Chapter 7 Linear - Mangham Math

Created by Lance Mangham, 6th grade math, Carroll ISD

ACCELERATED MATHEMATICS

CHAPTER 7

NON-PROPORTIONAL LINEAR RELATIONSHIPS

TOPICS COVERED:

• Representing Linear Non-Proportional Equations

• Slope & y-Intercept

• Graphing Using Slope & y-Intercept

• Proportional vs. Non-Proportional

• Systems of Linear Equations

Contents modified from my.hrw.com, 8th grade Edition online practice


x y

0 0

1 12

2 24

3 36

4 48

x y

0 25

1 35

2 45

3 55

4 65

LINEAR RELATIONSHIPS

Proportional Non-Proportional Constant of Proportionality

Constant of Variation

Direct Variation

5

y kx

y x

=

=

You save $25 each month.

2 4

4 7

y mx b

y x

y x

= +

= +

= − + −

You start with $50 and then save $25 each month.

m = slope b = y-intercept


Linear Relationships: Proportional vs. Non-Proportional Verbal Examples

Proportional: Mr. Mangham started the year with $0. Each week he earned $25.

Non-Proportional: Mr. Mangham started the year with $75. Each week he earned $25.

How to tell the difference: A proportional situation always starts at zero (in this case $0 at the first of the year). A non-proportional situation does not start at zero (in this case $75 at the first of the year).

Graph Examples

Proportional:

Non-Proportional:

How to tell the difference: A proportional graph is a straight line that always goes through the origin. A non-proportional graph is a straight line that does not go through the origin.

Table Examples

Proportional:

Weeks 0 1 2 3 4

Money ($) 0 25 50 75 100

Money

Weeks --- 25 25 25 25

Non-Proportional:

Weeks 0 1 2 3 4

Money ($) 75 100 125 150 175

Money

Weeks --- 100 62.5 50 43.75

How to tell the difference: A proportional table has a constant of proportionality in that y divided by x always equals the same value. A non-proportional table will have different values when y is divided by x.

Equation Examples

Proportional: 25y x= Non-proportional: 25 75y x= +

How to tell the difference: A proportional equation is always in the form y kx= , where k is the unit rate or constant of

proportionality. A non-proportional equation is always in the form y mx b= + , where m is the

constant rate of change or slope. The key difference is the added b on the end.

0

25

50

75

100

125

0 1 2 3 4

Mr. Mangham's Money

0

25

50

75

100

125

150

175

200

0 1 2 3 4

Mr. Mangham's Money


Activity 7-1: Representing Linear Non-Proportional Relationships (4.1) Name:

You can use an equation to describe the relationship between two quantities in a real-world situation. You can use a table to show some values that make the equation true. 1. The equation 3 2y x= + gives the total change, y, for bowling

x games at Baxter Bowling Lanes based on the prices shown. Make a table of values for this situation.

x (number of games) 1 2 3 4

y (cost in dollars)

2. Francisco makes $12 per hour doing part-time work on Saturdays. He spends $4 on transportation to and from work. The equation 12 4y x= − gives his earnings y, after transportation costs, for working x

hours. Make a table of values for this situation.

x (number of hours)

y (earnings in dollars)

3. The entrance fee for Mountain World theme park is $20. Visitors purchase additional $2 tickets for rides, games, and food. The equation 2 20y x= + gives the total cost, y, to

visit the park, including purchasing x tickets.

x (number of tickets) 0 2 4 6 8

y (total cost in dollars)

y

x

4. Plot the ordered pairs from the table. Describe the shape of the graph. 5. Find the rate of change between each point and the next. Is the rate constant? 6. Explain why the relationship between number of tickets and total cost is not proportional. 7. Would it make sense to add more points to the graph from 0x = to 10x = ? Would it make sense to connect the points with a line? Explain.

BAXTER BOWLING

LANES

$3 per game

$2 shoe rental

0

4

8

12

16

20

24

28

32

36

40

0 1 2 3 4 5 6 7 8 9 10

Co

st (

$)

Number of Tickets

Theme Park Costs



A linear equation is an equation whose solutions are ordered pairs that form a straight line when graphed on a coordinate plane. Linear equations can be written in the form y mx b= + . When 0b ≠ , the

relationship between x and y is nonproportional.

The diameter of a Douglas fir tree is currently 10 inches when measured at chest height. Over the next

50 years, the diameter is expected to increase by an average growth rate of 2

5 inch per year. The

equation 2

105

y x= + gives y, the diameter of the tree in inches, after x years.

