1/2 y-intercept = -3

Explanation:
y = 1/2 x – 3
The y-intercept is b = -3. Plot the point that contains the y-intercept (0, -3)
The slope m = 1/2. Use the slope to find a second point. From (0, -3) count 1 unit up and 2 unit right. The new point is (2, -2)
Draw a line through the points

Question 2.
y = −3x + 2
slope = _____ y-intercept = _____

Type below:
_____________

Answer:
slope = -3 y-intercept = 2

Explanation:
y = -3x + 2
The y-intercept is b = 2. Plot the point that contains the y-intercept (0, 2)
The slope m = -3/1. Use the slope to find a second point. From (0, 2) count 3 unit down and 1 unit right. The new point is (1, -1)
Draw a line through the points

Nonproportional Relationships Worksheet Question 3.
A friend gives you two baseball cards for your birthday. Afterward, you begin collecting them. You buy the same number of cards once each week. The equation y = 4x + 2 describes the number of cards, y, you have after x weeks.
a. Find and interpret the slope and the y-intercept of the line that represents this situation. Graph y = 4x + 2. Include axis labels.

Type below:
_____________

Answer:

Explanation:
y = 4x + 2
The y-intercept is b = 2. Plot the point that contains the y-intercept (0, 2)
The slope m = 4. Use the slope to find a second point. From (0, 2) count 4 unit up and 1 unit right. The new point is (1, 6)
Draw a line through the points

Question 3.
b. Discuss which points on the line do not make sense in this situation. Then plot three more points on the line that do make sense.
Type below:
_____________

Answer:

Explanation:
The points with a negative value of x or y do not make sense as the number of cards or weeks cannot be negative.

Essential Question Check-In

Question 4.
Why might someone choose to use the y-intercept and the slope to graph a line?
Type below:
_____________

Answer:
When the relationship is given in the form y = mx + b, the y-intercept (b) and the slope (m) are easily accessible and easily calculable. Therefore, it is a good practice to use them to graph the line.

Independent Practice – Graphing Linear Nonproportional Relationships Using Slope and Y-intercept – Page No. 111

Question 5.
Science
A spring stretches in relation to the weight hanging from it according to the equation y = 0.75x + 0.25 where x is the weight in pounds and y is the length of the spring in inches.
a. Graph the equation. Include axis labels.
Type below:
_____________

Answer:

Explanation:
y = 0.75x + 0.25
Slope m = 0.75 and y-intercept = 0.25
Plot the point that contains the y-intercept (0, 0.25)
The slope is m = 0.75/1. Use the slope to find a second point. From (0,0.25) count 0.75 unit up and 1 unit right. The new point is (1, 1)

Question 5.
b. Interpret the slope and the y-intercept of the line.
Type below:
_____________

Answer:
The slope represents the increase in the length of spring in inches for each increase of pound of weight. y-intercept represents the unstretched length of the spring When there is no weight attached.

Question 5.
c. How long will the spring be if a 2-pound weight is hung on it? Will the length double if you double the weight? Explain
Type below:
_____________

Answer:
When there is a 2-pound weight hung, the length of the spring would be 1.75 inches. No, When there is a 4-pound weight hung, the length of the spring would be 3.25 inches and not 3.5 inches.

Look for a Pattern

Identify the coordinates of four points on the line with each given slope and y-intercept.

Question 6.
slope = 5, y-intercept = -1
Type below:
_____________

Answer:
(2, 9)
(3, 14)

Explanation:
slope = 5, y-intercept = -1
Plot the point that contains the y-intercept (0, -1)
The slope is m = 5/1. Use the slope to find a second point. From (0, -1) count 5 unit up and 1 unit right. The new point is (1, 4)
Follow the same procedure to find the remaining three points.
(2, 9)
(3, 14)

Question 7.
slope = -1, y-intercept = 8
Type below:
_____________

Answer:
(2, 6)
(3, 5)

Explanation:
slope = -1, y-intercept = 8
Plot the point that contains the y-intercept (0, 8)
The slope is m = -1/1. Use the slope to find a second point. From (0, 8) count 1 unit down and 1 unit right. The new point is (1, 7)
Follow the same procedure to find the remaining three points.
(2, 6)
(3, 5)

Question 8.
slope = 0.2, y-intercept = 0.3
Type below:
_____________

Answer:
(2, 0.7)
(3, 0.9)

Explanation:
slope = 0.2, y-intercept = 0.3
Plot the point that contains the y-intercept (0, 0.3)
The slope is m = 0.2/1. Use the slope to find a second point. From (0, 0.3) count 0.2 unit up and 1 unit right. The new point is (1, 0.5)
Follow the same procedure to find the remaining three points.
(2, 0.7)
(3, 0.9)

Question 9.
slope = 1.5, y-intercept = -3
Type below:
_____________

Answer:
(2, 0)
(3, 1.5)

Explanation:
slope = 1.5, y-intercept = -3
Plot the point that contains the y-intercept (0, -3)
The slope is m = 1.5/1. Use the slope to find a second point. From (0, -3) count 1.5 unit up and 1 unit right. The new point is (1, -1.5)
Follow the same procedure to find the remaining three points.
(2, 0)
(3, 1.5)

Question 10.
slope = −\(\frac{1}{2}\), y-intercept = 4
Type below:
_____________

Answer:
(4, 2)
(6, 1)

Explanation:
slope = −\(\frac{1}{2}\), y-intercept = 4
Plot the point that contains the y-intercept (0, 4)
The slope is m = −\(\frac{1}{2}\)/1. Use the slope to find a second point. From (0, 4) count 1 unit down and 2 unit right. The new point is (2, 3)
Follow the same procedure to find the remaining three points.
(4, 2)
(6, 1)

Question 11.
slope = \(\frac{2}{3}\), y-intercept = -5
Type below:
_____________

Answer:
(6, -1)
(9, 1)

Explanation:
slope = \(\frac{2}{3}\), y-intercept = -5
Plot the point that contains the y-intercept (0, -5)
The slope is m = \(\frac{2}{3}\). Use the slope to find a second point. From (0, -5) count 2 unit up and 3 unit right. The new point is (3, -3)
Follow the same procedure to find the remaining three points.
(6, -1)
(9, 1)

Question 12.
A music school charges a registration fee in addition to a fee per lesson. Music lessons last 0.5 hour. The equation y = 40x + 30 represents the total cost y of x lessons. Find and interpret the slope and y-intercept of the line that represents this situation. Then find four points on the line.
Type below:
_____________

Answer:
y = 40x + 30
Slope = 40
y-intercept = 30
The slope represents the fee of the classes per lesson and the y-intercept represents the registration fee.
Plot the point that contains the y-intercept (0, 30)
The slope is m = 40/1. Use the slope to find a second point. From (0, 30) count 40 units up and 1 unit right. The new point is (1, 70)
Follow the same procedure to find the remaining three points.
(2, 110)
(3, 150)

Graphing Linear Nonproportional Relationships Using Slope and Y-intercept – Page No. 112

Question 13.
A public pool charges a membership fee and a fee for each visit. The equation y = 3x + 50 represents the cost y for x visits.
a. After locating the y-intercept on the coordinate plane shown, can you move up three gridlines and right one gridline to find a second point? Explain.

Type below:
_____________

Answer:
Yes

Explanation:
Yes; Since the horizontal and vertical gridlines each represent 25 units, hence moving up 3 gridlines and right 1 gridline represents a slope of 75/25 or 3

Question 13.
b. Graph the equation y = 3x + 50. Include axis labels. Then interpret the slope and y-intercept.
Type below:
_____________

Answer:

The slope represents the fee per visit and the y-intercept represents the membership fee.

Explanation:
Slope = 3
y-intercept = 50
The slope represents the fee of the classes per lesson and the y-intercept represents the registration fee.
Plot the point that contains the y-intercept (0, 50)
The slope is m = 3/1. Use the slope to find a second point. From (0, 50) count 3 units up and 1 unit right. The new point is (1, 53)

Question 13.
c. How many visits to the pool can a member get for $200?
______ visits

Answer:
50 visits

Explanation:
You would get 50 visits for $200

H.O.T.

Focus on Higher Order Thinking

Question 14.
Explain the Error
A student says that the slope of the line for the equation y = 20 − 15x is 20 and the y-intercept is 15. Find and correct the error.
Type below:
_____________

Answer:
The slope is -15 as it represents the change in y per unit change in x. The y-intercept is 20 when x = 0.

Explanation:
y = 20 − 15x
The slope is -15 as it represents the change in y per unit change in x. The y-intercept is 20 when x = 0.

Question 15.
Critical Thinking
Suppose you know the slope of a linear relationship and a point that its graph passes through. Can you graph the line even if the point provided does not represent the y-intercept? Explain.
Type below:
_____________

Answer:
Yes. You can plot the given point and use the slope to find a second point. Connect the points by drawing a line.

Question 16.
Make a Conjecture
Graph the lines y = 3x, y = 3x − 3, and y = 3x + 3. What do you notice about the lines? Make a conjecture based on your observation.

Type below:
_____________

Answer:

Explanation:
let’s tale the example
y = 3x
y = 3x – 3
y = 3x + 3
We notice that the lines are parallel to each other: the slopes of the lines are equal but the y-intersection point differs.

Guided Practice – Proportional and Nonproportional Situations – Page No. 117

Determine if each relationship is a proportional or nonproportional situation. Explain your reasoning.

Question 1.

Look at the origin.
_____________

Answer:
Proportional relationship

Explanation:
Proportional relationship
The graph passes through the origin. Graph of a proportional relationship must pass through the origin

Question 2.

_____________

Answer:
Non-proportional relationship

Explanation:
The graph does not pass through the origin. The graph of a proportional relationship must pass through the origin
Non-proportional relationship

Lesson 4 Skills Practice Proportional and Nonproportional Relationships Answer Key Question 3.
q = 2p + \(\frac{1}{2}\)
Compare the equation with y = mx + b.
_____________

Answer:
q = 2p + \(\frac{1}{2}\)
The equation is in the form y = mx + b, with p being used es the variable instead of x and q instead of y. The value of m is 2, and the value b is 1/2. Since b is not 0, the relationship presented through the above equation is non-proportional.

Question 4.
v = \(\frac{1}{10}\)u
_____________

Answer:
Proportional relationship

Explanation:
v = \(\frac{1}{10}\)u
Compare with the form of equation y = mx + b. The equation represents a proportional relationship if b = 0
Proportional relationship

Proportional and Nonproportional Situations – Page No. 118

The tables represent linear relationships. Determine if each relationship is a proportional or nonproportional situation.

Question 5.

Find the quotient of y and x.
_____________

Answer:
proportional relationship

Explanation:
Find the ratio y/x
12/3 = 4
36/9 = 4
84/21 = 4
Since the ratio is constant, the relationship is proportional.

Lesson 4 Skills Practice Proportional and Nonproportional Relationships Question 6.

_____________

Answer:
non-proportional

Explanation:
Find the ratio y/x
4/22 = 2/11
8/46 = 4/23
10/58 = 5/29
Since the ratio is not constant, the relationship is non-proportional.

Question 7.
The values in the table represent the number of households that watched three TV shows and the ratings of the shows. The relationship is linear. Describe the relationship in other ways.

Type below:
_____________

Answer:
proportional relationship

Explanation:
Find the ratio y/x
12/15,000,000 = 0.0000008
16/20,000,000 = 0.0000008
20/25,000,000 = 0.0000008
Since the ratio is constant, the relationship is proportional.

Essential Question Check-In

Question 8.
How are using graphs, equations, and tables similar when distinguishing between proportional and nonproportional linear relationships?
Type below:
_____________

Answer:
The ratio between y to x is constant when the relationship is proportional. Graphs, tables, and equations all can be used to find the ratio. The ratio is not constant when the relationship is non-proportional.

Independent Practice – Proportional and Nonproportional Situations – Page No. 119

Question 9.
The graph shows the weight of a cross-country team’s beverage cooler based on how much sports drink it contains.

a. Is the relationship proportional or nonproportional? Explain.
_____________

Answer:
Non-proportional

Explanation:
The graph does not pass through the origin. Graph of a proportional relationship must pass through the origin
Non-proportional

Question 9.
b. Identify and interpret the slope and the y-intercept.
Type below:
_____________

Answer:
Slope m = (y2 -y1)/(x2 – x1) = (12 – 10)/(4 – 0) = 0.5
y-intercept is the weight of the empty cooler, which is 10 lbs.

Explanation:
Find the slope using two points from the grapgh by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (0, 10) and (x2, y2) = (4, 12)
Slope m = (y2 -y1)/(x2 – x1) = (12 – 10)/(4 – 0) = 0.5
From the graph when x = 0
y-intercept (b) = 10
y-intercept is the weight of the empty cooler, which is 10 lbs.

In 10–11, tell if the relationship between a rider’s height above the first floor and the time since the rider stepped on the elevator or escalator is proportional or nonproportional. Explain your reasoning.

Question 10.
The elevator paused for 10 seconds after you stepped on before beginning to rise at a constant rate of 8 feet per second.

_____________

Answer:
Non-proportional

Explanation:
As there is a pause of 10 seconds, it would be the y-intercept of the graph (when x = 0)
Non-proportional

Representing Linear Nonproportional Relationships Lesson 4.1 Answer Key Question 11.
Your height, h, in feet above the first floor on the escalator is given by h = 0.75t, where t is the time in seconds.
_____________

Answer:
Proportional

Explanation:
Comparing with y = mx + b, where b = 0
Proportional

Analyze Relationships

Compare and contrast the two graphs.

Question 12.
Graph A       Graph B
y = \(\frac{1}{3}\) x        y = \(\sqrt { x } \)


Type below:
_____________

Answer:
Graph A represents a linear relationship while Graph B represents an exponential relationship. They both pass through the origin and the value of y increases with an increase in x.

Proportional and Nonproportional Situations – Page No. 120

Question 13.
Represent Real-World Problems
Describe a real-world situation where the relationship is linear and nonproportional.
Type below:
_____________

Answer:
The entrance fee to the amusement park is $8 and there is a fee of $2 per ride.

H.O.T.

Focus on Higher Order Thinking

Question 14.
Mathematical Reasoning
Suppose you know the slope of a linear relationship and one of the points that its graph passes through. How can you determine if the relationship is proportional or nonproportional?
Type below:
_____________

Answer:
Use the graph and the given point to determine the second point. Connect the two points by a straight line. If the graph passes through the origin, the relationship is proportional and if the graph does not pass through the origin, the relationship is non-proportional.

Lesson 4 Proportional and Nonproportional Relationships Question 15.
Multiple Representations
An entrant at a science fair has included information about temperature conversion in various forms, as shown. The variables F, C, and K represent temperatures in degrees Fahrenheit, degrees Celsius, and kelvin, respectively.

a. Is the relationship between kelvins and degrees Celsius proportional? Justify your answer in two different ways.
_____________

Answer:
No, the relationship is not proportional.

Explanation:
Compare the equation B to the form: y = mx + b. Since b is not equal to 0, the relationship is non-proportional.
Find the ratio between the Kelvin and Degrees Celsius. Since the ratio is not constant, the relationship is non-proportional.
281.15/8 = 35.14
288.15/15 = 19.21
309.15/36 = 8.59
No, the relationship is not proportional.

Question 15.
b. Is the relationship between degrees Celsius and degrees Fahrenheit proportional? Why or why not?
_____________

Answer:
No, the relationship is not proportional.

Explanation:
Compare the equation A to the form: y = mx + b. Since b is not equal to 0, the relationship is non-proportional.
No, the relationship is not proportional.

4.1 Representing Linear Nonproportional Relationships – Model Quiz – Page No. 121

Lesson 4.1 Representing Linear Nonproportional Relationships Question 1.
Complete the table using the equation y = 3x + 2.

Type below:
_____________

Answer:

Explanation:
Given y = 3x + 2

x = -1; y = 3(-1) + 2 = -3 + 2 = -1
x = 0; y = 3(0) +2 = 2
x = 1; y = 3(1) + 2 = 3 + 2 = 5
x = 2; y = 3(2) + 2 = 6 + 2 = 8
x = 3: y = 3(3) + 2 = 9 + 2 = 11

4.2 Determining Slope and Y-intercept

Question 2.
Find the slope and y-intercept of the line in the graph.

Type below:
_____________

Answer:
Slope = 3
y-intercept (b) = 1

Explanation:
Find the slope using two points from the grapgh by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (0, 1) and (x2, y2) = (1, 4)
Slope m = (y2 -y1)/(x2 – x1) = (4 – 1)/(1 – 0) = 3/1
From the graph when x = 0
y-intercept (b) = 1

4.3 Graphing Linear Nonproportional Relationships

Question 3.
Graph the equation y = 2x − 3 using slope and y-intercept.

Type below:
_____________

Answer:

Explanation:
Slope = 2
y-intercept = -3
Plot the point that contains the y-intercept (0, -3)
The slope is m = 2/1. Use the slope to find a second point. From (0, -3) count 2 unit up and 1 unit right. The new point is (1, -1)
Draw a line through the points

4.4 Proportional and Nonproportional Situations

Question 4.
Does the table represent a proportional or a nonproportional linear relationship?

_____________

Answer:
Since the ratio is constant, the table represents a proportional linear relationship.

Explanation:
Find the ratio y/x
4/1 = 4
8/2 = 4
12/3 = 4
16/4 = 4
20/5 = 4
Since the ratio is constant, the table represents a proportional linear relationship.

Question 5.
Does the graph in Exercise 2 represent a proportional or a nonproportional linear relationship?
_____________

Answer:
It represents a non-proportional linear relationship

Explanation:
The line of the graph does not pass through the origin. The graph of a proportional relationship must pass through the origin.
It represents a non-proportional linear relationship

Question 6.
Does the graph in Exercise 3 represent a proportional or a nonproportional relationship?
_____________

Answer:
It represents a non-proportional linear relationship

Explanation:
The line of the graph does not pass through the origin. The graph of a proportional relationship must pass through the origin
It represents a non-proportional linear relationship

Essential Question

Question 7.
How can you identify a linear nonproportional relationship from a table, a graph, and an equation?
Type below:
_____________

Answer:
In a table, the ratio of y/x is not constant for a non-proportional relationship.
In a graph, the line of the graph does not pass through the origin for a non-proportional relationship.
In an equation, the b is not equal to y = mx +b for a non-proportional relationship.

Selected Response – Mixed Review – Page No. 122

Question 1.
The table below represents which equation?

Options:
a. y = −x − 10
b. y = −6x
c. y = −4x − 6
d. y = −4x + 2

Answer:
c. y = −4x − 6

Explanation:
From the table, you can see that the y-intercept (when x = 0) is b = -6. Comparable to y = mx + b
The table is represented by Option C y = -4x – 6

Question 2.
The graph of which equation is shown below?

Options:
a. y = −2x + 3
b. y = −2x + 1.5
c. y = 2x + 3
d. y = 2x + 1.5

Answer:
a. y = −2x + 3

Explanation:
From the table, you can see that the y-intercept (when x = 0) is b = 3. Comparable to y = mx + b
The Option B and D are rejected.
Since the graph is slanting downwards, the slope is negative.
Option C is rejected
The graph represents y = -2x + 3

Question 3.
The table below represents a linear relationship.

What is the y-intercept?
Options:
a. -4
b. -2
c. 2
d. 3

Answer:
b. -2

Explanation:
Find the rate of change
(7 – 4)/(3 – 2) = (10 – 7)/(4 – 3) = 3
Find the value of y for x = 0
Works backward from x = 2 to x = 1
x = 2 – 1 = 1
y = 4 – 3 = 1
x = 1 – 1 = 0
y = 1 – 3 = -2
y-intercept = -2

Question 4.
Which equation represents a nonproportional relationship?
Options:
a. y = 3x + 0
b. y = −3x
c. y = 3x + 5
d. y = \(\frac{1}{3}\)x

Answer:
c. y = 3x + 5

Explanation:
For a non-proportional relationship, the equation is y = mx + b and b is not equal to 0.
Option C represents a non-proportional relationship y = 3x + 5

Question 5.
The table shows a proportional relationship. What is the missing y-value?

Options:
a. 16
b. 20
c. 18
d. 24

Answer:
c. 18

Explanation:
Find the ratio y/x
6/4 = 3/2
Since the relationship is proportional, the ratio is constant.
Using the ratio to find missing y
3/2 = y/12
y = 3/2 × 12 = 18

Question 6.
What is 0.00000598 written in scientific notation?
Options:
a. 5.98 × 10^-6
b. 5.98 × 10^-5
c. 59.8 × 10^-6
d. 59.8 × 10^-7

Answer:
c. 59.8 × 10^-6

Explanation:
0.00000598
Move the decimal 6 points
59.8 × 10^-6

Mini-Task

Question 7.
The graph shows a linear relationship.

a. Is the relationship proportional or nonproportional?
____________

Answer:
It represents a non-proportional linear relationship

Explanation:
The line of the graph does not pass through the origin. The graph of a proportional relationship must pass through the origin.
It represents a non-proportional linear relationship

Question 7.
b. What is the slope of the line?
_______

Answer:
Slope m = -2

Explanation:
Find the slope using two points from the graph by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (0, -2) and (x2, y2) = (2, 1)
Slope m = (y2 -y1)/(x2 – x1) = (-3 -1)/(0 + 2) = -4/2 = -2

Question 7.
c. What is the y-intercept of the line?
_______

Answer:
y-intercept (b) = -3

Explanation:
From the graph when x = 0
y-intercept (b) = -3

Question 7.
d. What is the equation of the line?
Type below:
____________

Answer:
y = -2x – 3

Explanation:
Substitute m and b in the form: y = mx + b
y = -2x – 3

Conclusion:

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Lesson 1: Representing Proportional Relationships

• · Representing Proportional Relationships – Page No. 74
• · Representing Proportional Relationships – Page No. 75
• · Representing Proportional Relationships Lesson Check – Page No. 76

Lesson 2: Rate of Change and Slope

• · Rate of Change and Slope – Page No. 80
• · Rate of Change and Slope – Page No. 81
• · Rate of Change and Slope Lesson Check – Page No. 82

Lesson 3: Interpreting the Unit As Slope

• · Interpreting the Unit As Slope – Page No. 86
• · Interpreting the Unit As Slope – Page No. 87
• · Interpreting the Unit As Slope Lesson Check – Page No. 88

Lesson 4: Representing Proportional Relationships – Model Quiz

• · Representing Proportional Relationships – Model Quiz – Page No. 89

Mixed Review

• Mixed Review – Page No. 90

Guided Practice – Representing Proportional Relationships – Page No. 74

Question 1.
Vocabulary
A proportional relationship is a relationship between two quantities in which the ratio of one quantity to the other quantity is/is not constant.
______ constant

Answer:
is constant

Explanation:
The rent would be proportional so therefore it is.