1. Complete the table.

x (years) 0 10 20 30 50

y (diameter in inches)

2. Plot the ordered pairs from the table and draw a line connecting the points to represent all the possible solutions. 3. Is this relationship linear? 4. Is it proportional? Make a table of values for each equation.

5. 2 5y x= + 6. 3

58

y x= −

x −2 −1 0 1 2

y

0

4

8

12

16

20

24

28

32

36

40

44

48

0 10 20 30 40 50 60

Dia

me

ter

(in

)

Time (yr)

Fir tree Growth

x −8 −4 8

y



Explain why each relationship is not proportional. 1. 2.

x 0 2 4 6 8

y 3 7 11 15 19

y

x

3. Complete the table for the equation and then graph the equation. 1y x= −

x 2− 1− 0 1 2

y

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

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

5. 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.



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 8 12y x= + , where x represents the number of years the

subscription is renewed and y represents the total cost. 1. Make a table for this situation.

2. Draw a graph to represent the situation. 3. Explain why this relationship is not proportional. 4. Does it make sense to connect the points on the graph with a solid line? Explain. 5. What is required of a proportional relationship that is not required of a general line relationship? 6. Explain how you can identify a linear nonproportional relationship from a table? From a graph? From an equation? 7. 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. 8. Two parallel lines are graphed on a coordinate plane. How many of the parallel lines could represent proportional relationships? Explain.

x 1 2 3 4 5

y 90 150 210 270 330

0

4

8

12

16

20

24

28

32

36

40

44

48

52

56

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Co

st (

$)

Years



Make a table of values for each equation.

1. 4 3y x= + 2. 1

24

y x= −

x −2 −1 0 1 2

y

3. 0.5 1y x= − + 4. 0.75 4y x= − +

x −4 −2 0 2 4

y

Make a table of values and graph the solutions of each equation.

5. 2 1y x= + 6. 1

32

y x= − −

x −2 −1 0 1 2

y

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

7. The relationship between the height of a tree and the time since the tree was planted.

8. The relationship between the number of $12 DVDs you buy and the total cost.

x −2 −1 0 1 2

y

x −4 −2 0 2 4

y

x −8 −4 0 4 8

y


Activity 7-6: Determining Slope and y-intercept (4.2) Name:

The graph of every nonvertical line crosses the y-axis. The y-intercept is the y-coordinate of the point where the graph intersects the y-axis. The x-coordinate of this point is always 0.

The graph represents the linear equation 2

43

y x= − +

1. What is the slope from (0,4) to (-3,6)?

2. What is the slope from (0,4) to (6,0)?

3. What do you notice about the two slopes and how they relate to the linear equation?

4. Find the value of y when 0x = using the equation

24

3y x= − + . Describe the point on the graph that

corresponds to this situation.

5. What do you notice about this y value and how it relates to the linear equation?

The linear equation below is written in the slope-intercept form of an equation. Its graph is the line with slope m and y-intercept b.

y mx b= +

A linear relationship has a constant rate of change. You can find the rate of change m and the initial value b for a linear situation from a table of values.

Example 1: A phone salesperson is paid a minimum weekly salary and a commission for each phone sold, as shown in the table. Determine the constant rate of change and the initial value.

Number of phones sold 0 10 20 30 40

Weekly Income ($) $480 $630 $780 $930

The rate of change is the same as the slope. y

mx

= =�

�

To find the initial value we need to know how much the salesperson makes when zero phones are sold. Work backwards from 10 to 0.x x= = The initial value is the same as the y-intercept. b =

-2

-1

0

1

2

3

4

5

6

7

-4 -3 -2 -1 0 1 2 3 4 5 6 7



1-2. Find the slope and y-intercept of the line represented by each table.

x 0 2 4 6 8

x 0 1 2 3 4

y 22 32 42 52

y 8 15 22 29

Slope m = y-intercept b = Slope m = y-intercept b =

3-6. Find the slope and y-intercept of the line in each graph.



-30

-25

-20

-15

-10

-5

0

5

10

15

20

25

30

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

-15

-12

-9

-6

-3

0

3

6

9

12

15

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6



REMINDER: Initial Value = y-intercept = b

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

x 0 2 4 6 8

x 0 5 10 15 20

y 7 13 19 25

y 120 100 80 60

Some carpet cleaning costs are shown in the table. The relationship is linear.

Rooms cleaned 1 2 3 4

Cost ($) 125 175 225 275

2. Find and interpret the rate of change (slope) and the initial value of for this situation. The total cost to pay for parking at a state park for the day and rent a paddleboat are shown below.

Number of hours 1 2 3 4

Cost ($) 17 29 41 53

3. Find the cost to park for a day and the hourly rate to rent a paddleboat. 4. What will Lin pay if she rents a paddleboat for 3.5 hours and splits the total cost with a friend? Explain.