Question 2.
Vocabulary
When writing an equation of a proportional relationship in the form y = kx, k represents the __________________________.
______________

Answer:
constant of proportionality

Explanation:
When writing an equation of a proportional relationship in the form y = kx, k represents the constant of proportionality.

Representing Proportional Relationships With Equations Answer Key Question 3.
Write an equation that describes the proportional relationship between the number of days and the number of weeks in a given length of time.
a. Complete the table.

Type below:
______________

Answer:

Question 3.
b. Let x represent _____.
Let y represent _____.
The equation that describes the relationship is _____.
Type below:
______________

Answer:
Let x represent the time in weeks.
Let y represent the time in days.
The equation that describes the relationship is y = 7x.

Each table or graph represents a proportional relationship. Write an equation that describes the relationship.

Question 4.
Physical Science
The relationship between the numbers of oxygen atoms and hydrogen atoms in water.

Type below:
______________

Answer:
y = 2x

Explanation:
x represents the Oxygen atoms
y represents the Hydrogen atoms
For every point of the x-axis, the y-axis is varying with 2x times.
y = 2x

Question 5.

Type below:
______________

Answer:
y = 30x

Explanation:
x represents the Distance in inches
y represents the Actual Distance in miles
For every point of the x-axis, the y-axis is varying with 30x times.
y = 30x

Essential Question Check-In

Question 6.
If you know the equation of a proportional relationship, how can you draw the graph of the equation?
Type below:
______________

Answer:
Use the equation to make a table with x-values and y-values. Then graph the points (x, y) and draw a line through the points.

Independent Practice – Representing Proportional Relationships – Page No. 75

The table shows the relationship between temperatures measured on the Celsius and Fahrenheit scales.

Question 7.
Is the relationship between the temperature scales proportional? Why or why not?
______________

Answer:
No. The ratios of the numbers in each column are not equal.

Explanation:
Fahrenheit Temperature/Celsius Temperature = 50/10 = 5
86/30 = 2.87
122/50 = 2.44
The relationship is not propotional as the ratio is not constant.

Question 8.
Describe the graph of the Celsius-Fahrenheit relationship.
Type below:
______________

Answer:
A line starting at (0, 32) and slanting upward to the right.

Explanation:
The graph is a straight line with a y-intercept of 32.

Representing Proportional Relationships Worksheet Question 9.
Analyze Relationships
Ralph opened a savings account with a deposit of $100. Every month after that, he deposited $20 more.
a. Why is the relationship described not proportional?
Type below:
______________

Answer:
The account had a balance of $100, to begin with.

Question 9.
b. How could the situation be changed to make the situation proportional?
Type below:
______________

Answer:
Have Ralph open the account with no money, to begin with, and then put $20 every month.

Question 10.
Represent Real-World Problems
Describe a real-world situation that can be modeled by the equation y = \(\frac{1}{20}\)x. Be sure to describe what each variable represents.
Type below:
______________

Answer:
If x is the number of nickels you have, y = \(\frac{1}{20}\)x is the amount of money you have in dollars.

Look for a Pattern

The variables x and y are related proportionally.

Question 11.
When x = 8, y = 20. Find y when x = 42.
_______

Answer:
y = 105

Explanation:
x = 8, y = 20
y/x = 20/8
y = 20x/8
when x = 42
y = (20 × 42)/8
y = 105

Using Proportional Relationships Answer Key Question 12.
When x = 12, y = 8. Find x when y = 12.
_______

Answer:
x = 18

Explanation:
x/y = 12/8
x = 12y/8
when y = 12
x = (12 × 12)/8
x = 18

Representing Proportional Relationships – Page No. 76

Question 13.
The graph shows the relationship between the distance that a snail crawls and the time that it crawls.
a. Use the points on the graph to make a table.


Type below:
______________

Answer:

Question 13.
b. Write the equation for the relationship and tell what each variable represents.
Type below:
______________

Answer:
y = 1/10 x, where y is the time in minutes and x is the distance in inches

Question 13.
c. How long does it take the snail to crawl 85 inches?
_______ minutes

Answer:
8.5 minutes

H.O.T.

Focus on Higher Order Thinking

Question 14.
Communicate Mathematical Ideas
Explain why all of the graphs in this lesson show the first quadrant but omit the other three quadrants.
Type below:
______________

Answer:
All of the graphs represent real-world data for which both x and y take on only nonnegative values, which graph in the first quadrant or on the axes. If either x or y or both could be negative, then other quadrants would be needed.

Representing Proportional Relationships Question 15.
Analyze Relationships
Complete the table.

a. Are the length of a side of a square and the perimeter of the square related proportionally? Why or why not?
______________

Answer:
Yes. The ratio of the perimeter of a square to its side length is always 4.

Question 15.
b. Are the length of a side of a square and the area of the square related proportionally? Why or why not?
______________

Answer:
No. The ratio of the area of a square to its side length is not constant

Question 16.
Make a Conjecture
A table shows a proportional relationship where k is the constant of proportionality. The rows are then switched. How does the new constant of proportionality relate to the original one?
Type below:
______________

Answer:
It is the reciprocal of the original constant of proportionality

Guided Practice – Rate of Change and Slope – Page No. 80

Tell whether the rates of change are constant or variable.

Question 1.
building measurements _____

______________

Answer:
constant

Explanation:
Input variable: Feet
Output variable: Yard
For every point of the Yard, the Feet is increasing 3 times.
So, the answer is constant.

Question 2.
computers sold _____

______________

Answer:
variable

Explanation:
Identify the input and output variables
Input variable: Week
Output variable: The number sold
x-axis and y-axis points are not varying constantly. So, the answer is variable.

Question 3.
distance an object falls _____

______________

Answer:
variable

Explanation:
Identify the input and output variables
Input variable: Time
Output variable: Distance
x-axis and y-axis points are not varying constantly. So, the answer is variable.

Question 4.
cost of sweaters _____

______________

Answer:
constant

Explanation:
Identify the input and output variables
Input variable: Number
Output variable: Cost
x-axis and y-axis points are varying constantly. So, the answer is constant.

Erica walks to her friend Philip’s house. The graph shows Erica’s distance from home over time.

Question 5.
Find the rate of change from 1 minute to 2 minutes.

\(\frac{\text { change in distance }}{\text { change in time }}=\frac{400-?}{2-?}=\frac{?}{?}=?\) ft per min
________ feet per minute

Answer:
200 feet per minute

Explanation:
(400 – 200)/(2 – 1)
200/1
200 feet per minute

Question 6 (request help)
Find the rate of change from 1 minute to 4 minutes.
______ feet per minute

Answer:
200 ft per min

Explanation:
change in distance/change in time
(800 – 200)/(4 – 1)
600/3 = 200 ft per min

Find the slope of each line.

Question 7.

slope = _____
________

Answer:
slope = -2

Explanation:
From the given image, the x-axis is separated by 1 unit.
the y-axis is separated with 1 unit.
For every 1 unit of the x-axis, the slope is taken -2 units on the y-axis
The slope of the line is -2

Lesson 8 Graph Proportional Relationships and Define Slope Question 8.

slope = _____
\(\frac{□}{□}\)

Answer:
\(\frac{3}{2}\)

Explanation:
From the given image, the slope is 3/2

Essential Question Check-In

Question 9.
If you know two points on a line, how can you find the rate of change of the variables being graphed?
Type below:
_____________

Answer:
Find the coordinates of two points on the line. Then divide the change in y-values from one point to the next by the change in x-values.

Independent Practice – Rate of Change and Slope – Page No. 81

Question 10.
Rectangle EFGH is graphed on a coordinate plane with vertices at E(-3, 5), F(6, 2), G(4, -4), and H(-5, -1).
a. Find the slopes of each side.
Type below:
_____________

Answer:
Slope EF = 1/3
slope FG = 3
slope GH = -1/3
slope HE =3

Question 10.
b. What do you notice about the slopes of opposite sides?
Type below:
_____________

Answer:
They are the same.

Question 10.
c. What do you notice about the slopes of adjacent sides?
Type below:
_____________

Answer:
They are negative reciprocals of one another.

Question 11.
A bicyclist started riding at 8:00 A.M. The diagram below shows the distance the bicyclist had traveled at different times. What was the bicyclist’s average rate of speed in miles per hour?

_______ miles per hour

Answer:
15 miles per hour

Explanation:
Total distance traveled by bicyclist = 4.5 mile + 7.5 mile = 12 mile
Total time taken by bicyclist = (8:48 A.M – 8:00 A.M) = 48 min = 0.8 hr
conversion used : ( 1 hour = 60 minute)
Average rate of speed = total distance/total time = 15 mile/hr

Lesson 3.2 Rate of Change and Slope Answer Key Question 12.
Multistep
A line passes through (6, 3), (8, 4), and (n, -2). Find the value of n.
_______

Answer:
n = -4

Explanation:
A line passes through (6, 3), (8, 4), and (n, -2).
From the given information, for every 2 points on the x-axis, the y-values are changing one point.
(4, 2), (2, 1), (0, 0), (-2, -1), (-4, -2)

Question 13.
A large container holds 5 gallons of water. It begins leaking at a constant rate. After 10 minutes, the container has 3 gallons of water left.
a. At what rate is the water leaking?
_______ gallons per minute

Answer:
1 gallon every 5 minutes, or 0.2 gal/min

Explanation:
Rate = (5 – 3)/(0 – 10)
= 2/-10
= -0.2
The rate of water leaking is 1 gallon every 5 minutes, or 0.2 gal/min

Question 13.
b. After how many minutes will the container be empty?
_______ minutes

Answer:
25 minutes

Explanation:
Number of minutes = 5/0.2 = 25
It will take 25 minutes for the container to be empty.

Question 14.
Critique Reasoning
Billy found the slope of the line through the points (2, 5) and (-2, -5) using the equation \(\frac{2-(-2)}{5-(-5)}=\frac{2}{5}\). What mistake did he make?
Type below:
_____________

Answer:
He used the change in x over the change in y instead of the change in y over the change in x.

Rate of Change and Slope – Page No. 82

Question 15.
Multiple Representations
Graph parallelogram ABCD on a coordinate plane with vertices at A(3, 4), B(6, 1), C(0, -2), and D(-3, 1).
a. Find the slope of each side.

Type below:
_____________

Answer:
slope AB = -1;
slope BC = 1/2
slope CD = -1;
slope DA = 1/2

Explanation:

Question 15.
b. What do you notice about the slopes?
Type below:
_____________

Answer:
The slopes of the opposite sides are the same.

Question 15.
c. Draw another parallelogram on the coordinate plane. Do the slopes have the same characteristics?
Type below:
_____________

Answer:
Yes; opposite sides still have the same slope.

H.O.T.

Focus on Higher Order Thinking

Question 16.
Communicate Mathematical Ideas
Ben and Phoebe are finding the slope of a line. Ben chose two points on the line and used them to find the slope. Phoebe used two different points to find the slope. Did they get the same answer? Explain.
_____________

Answer:
Yes. The slope of a line is constant. Therefore, the slope that you calculate will be the same no matter which two points you choose.

Proportional Relationships and Slope Question 17.
Analyze Relationships
Two lines pass through the origin. The lines have slopes that are opposites. Compare and contrast the lines.
Type below:
_____________

Answer:
One line has a positive slope and one has a negative slope. The lines are equally steep, but one slants upward left to right while the other slants downward left to right. The lines cross at the origin.

Question 18.
Reason Abstractly
What is the slope of the x-axis? Explain.
_____________

Answer:
Zero. The rise along the x-axis is zero, while the run along the x-axis is not zero. The slope zero/run or zero.

Guided Practice – Interpreting the Unit As Slope – Page No. 86

Give the slope of the graph and the unit rate.

Question 1.
Jorge: 5 miles every 6 hours

\(\frac{□}{□}\) miles per hour

Answer:
\(\frac{5}{6}\) miles per hour

Explanation:

Slope = rise/run = 5/6
The unit rate a the distance traveled and the slope a the graph of the relationship is equal, 5/6 miles per hour
Calculate miles PER hour
5/6

Interpreting the Unit Rate as Slope Lesson 3.3 Answer Key Question 2.
Akiko

\(\frac{□}{□}\) miles per hour

Answer:
\(\frac{5}{4}\) miles per hour

Explanation:
Calculate miles PER hour
5 miles/4hours = 5/4 miles per hour

Question 3.
The equation y = 0.5x represents the distance Henry hikes, in miles, over time, in hours. The graph represents the rate which Clark hikes. Determine which hiker is faster. Explain.

___________

Answer:
Clark is faster.

Explanation:
From the equation, Henry’s rate is equal to 0.5, or 1/2 mile per hour. Clark’s rate is the slope of the line, which is 3/2 or 1.5 miles per hour.

Write an equation relating the variables in each table.

Question 4.

Type below:
___________

Answer:
y = 15x

Explanation:
y/x = 15/1
y = 15x
Multiply 15 with the x values to get the y values.
y = 15x

Constant of Proportionality Worksheet With Answers Question 5.

Type below:
___________

Answer:
y = 3/8 x

Explanation:
y/x = 6/16
y/x = 3/8
y = 3/8 x

Essential Question Check-In

Question 6.
Describe methods you can use to show a proportional relationship between two variables, x and y. For each method, explain how you can find the unit rate and the slope.
Type below:
___________

Answer:
The ratio of y to x gives the unit rate and slope.

Explanation:
If the equation can be written as y = mx, then m is the unit rate and the slope. Graph: When the line passes through the origin, then the value of r at the point (1, r) is the unit rate and the slope.

Independent Practice – Interpreting the Unit As Slope – Page No. 87

Question 7.
A Canadian goose migrated at a steady rate of 3 miles every 4 minutes.
a. Fill in the table to describe the relationship.

Type below:
___________

Answer:

Explanation:
Canadian goose migrated at a steady rate of 3 miles every 4 minutes
y/x = 3/4; x/y = 4/3
y = 3/4 x; x = 4/3 y
If x = 8, y = 3/4 × 8 = 6
If y = 9, x = 4/3 × 9 = 12
If y = 12, x = 4/3 × 12 = 16
If x = 20, y = 3/4 × 20 = 15

Question 7.
b. Graph the relationship.

Type below:
___________

Answer:

Explanation:
The points are (4, 3), (8, 6), (12, 9), (16, 12), (20, 15)

Question 7.
c. Find the slope of the graph and describe what it means in the context of this problem.
Slope: \(\frac{□}{□}\) miles per minute

Answer:
Slope: \(\frac{3}{4}\) miles per minute

Explanation:
The unit rate of migration of the goose and the slope of the graph both equal 3/4 mi/min

Question 8.
Vocabulary
A unit rate is a rate in which the first quantity / second quantity in the comparison is one unit.
___________

Answer:
second quantity

Explanation:
A unit rate is a rate in which the “second quantity” in the comparison is one unit

Question 9.
The table and the graph represent the rate at which two machines are bottling milk in gallons per second.

a. Determine the slope and unit rate of each machine.
Type below:
___________

Answer:
Machine 1: slope = unit rate = 0.6/1 = 0.6 gal/s
Machine 2: slope = unit rate = 3/4 = 0.75 gal/s

Question 9.
b. Determine which machine is working at a faster rate.
___________

Answer:
Machine 2 is working at a faster rate since 0.75 > 0.6

Interpreting the Unit As Slope – Page No. 88

Question 10.
Cycling
The equation y = \(\frac{1}{9}\) x represents the distance y, in kilometers, that Patrick traveled in x minutes while training for the cycling portion of a triathlon. The table shows the distance y Jennifer traveled in x minutes in her training. Who has the faster training rate?

___________

Answer:
Jennifer has a faster training rate

Explanation:
Patrick’s rate is 1/9 kilometer per minute. Jennifer’s rate is 1/8 kilometer per minute. 1/9 < 1/8. So, Jennifer has a faster training rate.

H.O.T.

Focus on Higher Order Thinking

Question 11.
Analyze Relationships
There is a proportional relationship between minutes and dollars per minute, shown on a graph of printing expenses. The graph passes through the point (1, 4.75). What is the slope of the graph? What is the unit rate? Explain.
______ dollars per minute

Answer:
$4.75/min

Explanation:
slope = unit rate = 4.75.
If the graph of a proportional relationship passes through the point (1, r), then r equals the slope and the unit rate, which is $4.75/min.

Question 12.
Draw Conclusions
Two cars start at the same time and travel at different constant rates. A graph for Car A passes through the point (0.5, 27.5), and a graph for Car B passes through (4, 240). Both graphs show distance in miles and time in hours. Which car is traveling faster? Explain.
___________

Answer:
Car B

Explanation:
The slope and unit rate of speed of Car A is (27.5 – 0)/(0.5 – 0) = 27.5/0.5 = 55 mi/h.
The slope and unit rate of speed of Car B is (240 – 0)/(4 – 0) = 240/4 = 60 mi/h.
60 > 55, so Car B is traveling faster.

Question 13.
Critical Thinking
The table shows the rate at which water is being pumped into a swimming pool. Use the unit rate and the amount of water pumped after 12 minutes to find how much water will have been pumped into the pool after 13 \(\frac{1}{2}\) minutes. Explain your reasoning.

______ gallons

Answer:
243 gallons

Explanation:
The unit rate is 36/2 = 18gal/min.
So, 1 1/2 minutes after 12 minutes, an additional 18 × 1 1/2 = = 27 gallons will be pumped in.
So, the total is 216 + 27 = 243 gal.

3.1 Representing Proportional Relationships – Model Quiz – Page No. 89

Question 1.
Find the constant of proportionality for the table of values.

______

Answer:
1.5

Explanation:
constant of proportionality
k = y/x = 3/2 = 1.5
k = y/x = 4.5/3 = 1.5
k = y/x = 6/4 = 1.5
k = y/x = 7.5/5 = 1.5

Proportional Relationship 8th Grade Question 2.
Phil is riding his bike. He rides 25 miles in 2 hours, 37.5 miles in 3 hours, and 50 miles in 4 hours. Find the constant of proportionality and write an equation to describe the situation.
Type below:
___________

Answer:
The constant of proportionality is 12.5 miles per hour.

Explanation:
The equation is d = 12.5 × t
25 miles ÷ 2 hours = 12.5 miles/hour
A direct proportionality d = 12.5 × t

3.2 Rate of Change and Slope

Find the slope of each line.

Question 3.

______

Answer:
Slope = 3

Question 4.

______

Answer:
Slope = -5

3.3 Interpreting the Unit Rate as Slope

Question 5.
The distance Train A travels is represented by d = 70t, where d is the distance in kilometers and t is the time in hours. The distance Train B travels at various times is shown in the table. What is the unit rate of each train? Which train is going faster?

____________

Answer:
Train A: 70 km per hour; Train B: 75 km per hour; Train B is faster.

Explanation:
The distance Train A travels is represented by d = 70t
where d is the distance in kilometers and t is the time in hours
The speed of train A is 70 kilometer per hour
To find the speed of train B use the table
Slope = (300 – 150)/(4 – 2) = 75
The speed of train B is 75 km per hour
The speed of train B is more

Essential Question

Question 6.
What is the relationship among proportional relationships, lines, rates of change, and slope?
Type below:
____________

Answer:
The relationship between the x-axis and y-axis of any graph is a proportional Relationship which is defined by slope i.e. calculating the rate of change of the plotted line.
1) Here proportional change and rate of change are algebraic quantities which specify how one quantity changes with respect to another.
2) Line and Slope are geometric quantities which describe the graph of any equation.

Selected Response – Mixed Review – Page No. 90

Question 1.
Which of the following is equivalent to 5^-1?
Options:
a. 4
b. \(\frac{1}{5}\)
c. −\(\frac{1}{5}\)
d. -5

Answer:
b. \(\frac{1}{5}\)

Explanation:
5^-1
1/5

Question 2.
Prasert earns $9 an hour. Which table represents this proportional relationship?

Options:
a. A
b. B
c. C
d. D

Answer:
a. A

Explanation:
a. 36/4 = 9; 54/6 = 9; 72/8 = 9
b. 36/4 = 9; 45/6 = 7.5; 54/8 = 6.75
c. 9/2 = 4.5
d. 18/2 = 9; 27/3 = 9; 54/4 = 13.5

Chapter 3 Math Test 8th Grade Answers Question 3.
A factory produces widgets at a constant rate. After 4 hours, 3,120 widgets have been produced. At what rate are the widgets being produced?
Options:
a. 630 widgets per hour
b. 708 widgets per hour
c. 780 widgets per hour
d. 1,365 widgets per hour

Answer:
c. 780 widgets per hour

Explanation:
A factory produces widgets at a constant rate. After 4 hours, 3,120 widgets have been produced.
3,120/4 = 780 widgets per hour

Question 4.
A full lake begins dropping at a constant rate. After 4 weeks it has dropped 3 feet. What is the unit rate of change in the lake’s level compared to its full level?
Options:
a. 0.75 feet per week
b. 1.33 feet per week
c. -0.75 feet per week
d. -1.33 feet per week

Answer:
c. -0.75 feet per week

Explanation:
A full lake begins dropping at a constant rate. After 4 weeks it has dropped 3 feet.
(-3 ft)/(4 weeks) = -3/4 ft/wk = -0.75 ft/wk

Question 5.
What is the slope of the line below?

Options:
a. -2
b. −\(\frac{1}{2}\)
c. \(\frac{1}{2}\)
d. 2

Answer:
c. \(\frac{1}{2}\)

Explanation:
(-1.5, 1.5) & (1.5, 0)
(0 – 1..5)/(1.5 – (-1.5))
1.5/3
= 1/2

Question 6.
Jim earns $41.25 in 5 hours. Susan earns $30.00 in 4 hours. Pierre’s hourly rate is less than Jim’s, but more than Susan’s. What is his hourly rate?
Options:
a. $6.50
b. $7.75
c. $7.35
d. $8.25

Answer:
b. $7.75

Explanation:
Jim earns $41.25 in 5 hours.
$41.25/5 = 8.25
Jim’s unit rate is $8.25 per hour
30/4 = 7.5
Pierre’s hourly rate is is less than $8.25 but more than $7.50
$7.75

Mini-Task

Question 7.
Joelle can read 3 pages in 4 minutes, 4.5 pages in 6 minutes, and 6 pages in 8 minutes.
a. Make a table of the data.

Type below:
______________

Answer:

Explanation:
Joelle can read 3 pages in 4 minutes,
y/x = 3/4
y = 3/4 x
If x = 6, y = 3/4 × 6 = 4.5
If x = 8, y = 3/4 × 8 = 6

Question 7.
b. Use the values in the table to find the unit rate.
_______ pages per minute

Answer:
0.75 pages per minute

Explanation:
3/4 = 0.75
4.5/6 = 0.75
6/8 = 0.75
The unit rate is 0.75 pages per minute

Question 7.
c. Graph the relationship between minutes and pages read.
Type below:
______________

Answer:

Conclusion:

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Lesson 1: Equations with the Variable on Both Sides

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

Lesson 2: Equations with Rational Numbers

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

Lesson 3: Equations with the Distributive Property

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

Lesson 4: Equations with Many Solutions or No Solution

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

Lesson 5: Equations with the Variable on Both Sides

• · Equations with the Variable on Both Sides – Model Quiz – Page No. 221

Reviews

• • Mixed Review – Page No. 222

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

Use algebra tiles to model and solve each equation.

Question 1.
x + 4 = -x – 4
x = ________

Answer:
x = -4

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

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

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

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

x = -4

Question 2.
2 – 3x = -x – 8
x = ________

Answer:

Explanation:
Given 2 – 3x = -x – 8
Model 2-3x on the left side of the mat and -x-8 on the right side.

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

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

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

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

Solving Linear Equations Grade 8 Question 3.
At Silver Gym, membership is $25 per month, and personal training sessions are $30 each. At Fit Factor, membership is $65 per month, and personal training sessions are $20 each. In one month, how many personal training sessions would Sarah have to buy to make the total cost at the two gyms equal?
________ sessions

Answer:
4 sessions

Explanation:
At Silver Gym, membership is $25 per month, and personal training sessions are $30 each.
Membership + Personal training session = 25 + 30x
At Fit Factor, membership is $65 per month, and personal training sessions are $20 each.
Membership + Personal training session = 65 + 20x
Membership at Silver Gym = Membership at Fit Factor
25 + 30x = 65 + 20x
30x – 20x = 65 – 25
10x = 40
x = 4
Sarah would have to buy 4 sessions for the total cost at the two gyms to be equal.

Question 4.
Write a real-world situation that could be modeled by the equation 120 + 25x = 45x.
Type below:
_______________

Answer:
120 + 25x = 45x
Sarah offers a plan to tutor a student at $25 per her plus a one-time registration fee of $ 120.
Surah offers an alternative plan to tutor a student at $45 per hour and no registration fee.
120 + 25x = 45x

Question 5.
Write a real-world situation that could be modeled by the equation 100 – 6x = 160 – 10x.
Type below:
_______________

Answer:
100 – 6x = 160 – 10x
The initial water in Tank A is 100 gallons and leaks at 6 gallons per week.
The initial water in Tank B is 160 gallon and leaks at 10 gallons per week
100 – 6x = 160 – 10x

Essential Question Check-In

Question 6.
How can you solve an equation with the variable on both sides?
Type below:
_______________

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

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

Question 7.
Derrick’s Dog Sitting and Darlene’s Dog Sitting are competing for new business. The companies ran the ads shown.

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

Answer:
3 hours

Explanation:
Hourly rate + One time fee = 5x + 12
Hourly rate + One time fee = 3x + 18
5x + 12 = 3x + 18
5x – 3x = 18 – 12
2x = 6
x = 3
The cost of the two dog sitting would be the same for 3 hrs.

Question 7.
b. Analyze Relationships
Which dog-sitting service is more economical to use if you need 5 hours of service? Explain.
____________

Answer:
Darlene’s Dog Sitting would be cheaper

Explanation:
Let y be the cost of dog sitting after x hours for both companies
y = 5x +12
y = 3x +18
Substitute x = 5
y = 5(5) + 12 = 37
y = 3 (5) + 18 = 33
Compare the cost for both companies for x = 5hr.
$37 > $33
Darlene’s Dog Sitting would be cheaper

Linear Equations 8th Grade Question 8.
Country Carpets charges $22 per square yard for carpeting and an additional installation fee of $100. City Carpets charges $25 per square yard for the same carpeting and an additional installation fee of $70.
a. Write and solve an equation to find the number of square yards of carpeting for which the total cost charged by the two companies will be the same.
_______ square yards

Answer:
10 square yards

Explanation:
Unit square rate + One time installation fee = 22x + 100
Unit square rate + One time installation fee = 25x + 70
22x + 100 = 25x + 70
25x – 22x = 100 – 70
3x = 30
x = 10
The total cost charged by the two companies will be the same for 10 square yards of carpeting.

Question 8.
b. Justify Reasoning
Mr. Shu wants to hire one of the two carpet companies to install carpeting in his basement. Is he more likely to hire Country Carpets or City Carpets? Explain your reasoning.
___________

Answer:
City Carpets are cheaper when x < 10
y = 25(9) + 70 = 295
y = 22(9) + 100 = 298
Country Carpets are cheaper when x > 10
y = 25(11) + 70 = 345
y = 25(11) + 100 = 342
If Mr.Shu needs the carpeting done for less than 10 square yards, he will hire City Carpets and if he needs carpeting for more than 10 square yards, he will hire Country Carpets.

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

Question 9.
Two less than 3 times a number is the same as the number plus 10.
________

Answer:
3x – 2 = x + 10
x = 6

Explanation:
Two less than 3 times a number is the same as the number plus 10.
Two less than 3 times x is the same as the x plus 10.
Two less than 3x is the same as the x + 10
3x – 2 is the same as x + 10
3x – 2 = x + 10
3x – x = 10 + 2
2x = 12
x = 6

Question 10.
A number increased by 4 is the same as 19 minus 2 times the number.
______

Answer:
x + 4 = 19 – 2x
x = 5

Explanation:
A number increased by 4 is the same as 19 minus 2 times the number.
x increased by 4 is the same as 19 minus 2x.
x + 4 is the same as 19 – 2x
x + 4 = 19 – 2x
x + 2x = 19 – 4
3x = 15
x = 15/3
x = 5

Question 11.
Twenty less than 8 times a number is the same as 15 more than the number.
Type below:
____________

Answer:
8x – 20 = x + 15
x = 5

Explanation:
Twenty less than 8 times a number is the same as 15 more than the number.
Twenty less than 8 times x is the same as 15 more than the x.
Twenty less than 8x is the same as 15 more than the x
8x – 20 is the same as x + 15
8x – 20 = x + 15
8x – x = 15 + 20
7x = 35
x = 35/7 = 5
x = 5

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

Question 12.
The charges for an international call made using the calling card for two phone companies are shown in the table.

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

Answer:
10 minutes

Explanation:
Cost of minutes + One time fee = 3x + 35
Cost of minutes + One time fee = 2x + 45
3x + 35 = 2x + 45
3x – 2x = 45 – 35
x = 10
The cost would be same for 10 minutes.

Question 12.
b. Analyze Relationships
When is it better to use the card from Company B?
Type below:
____________

Answer:
y = 3x + 35
y = 3(11) + 35 = $68
y = 2x + 45
y = 2(11) + 45 = $67
Since when x > 10, Company B is cheaper so it should be better to use when the length of the call is greater than 10 minutes.

H.O.T.

Focus on Higher Order Thinking

Question 13.
Draw Conclusions
Liam is setting up folding chairs for a meeting. If he arranges the chairs in 9 rows of the same length, he has 3 chairs left over. If he arranges the chairs in 7 rows of that same length, he has 19 left over. How many chairs does Liam have?
______ chairs

Answer:
75 chairs

Explanation:
Number of chairs in 9 row + leftover chairs = 9x + 3
Number of chairs in 7 row + leftover chairs = 7x + 19
9x + 3 = 7x + 19
9x – 7x = 19 – 3
2x = 16
x = 16/2
x = 8
Total number of chairs = 9(8) + 3 = 75

Solving Linear Equations 8th Grade Question 14.
Explain the Error
Rent-A-Tent rents party tents for a flat fee of $365 plus $125 a day. Capital Rentals rents party tents for a flat fee of $250 plus $175 a day. Delia wrote the following equation to find the number of days for which the total cost charged by the two companies would be the same:
365x + 125 = 250x + 175
Find and explain the error in Delia’s work. Then write the correct equation.
Type below:
____________

Answer:
Delia’s equation
365x + 125 = 250x + 175
The error is that she attached the variable with the flat fee (which is constant) and put the daily rent as a constant (which is variable).
Correct equation
125x + 365 = 175x + 250

Question 15.
Persevere in Problem-Solving
Lilliana is training for a marathon. She runs the same distance every day for a week. On Monday, Wednesday, and Friday, she runs 3 laps on a running trail and then runs 6 more miles. On Tuesday and Sunday, she runs 5 laps on the trail and then runs 2 more miles. On Saturday, she just runs laps. How many laps does Lilliana run on Saturday?
________ laps

Answer:
2 laps

Explanation:
Miles covered by lap + Addition number of miles = 3x + 6
Miles covered by lap + Addition number of miles = 5x + 2
3x + 6 = 5x + 2
5x – 3x = 6 – 2
2x = 4
x = 4/2
x = 2

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

Question 1.
Sandy is upgrading her Internet service. Fast Internet charges $60 for installation and $50.45 per month. Quick Internet has free installation but charges $57.95 per month.
a. Write an equation that can be used to find the number of months at which the Internet service would cost the same.
Type below:
____________

Answer:
50.45x + 60 = 57.95x

Explanation:
Write an equation for Fast Internet, where x is the number of months.
Charge per Month × Number of Months + Installation Fee
50.45x + 60
Write an equation for Quick Internet, where x is the number of months.
Charge per Month × Number of Months + Installation Fee
57.95x
50.45x + 60 = 57.95x

Question 1.
b. Solve the equation.
_______ hours

Answer:
8

Explanation:
50.45x + 60 = 57.95x
57.95x – 50.45x = 60
7.5x = 60
x = 60/7.5
x = 8
The total cost will be the same for 8 months.

Solve.

Question 2.
\(\frac{3}{4}\) n – 18 = \(\frac{1}{4}\) n – 4
______

Answer:
n = 28

Explanation:
3/4 . n – 18 = 1/4 . n – 4
Determine the least common multiple of the denominators
LCM is 4
Multiply both sides of the equation by the LCM
4(3/4 . n – 18) = 4(1/4 . n – 4)
3n – 72 = n – 16
3n – n = -16 + 72
2n = 56
n = 56/2
n = 28

Question 3.
6 + \(\frac{4}{5}\) b = \(\frac{9}{10}\) b
_______

Answer:
b = 60

Explanation:
6 + \(\frac{4}{5}\) b = \(\frac{9}{10}\) b
LCM is 10
10(6 + \(\frac{4}{5}\) b) = 10(\(\frac{9}{10}\) b)
60 + 8b = 9b
9b – 8b = 60
b = 60

Solving Linear Equations Worksheets Grade 8 Pdf Question 4.
\(\frac{2}{11}\) m + 16 = 4 + \(\frac{6}{11}\) m
_______

Answer:
m = 33

Explanation:
\(\frac{2}{11}\) m + 16 = 4 + \(\frac{6}{11}\) m
The LCM is 11
11(\(\frac{2}{11}\) m + 16) = 11(4 + \(\frac{6}{11}\) m)
2m + 176 = 44 + 6m
6m – 2m = 176 – 44
4m = 132
m = 132/4
m = 33

Question 5.
2.25t + 5 = 13.5t + 14
_______

Answer:
t = -0.8

Explanation:
2.25t + 5 = 13.5t + 14
13.5t – 2.25t = 5 – 14
11.25t = -9
t = -9/11.25
t = -0.8

Question 6.
3.6w = 1.6w + 24
_______

Answer:
w = 12

Explanation:
3.6w = 1.6w + 24
3.6w – 1.6w = 24
2w = 24
w = 24/2
w = 12

Question 7.
-0.75p – 2 = 0.25p
_______

Answer:
p = -2

Explanation:
-0.75p – 2 = 0.25p
-2 = 0.25p + 0.75p
-2 = p
p = -2

Question 8.
Write a real-world problem that can be modeled by the equation 1.25x = 0.75x + 50.
Type below:
______________

Answer:
1.25x = 0.75x + 50.
Cell offers Plan A for no base fee and $1.25 per minute.
Cell offers Plan B for a $50 base fee and $0.75 per minute.
The equation shows when the total cost of the plan would be equal.

Essential Question Check-In

Question 9.
How does the method for solving equations with fractional or decimal coefficients and constants compare with the method for solving equations with integer coefficients and constants?
Type below:
______________

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

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

Question 10.
Members of the Wide Waters Club pay $105 per summer season, plus $9.50 each time they rent a boat. Nonmembers must pay $14.75 each time they rent a boat. How many times would a member and a non-member have to rent a boat in order to pay the same amount?
_______ times

Answer:
20 times

Explanation:
Members of the Wide Waters Club pay $105 per summer season, plus $9.50 each time they rent a boat.
9.5x + $105
Nonmembers must pay $14.75 each time they rent a boat.
9.5x + $105 = 14.75x
9.5x – 14.75x = $105
5.25x = 105
x = 105/5.25
x = 20
The cost for members and non-members will be the same for 8 visits.

Question 11.
Margo can purchase tile at a store for $0.79 per tile and rent a tile saw for $24. At another store, she can borrow the tile saw for free if she buys tiles there for $1.19 per tile. How many tiles must she buy for the cost to be the same at both stores?
_______ tiles

Answer:
60 tiles

Explanation:
Margo can purchase tile at a store for $0.79 per tile and rent a tile saw for $24.
0.79x + 24
At another store, she can borrow the tile saw for free if she buys tiles there for $1.19 per tile.
1.19x
0.79x + 24 = 1.19x
1.19x – 0.79x = 24
0.4x = 24
x = 24/0.4
x = 60
Margo should buy 60 tiles for the cost to be the same at both stores.

Question 12.
The charges for two shuttle services are shown in the table. Find the number of miles for which the cost of both shuttles is the same.

_______ miles

Answer:
40 miles

Explanation:
0.1x + 10
0.35x
0.1x + 10 = 0.35x
0.35x – 0.1x = 10
0.25x = 10
x = 10/0.25
x = 40
The cost of shuttles would be the same for 40 miles.

Question 13.
Multistep
Rapid Rental Car charges a $40 rental fee, $15 for gas, and $0.25 per mile driven. For the same car, Capital Cars charges $45 for rental and gas and $0.35 per mile.
a. For how many miles is the rental cost at both companies the same?
_______ miles

Answer:
100 miles

Explanation:
0.25x + 40 + 15 = 0.35x + 45
0.35x – 0.25x = 55 – 45
0.1x = 10
x = 10/0.1
x = 100
The cost of car rentals would be the same for 100 miles.

Question 13.
b. What is that cost?
$ _______

Answer:
$80

Explanation:
Let y be the total cost. Substitute 100 miles in any one of the two equations
y = 0.35x + 45
y = 0.35(100) + 45 = $80
Total cost would be $80.

Question 14.
Write an equation with the solution x = 20. The equation should have the variable on both sides, a fractional coefficient on the left side, and a fraction anywhere on the right side.
Type below:
______________

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

Explanation:
Write an equation with the solution x = 20. The equation should have the variable on both sides, a fractional coefficient on the left side, and a fraction anywhere on the right side.
1/3 . x = 1/3 . 20
1/3 . x + x = 1/3 . 20 + x
4/3x = 20/3 + x
4/3x + 10 = 20/3 + x + 10
4/3x + 10 = 50/3 + x

Question 15.
Write an equation with the solution x = 25. The equation should have the variable on both sides, a decimal coefficient on the left side, and a decimal anywhere on the right side. One of the decimals should be written in tenths, the other in hundredths.
Type below:
______________

Answer:
x=25
divide both sides by 25
x/25 = 1
convert 1/25 to decimal form 0.04
0.04x = 1
add x on both sides
1.04x = 1 + x
add 0.1 on both sides
1.04x + 0.1 = x + 1.1

Question 16.
Geometry
The perimeters of the rectangles shown are equal. What is the perimeter of each rectangle?

Perimeter = _______

Answer:
Perimeter = 3.2

Explanation:
Perimeter of the first rectangle
P = 2(n + n + 0.6) = 2(2n + 0.6) = 4n + 1.2
Perimeter of the second rectangle
P = 2(n + 0.1 + 2n) = 2(3n + 0.1) = 6n + 0.2
the perimeter is equal
4n + 1.2 = 6n + 0.2
6n – 4n = 1.2 – 0.2
2n = 1
n = 1/2
n = 0.5
P = 4n + 1.2 = 4(0.5) + 1.2 = 3.2

Question 17.
Analyze Relationships
The formula F = 1.8C + 32 gives the temperature in degrees Fahrenheit (F) for a given temperature in degrees Celsius (C). There is one temperature for which the number of degrees Fahrenheit is equal to the number of degrees Celsius. Write an equation you can solve to find that temperature and then use it to find the temperature
Type below:
______________

Answer:
x = 1.8x + 32

Explanation:
F = 1.8C +32
let x be the temperature such that it is the same in both Celsius and in Fahrenheit
Then the required equation is
x = 1.8x + 32
subtract 1.8x from both sides
-0.8x = 32
divide by -0.8 on both sides
x = -40
So -40 degree celsius

Equations with Rational Numbers – Page No. 208

Question 18.
Explain the Error
Agustin solved an equation as shown. What error did Agustin make? What is the correct answer?

x = _______

Answer:
x = -12

Explanation:
Agustin did not multiply by 12 on both sides in step 2. He only partially multiplied the variable and left the constants as such, which doesn’t make any sense.
The correct solution is
12(x/3 – 4) = 12(3x/4 + 1)
4x – 48 = 9x + 12
subtract 12 on both sides
4x – 60 = 9x
subtract 4x on both sides
-60 = 5x
x = -12

H.O.T.

Focus on Higher Order Thinking

Question 19.
Draw Conclusions
Solve the equation \(\frac{1}{2} x-5+\frac{2}{3} x=\frac{7}{6} x+4\). Explain your results.
Type below:
_____________

Answer:
\(\frac{1}{2} x-5+\frac{2}{3} x=\frac{7}{6} x+4\)
The least common multiple of the denominators: LCM(2, 3, 6) = 6
6(\(\frac{1}{2} x-5+\frac{2}{3} x=\frac{7}{6} x+4\))
6.1/2x – 6.5 + 6.2/3x = 6.7/6x +6.4
3x – 30 + 4x = 7x + 24
7x – 30 = 7x + 24
-30 = 24
This is not true. The equation has no solution.

Question 20.
Look for a Pattern
Describe the pattern in the equation. Then solve the equation.
0.3x + 0.03x + 0.003x + 0.0003x + .. = 3
x = ______

Answer:
x = 9

Explanation:
0.3x + 0.03x + 0.003x + 0.0003x + .. = 3
0.3x = 3
0.9x = 9
x = 9

Question 21.
Critique Reasoning
Jared wanted to find three consecutive even integers whose sum was 4 times the first of those integers. He let k represent the first integer, then wrote and solved this equation : k + (k + 1) + (k + 2) = 4k. Did he get the correct answer? Explain.
__________

Answer:
No, it is wrong on two accounts.
First, he has not specified if k is even or not. An easy way of doing so would assume x to be any integer and k=2a
This ensures that k is an even integer.
Nest the question asks for 3 consecutive even integers, Jared just took 3 consecutive integers, and thus at least 1 of them is odd.
So correct representation would be
k + (k+2) + (k + 4) = 4k
which upon solving yields k=6

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

Solve each equation.

Question 1.
4(x + 8) – 4 = 34 – 2x
________

Answer:
x = 1

Explanation:
4(x + 8) – 4 = 34 – 2x
4x + 32 – 4 = 34 – 2x
4x + 2x = 34 – 28
6x = 6
x = 6/6
x = 1

7.3 Equations with the Distributive Property Question 2.
\(\frac{2}{3}\)(9 + x) = -5(4 – x)
________

Answer:
x = 6

Explanation:
\(\frac{2}{3}\)(9 + x) = -5(4 – x)
2/3(9 + x) = -5(4 – x)
3 (2/3(9 + x)) = 3(-5(4 – x))
2(9 + x ) = -15 (4 – x)
18 + 2x = -60 + 15x
15x – 2x = 18 + 60
13x = 78
x = 78/13
x = 6

Question 3.
-3(x + 4) + 15 = 6 – 4x
________

Answer:
x = 3

Explanation:
-3(x + 4) + 15 = 6 – 4x
-3x – 12 + 15 = 6 – 4x
-3x + 3 = 6 – 4x
-3x + 4x = 6 – 3
x = 3

Question 4.
10 + 4x = 5(x – 6) + 33
________

Answer:
x = 7

Explanation:
10 + 4x = 5(x – 6) + 33
10 + 4x = 5x – 30 + 33
10 + 4x = 5x + 3
5x – 4x = 10 – 3
x = 7

Question 5.
x – 9 = 8(2x + 3) – 18
________

Answer:
x = -1

Explanation:
x – 9 = 8(2x + 3) – 18
x – 9 = 16x + 24 – 18
x – 9 = 16x + 6
16x – x = -9 – 6
15x = – 15
x = -15/15
x = -1

Question 6.
-6(x – 1) – 7 = -7x + 2
________

Answer:
x = 3

Explanation:
-6(x – 1) – 7 = -7x + 2
-6x + 6 – 7 = -7x + 2
-6x – 1 = -7x + 2
-7x + 6x = -1 -2
-x = -3
x = 3

Question 7.
\(\frac{1}{10}\)(x + 11) = -2(8 – x)
________

Answer:
x = 9

Explanation:
\(\frac{1}{10}\)(x + 11) = -2(8 – x)
10(\(\frac{1}{10}\)(x + 11)) = 10 (-2(8 – x))
x + 11 = -20(8 – x)
x + 11 = -160 + 20x
20x – x = 11 + 160
19x = 171
x = 171/19 = 9

Question 8.
-(4 – x) = \(\frac{3}{4}\)(x – 6)
________

Answer:
x = -2

Explanation:
-(4 – x) = \(\frac{3}{4}\)(x – 6)
4(-(4 – x)) = 4 (3/4(x – 6))
-16 + 4x = 3x – 18
4x – 3x = -18 + 16
x = -2

Question 9.
-8(8 – x) = \(\frac{4}{5}\)(x + 10)
________

Answer:
x = 10

Explanation:
-8(8 – x) = \(\frac{4}{5}\)(x + 10)
5(-8(8 – x)) = 5(\(\frac{4}{5}\)(x + 10))
-40(8 – x) = 4(x + 10)
-320 + 40x = 4x + 40
40x – 4x = 40 + 320
36x = 360
x = 360/36
x = 10

Question 10.
\(\frac{1}{2}\)(16 – x) = -12(x + 7)
________

Answer:
x = 8

Explanation:
\(\frac{1}{2}\)(16 – x) = -12(x + 7)
2 (\(\frac{1}{2}\)(16 – x)) = 2 (-12(x + 7))
16 – x = -24 (x + 7)
16 – x = -24x – 168
24x – x = -168 – 16
23x = 184
x = 184/23
x = 8

Lesson 7.3 Equations with the Distributive Property Answer Key Question 11.
Sandra saves 12% of her salary for retirement. This year her salary was $3,000 more than in the previous year, and she saved $4,200.What was her salary in the previous year?
Write an equation _____
Sandra’s salary in the previous year was _____
Salary = $ _____

Answer:
Write an equation 0.12x + 360 = 4200
Sandra’s salary in the previous year was $32000
Salary = $3000

Explanation:
0.12(x + 3000) = 4200
0.12x + 360 = 4200
0.12x = 4200 – 360
0.12x = 3840
x = 3840/0.12
x = 32000
Sandra’s salary in the previous year was $32000

Essential Question Check-In

Question 12.
When solving an equation using the Distributive Property, if the numbers being distributed are fractions, what is your first step? Why?
Type below:
___________

Answer:
Multiply both sides by the denominator of the fraction

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

Question 13.
Multistep
Martina is currently 14 years older than her cousin Joey. In 5 years she will be 3 times as old as Joey. Use this information to answer the following questions.
a. If you let x represent Joey’s current age, what expression can you use to represent Martina’s current age?
Type below:
___________

Answer:
y = x + 14

Explanation:
y = x + 14
where x is Joey’s current age and t is Martna’s current age.

Question 13.
b. Based on your answer to part a, what expression represents Joey’s age in 5 years? What expression represents Martina’s age in 5 years?
Type below:
___________

Answer:
Ages in 5 years
Joey’s age = x + 5
Martina’s age = x + 14 + 5 = x + 19

Question 13.
c. What equation can you write based on the information given?
Type below:
___________

Answer:
3(x + 5) = x + 19

Explanation:
In 5 years, Martina will be three times as old as Joey
3(x + 5) = x + 19

Question 13.
d. What is Joey’s current age? What is Martina’s current age?
Joey’s current age ___________
Martina’s current age ___________

Answer:
Joey’s current age 2
Martina’s current age 16

Explanation:
3(x + 5) = x + 19
3x + 15 = x + 19
3x – x = 19 – 15
2x = 4
x = 2

Question 14.
As part of a school contest, Sarah and Luis are playing a math game. Sarah must pick a number between 1 and 50 and give Luis clues so he can write an equation to find her number. Sarah says, “If I subtract 5 from my number, multiply that quantity by 4, and then add 7 to the result, I get 35.” What equation can Luis write based on Sarah’s clues and what is Sarah’s number?
Type below:
___________

Answer:
x = 12

Explanation:
As part of a school contest, Sarah and Luis are playing a math game. Sarah must pick a number between 1 and 50 and give Luis clues so he can write an equation to find her number. Sarah says, “If I subtract 5 from my number, multiply that quantity by 4, and then add 7 to the result, I get 35.”
4 (x – 5) + 7 = 35
4x – 20 + 7 = 35
4x – 13 = 35
4x = 35 + 13
4x = 48
x = 48/4
x = 12

Question 15.
Critical Thinking
When solving an equation using the Distributive Property that involves distributing fractions, usually the first step is to multiply by the LCD to eliminate the fractions in order to simplify computation. Is it necessary to do this to solve \(\frac{1}{2}\)(4x + 6) = 13(9x – 24)? Why or why not?
___________

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

Question 16.
Solve the equation given in Exercise 15 with and without using the LCD of the fractions. Are your answers the same?
___________

Answer:
x = 11

Explanation:
\(\frac{1}{2}\)(4x + 6) = 13(9x – 24)
6(\(\frac{1}{2}\)(4x + 6)) = 6(13(9x – 24))
3(4x + 6) = 2(9x – 24)
12x + 18 = 18x – 48
18x – 12x = 18 + 48
6x = 66
x = 66/6
x = 11

Equations with the Distributive Property – Page No. 214

Question 17.
Represent Real-World Problems
A chemist mixed x milliliters of 25% acid solution with some 15% acid solution to produce 100 milliliters of a 19% acid solution. Use this information to fill in the missing information in the table and answer the questions that follow.

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

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

Explanation:

Question 17.
b. What equation can you use to solve for x based on your answer to part a?
Type below:
_____________

Answer:
0.25x + 0.15(100 – x) = 19

Question 17.
c. How many milliliters of the 25% solution and the 15% solution did the chemist use in the mixture?

Type below:
_____________

Answer:
0.25x + 0.15(100 – x) = 19
0.25x + 15 – 0.15x = 19
0.1x + 15 = 19
0.1x = 4
x = 4/0.1
x = 40
The chemist used 40ml of the 25% solution and 100 – 40 = 60ml of the 15% solution.

H.O.T.

Focus on Higher Order Thinking

Question 18.
Explain the Error
Anne solved 5(2x) – 3 = 20x + 15 for x by first distributing 5 on the left side of the equation. She got the answer x = -3. However, when she substituted -3 into the original equation for x, she saw that her answer was wrong. What did Anne do wrong, and what is the correct answer?
x = ________

Answer:
x = -1.8

Explanation:
Dado que 5 solo se multiplica por 2x, no tiene sentido usar la distribución aquí. Básicamente, distribuir 5 fue el problema
Solución correcta:
5 (2x) – 3 = 20x + 15
10x -3 = 20x + 15
restar 15 en ambos lados
10x – 18 = 20x
restar 10x de ambos lados
-18 = 10x
x = -1.8

Question 19.
Communicate Mathematical Ideas
Explain a procedure that can be used to solve 5[3(x + 4) – 2(1 – x)] – x – 15 = 14x + 45. Then solve the equation.
x = ________

Answer:
x = 1

Explanation:
5[3(x + 4) – 2(1 – x)] – x – 15 = 14x + 45
5[3x + 12 – 2 + 2x] – x – 15 = 14x + 45
5[5x + 10] – x – 15 = 14x + 45
25x + 50 – x – 15 = 14x + 45
24x + 35 = 14x + 45
24x – 14x = 45 – 35
10x = 10
x = 1

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

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

Question 1.

The statement is: _______

Answer:
The statement is: true

Explanation:
3x – 2 = 25 – 6x
3x + 6x -2 = 25 -6x + 6x
9x – 2 = 25
9x -2 + 2 = 25 + 2
9x = 27
x = 27/9
x = 3
The statement is true.

Solving Equations by Clearing Fractions Worksheet Question 2.

____________

Answer:
The statement is false.

Explanation:
2x – 4 = 2(x – 1) + 3
2x – 4 = 2x – 2 + 3
2x – 4 = 2x + 1
2x – 4 – 2x = 2x + 1 – 2x
-4 not equal to 1
The statement is false.

Question 3.
How many solutions are there to the equation in Exercise 2?
____________

Answer:
There is no solution to exercise 2.

Question 4.
After simplifying an equation, Juana gets 6 = 6. Explain what this means.
____________

Answer:
When 6 = 6, there are infinite solutions.

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

Question 5.
Start with a _____ statement.
Add the _____ to both sides.
Add the _____ to both sides.
Combine _____ terms.

Type below:
____________

Answer:
Start with a “true” statement
Add the “same variable” to both sides
Add the “same constant” to both sides
Combine “like” terms

Explanation:
Start with a “true” statement
10 = 10
Add the “same variable” to both sides
10 + x = 10 + x
Add the “same constant” to both sides
10 + x + 5 = 10 + x + 5
Combine “like” terms
15 + x = 15 + x

Essential Question Check-In

Question 6.
Give an example of an equation with an infinite number of solutions. Then make one change to the equation so that it has no solution.
Type below:
____________

Answer:
An equation with infinitely many solutions
x – 2x + 3 = 3 – x
-x + 3 = 3 – x
+x/3 = +x/3
An equation for no solution
x – 2x + 3 = 3 – x + 4
-x + 3 = 7 – x
-x/3 = -x/7

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

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

Question 7.
-(2x + 2) – 1 = -x – (x + 3)
____________

Answer:
The statement is true

Explanation:
-(2x + 2) – 1 = -x – (x + 3)
-2x – 2 – 1 = -x – x + 3
-2x – 3 = -2x + 3
-3 = -3
The statement is true

Question 8.
-2(z + 3) – z = -z – 4(z + 2)
____________

Answer:
The statement is false.

Explanation:
-2(z + 3) – z = -z – 4(z + 2)
-3z – 6 = -3z -8
-3z -6 + 3z = -3z – 8 + 3z
-6 not equal to -8
The statement is false.

Create an equation with the indicated number of solutions.

Question 9.
No solution:
3(x – \(\frac{4}{3}\)) = 3x + _____
Type below:
______________

Answer:
3(x – \(\frac{4}{3}\)) = 3x + ?
3x – 4 = 3x + ?
3x – 4 = 3x + 2
When there is no solution, the statement should be false. Any number except -4 would make the equation have no solutions.

Question 10.
Infinitely many solutions:
2(x – 1) + 6x = 4( _____ – 1) + 2
Type below:
______________

Answer:
2(x – 1) + 6x = 4( _____ – 1) + 2
2(x – 1) + 6x = 4( ? – 1) + 2
2x – 2 + 6x = 4(? – 1) + 2
8x – 2 = 4(? – 1) + 2
8x – 2 = 4(2x – 1) + 2
8x – 2 = 8x – 4 + 2
8x – 2 = 8x – 2
When there are infinitely many solutions, the statement should be true

Question 11.
One solution of x = -1:
5x – (x – 2) = 2x – ( _____ )
Type below:
______________

Answer:
Put x = -1 in the equation
-5 – (-1 – 2) = -2 – blank
simplifying
-2 = -2 – blank
add 2 on both sides
0 = blank

Question 12.
Infinitely many solutions:
-(x – 8) + 4x = 2( _____ ) + x
Type below:
______________

Answer:
-(x – 8) + 4x = 2( ?) + x
-x + 8 + 4x = 2(?) + x
3x + 8 = 2(?) + x
3x + 8 = 2 (x + 4) + x
3x + 8 = 2x + 8x + x
3x + 8 = 3x + 8
When there are infinitely many solutions, the statement should be true.

Question 13.
Persevere in Problem Solving
The Dig It Project is designing two gardens that have the same perimeter. One garden is a trapezoid whose nonparallel sides are equal. The other is a quadrilateral. Two possible designs are shown at the right.

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

Answer:
There are more than one value of x

Explanation:
Perimeter of the trapezoid
P = 2x – 2 + x + 1 + x + x + 1 = 5x
Perimeter of the quadrilateral
P = 2x – 9 + x + x + 8 + x + 1 = 5x
5x = 5x
There are more than one value of x

Question 13.
b. Why does your answer to part a make sense in this context?
Type below:
______________

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

Explanation:
Interpretation of part a in this context
The condition was that the two perimeters are to be equal. However, a specific number was not given, so there are an infinite number of possible perimeters

Question 13.
c. Suppose the Dig It Project wants the perimeter of each garden to be 60 meters. What is the value of x in this case? How did you find this?
______ meters

Answer:
12 meters

Explanation:
2x – 2 + x + 1 + x + x + 1 = 60
5x = 60
x = 60/5
x = 12

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

Question 14.
Critique Reasoning
Lisa says that the indicated angles cannot have the same measure. Marita disagrees and says she can prove that they can have the same measure. Who do you agree with? Justify your answer.

I agree with: ______________

Answer:
I agree with: Marita

Explanation:
9x – 25 + x = x + 50 + 2x – 12
10x – 25 = 3x + 38
10x – 3x = 38 + 25
7x = 63
x = 63/7
x = 9
When x = 9 the angles will be same and for any other value of x, the angles will not be the same.

Question 15.
Represent Real-World Problems
Adele opens an account with $100 and deposits $35 a month. Kent opens an account with $50 and also deposits $35 a month. Will they have the same amount in their accounts at any point? If so, in how many months and how much will be in each account? Explain.
______________

Answer:
Adele’s amount after x months
A = 100 + 35x
Kent’s amount after x months
A = 50 + 35x
100 + 35x = 50 + 35x
100 is not equal to 50
The statement is false, the amounts in two accounts would never be equal.

H.O.T.

Focus on Higher Order Thinking

Question 16.
Communicate Mathematical Ideas
Frank solved an equation and got the result x = x. Sarah solved the same equation and got 12 = 12. Frank says that one of them is incorrect because you cannot get different results for the same equation. What would you say to Frank? If both results are indeed correct, explain how this happened.
Frank is: ____________

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

Question 17.
Critique Reasoning
Matt said 2x – 7 = 2(x – 7) has infinitely many solutions. Is he correct? Justify Matt’s answer or show how he is incorrect.
Matt is: ____________

Answer:

Explanation:
2x – 7 = 2(x – 7)
2x – 7 = 2x – 14
-7 not equal to -14
The statement is false, there is no solution. Matt is incorrect.

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

Solve.

Question 1.
4a – 4 = 8 + a
_______

Answer:
a = 4

Explanation:
4a – 4 = 8 + a
4a – a = 8 + 4
3a = 12
a = 12/3
a = 4

Question 2.
4x + 5 = x + 8
_______

Answer:
x = 1

Explanation:
4x + 5 = x + 8
4x – x = 8 – 5
3x = 3
x = 3/3
x = 1

Homework and Practice Solving Linear Equations 7.1 Answer Key Question 3.
Hue is arranging chairs. She can form 6 rows of a given length with 3 chairs left over, or 8 rows of that same length if she gets 11 more chairs. Write and solve an equation to find how many chairs are in that row length.
_______ chairs

Answer:
7 chairs

Explanation:
Hue is arranging chairs. She can form 6 rows of a given length with 3 chairs left over, or 8 rows of that same length if she gets 11 more chairs.
6x + 3 = 8x – 11
8x – 6x = 3 + 11
2x = 14
x = 14/2
x = 7
There are 7 chairs in each row.

7.2 Equations with Rational Numbers

Solve.

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

Answer:
n = 4

Explanation:
\(\frac{2}{3} n-\frac{2}{3}=\frac{n}{6}+\frac{4}{3}\)
The LCM is 6.
6(2/3n – 2/3) = 6(n/6 + 4/3)
6(2/3n) -6(2/3) = 6(n/6) + 6(4/3)
4n – 4 = n + 8
4n – n = 8 + 4
3n = 12
n = 12/3
n = 4

Question 5.
1.5d + 3.25 = 1 + 2.25d
_______

Answer:
d = 3

Explanation:
1.5d + 3.25 = 1 + 2.25d
2.25d – 1.5d = 3.25 – 1
0.75d = 2.25
d = 2.25/0.75
d = 3

Question 6.
Happy Paws charges $19.00 plus $1.50 per hour to keep a dog during the day. Woof Watchers charges $14.00 plus $2.75 per hour. Write and solve an equation to find for how many hours the total cost of the services is equal.
_______ hours

Answer:
3.2 hours

Explanation:
Happy Paws charges $19.00 plus $1.50 per hour to keep a dog during the day.
1.5x + 19
Woof Watchers charges $14.00 plus $2.75 per hour.
2.75x + 15
1.5x + 19 = 2.75x + 15
2.75x – 1.5x = 19 – 15
1.25x = 4
x = 4/1.25
x = 3.2
The total cost of the services is equal after 3.2 hrs.

7.3 Equations with the Distributive Property

Solve.

Question 7.
14 + 5x = 3(-x + 3) – 11
_______

Answer:
x = -2

Explanation:
14 + 5x = 3(-x + 3) – 11
14 + 5x = -3x + 9 – 11
14 + 5x = -3x – 2
5x + 3x = -2 –  14
8x = – 16
x = -16/8
x = -2

Question 8.
\(\frac{1}{4}\)(x – 7) = 1 + 3x
_______

Answer:
x = -1

Explanation:
\(\frac{1}{4}\)(x – 7) = 1 + 3x
4(\(\frac{1}{4}\)(x – 7)) = 4(1 + 3x)
(x – 7) = 4 + 12x
12x – x = -7 – 4
11x = -11
x = -11/11
x = -1

Question 9.
-5(2x – 9) = 2(x – 8) – 11
_______

Answer:
x = 6

Explanation:
-5(2x – 9) = 2(x – 8) – 11
-10x + 45 = 2x – 16 – 11
-10x + 45 = 2x – 27
2x + 10x = 45 + 27
12x = 72
x = 72/12
x = 6

Question 10.
3(x + 5) = 2(3x + 12)
_______

Answer:
x = -3

Explanation:
3(x + 5) = 2(3x + 12)
3x + 15 = 6x + 24
6x – 3x = 15 – 24
3x = -9
x = -9/3
x = -3

7.4 Equations with Many Solutions or No Solution

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

Question 11.
5(x – 3) + 6 = 5x – 9
____________

Answer:
There are infinitely many solutions

Explanation:
5(x – 3) + 6 = 5x – 9
5x – 15 + 6 = 5x – 9
5x – 9 = 5x – 9
The statement is true. There are infinitely many solutions.

Question 12.
5(x – 3) + 6 = 5x – 10
____________

Answer:
There are no solutions

Explanation:
5(x – 3) + 6 = 5x – 10
5x – 15 + 6 = 5x – 10
5x – 9 = 5x – 10
-9 not equal to -10
The statement is false. There are no solutions.

Question 13.
5(x – 3) + 6 = 4x + 3
____________

Answer:
There is one solution

Explanation:
5(x – 3) + 6 = 4x + 3
5x – 15 + 6 = 4x + 3
5x – 9 = 4x + 3
5x – 4x = 3 + 9
x = 12
There is one solution

Selected Response – Mixed Review – Page No. 222

Question 1.
Two cars are traveling in the same direction. The first car is going 40 mi/h, and the second car is going 55 mi/h. The first car left 3 hours before the second car. Which equation could you solve to find how many hours it will take for the second car to catch up to the first car?
Options:
a. 55t + 3 = 40t
b. 55t + 165 = 40t
c. 40t + 3 = 55t
d. 40t + 120 = 55t

Answer:
d. 40t + 120 = 55t

Explanation:
Two cars are traveling in the same direction. The first car is going 40 mi/h, and the second car is going 55 mi/h. The first car left 3 hours before the second car.
3 × 40 + 40t = 120 + 40t
55t
40t + 120 = 55t

Question 2.
Which linear equation is represented by the table?

Options:
a. y = -x + 5
b. y = 2x – 1
c. y = x + 3
d. y = -3x + 11

Answer:
a. y = -x + 5

Explanation:
Find the slope using
m = (y2 – y1)/(x2 – x1)
where (x1, y1) = (3, 2), (x2, y2) = (1, 4)
Slope = (4 – 2)/(1 – 3) = -2/2 = -1

Question 3.
Shawn’s Rentals charges $27.50 per hour to rent a surfboard and a wetsuit. Darla’s Surf Shop charges $23.25 per hour to rent a surfboard plus $17 extra for a wetsuit. For what total number of hours are the charges for Shawn’s Rentals the same as the charges for Darla’s Surf Shop?
Options:
a. 3
b. 4
c. 5
d. 6

Answer:
b. 4

Explanation:
Shawn’s Rentals charges $27.50 per hour to rent a surfboard and a wetsuit.
27.5x
Darla’s Surf Shop charges $23.25 per hour to rent a surfboard plus $17 extra for a wetsuit.
23.25x + 17
23.25x + 17 = 27.5x
27.5x – 23.25x = 17
4.25x = 17
x = 17/4.25
x = 4
The charge would be equal after 4 hrs

Question 4.
Which of the following is irrational?
Options:
a. -8
b. 4.63
c. \(\sqrt { x } \)
d. \(\frac{1}{3}\)

Answer:
c. \(\sqrt { x } \)

Explanation:
\(\sqrt { x } \) is irrational

Question 5.
Greg and Jane left a 15% tip after dinner. The amount of the tip was $9. Greg’s dinner cost $24. Which equation can you use to find x, the cost of Jane’s dinner?
Options:
a. 0.15x + 24 = 9
b. 0.15(x + 24) = 9
c. 15(x + 24) = 9
d. 0.15x = 24 + 9

Answer:
b. 0.15(x + 24) = 9

Explanation:
Let x be the cost of Jane’s dinner. The amount of tip is the 15% of the total cost of dinner.
0.15(x + 24) = 9

Question 6.
For the equation 3(2x − 5) = 6x + k, which value of k will create an equation with infinitely many solutions?
Options:
a. 15
b. -5
c. 5
d. -15

Answer:
d. -15

Explanation:
3(2x – 5) = 6x + k
6x – 15 = 6x + k
6x – 15 = 6x – 15
The statement is true. k = -15

Question 7.
Which of the following is equivalent to 2⁻⁴?
Options:
a. \(\frac{1}{16}\)
b. \(\frac{1}{8}\)
c. -2
d. -16

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

Explanation:
2⁻⁴
1/2⁴
1/16

Mini-Task

Question 8.
Use the figures below for parts a and b.

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

Answer:
x=12

Explanation:
4x+10=3x+22
4x – 3x = 22 – 10
x = 12
Answer: x=12

Question 8.
b. What is the perimeter of each figure?
_______

Answer:
Both are 58

Explanation:
x + x + 5 + x + x + 5
12 + 12 + 5 + 12 + 12 + 5
58
x + 7 + x + 4 + x + 11
12 + 7 + 12 + 4 + 12 + 11
58

Conclusion:

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