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.

Lessons 1 2 3 4

Group ($) 55 85 115 145

Private ($) 75 125 175 225

5. Find the rate of change and the initial value for the group lessons. 6. Find the rate of change and the initial value for the private lessons. 7. Compare and contrast the rates of change and the initial values.



1. Prove why each relationship is not linear.

x 1 2 3 4 x 3 5 7 9

y 4.4 6.5 8.5 11.5 y 140 126 110 92

2. Your teacher asked your class to describe a real-world situation in which the 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 lost 5 of them every month since. What mistake did your partner make? How can you change the situation to match? 3. John has a job parking cars. He earns a fixed 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 that from his fixed salary? Justify your answer.

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

4. 5.

Find the slope and y-intercept of the line represented by each table. 6. 7.

x 0 2 4 6 8

y 2 3 4 5

x 0 3 6 9 12

y 10 19 28 37

0

100

200

300

400

500

600

700

800

900

0 10 20 30 40 50 60 70 80 90 100 110

We

ekl

y e

arn

ing

s

Cars parked


Activity 7-10: Graphing Linear Nonproportional Relationships (4.3) Name:

Example Graph 2

13

y x= −

Step 1: What is the y-intercept (b)? Plot that point.

Step 2: What is the slope (m)? Use the slope to find a second point.

Step 3: Draw a line through the points.

Example 2 Graph 5

32

y x= − +

Step 1: What is the y-intercept (b)? Plot that point.

Step 2: What is the slope (m)? Use the slope to find a second point.

Step 3: Draw a line through the points.

Graph each equation. 1. 2.

1

12

y x= + 3 4y x= − +



Analyzing a Graph

Many real-world situations can be represented by linear relationships. Example Ken has a weekly goal for the number of

calories he will burn by taking brisk

walks. The equation 300 2400y x= − +

represents the number of calories y Ken has left to burn after x hours of walking.

Graph the equation 300 2400y x= − + .

Step 1: What is the y-intercept? Step 2: What is the slope written as a fraction? Use the slope to locate a second point. Step 3: Draw a line through the two points. 1. After how many hours of walking will Ken have 600 calories left to burn? After how many hours will he reach his weekly goal? 2. What if Ken modifies his plans by slowing his speed. The equation for the modified plan is

200 2400y x= − + . Graph the new equation on the graph and compare to the previous graph.

3. Suppose Ken decides that instead of walking, he will jog, and jogging burns 600 calories per hour. How do you think this would change the graph? 4. Graph each equation using the slope and y-intercept.

1

32

y x= − 3 2y x= − +

Slope = y-intercept = Slope = y-intercept =

0

300

600

900

1200

1500

1800

2100

2400

2700

0 1 2 3 4 5 6 7 8 9 10

Ca

lori

es

rem

ain

ing

Time (h)



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 4 2y x= + describes the number of cards, y,

you have after x weeks. 1. Find and interpret the slope and the y-intercept of the line that represents this situation.

2. Graph the equation 4 2y x= + .

3. 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. A spring stretches in relation to the weight hanging from it

according to the equation 0.75 0.25y x= + where x is the

weight in pounds and y is the length of the spring in inches. 4. Graph the equation. 5. Interpret the slope and the y-intercept of the line. 6. How long will the spring be if a 2 pound weight is hung on it? 7. Will the length double if you double the weight? Explain. 8. Identify the coordinates of four points on the line with each given slope and y-intercept. A. Slope = 5, y-intercept = 1− B. Slope = 0.2, y-intercept = 0.3

C. Slope = 1− , y-intercept = 8 D. Slope = 1

2− , y-intercept = 4

9. A music school charges a registration fee in addition to a fee per lesson. Music lesson last 0.5 hour.

The equation 40 30y x= + represents the total cost of y of x lessons. Find and interpret the slope and y-

intercept of the line that represents this situation.

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.5 1 1.5 2 2.5 3 3.5 4

Len

gth

(in

)

Weight (lb)

0

2

4

6

8

10

12

14

16

18

20

22

24

0 1 2 3 4 5 6 7 8

Ca

rds

Weeks



A public pool charges a membership fee and a fee for

each visit. The equation 3 50y x= + represents the cost y

for x visits.

1. After locating the y-intercept on the coordinate plane shown, Danielle moves up three gridlines and right one gridline. Is this a correct slope based on the information presented in the problem?

2. Graph the equation 3 50y x= + . Then interpret the

slope and y-intercept. 3. How many visits to the pool can a member get for $200?

4. A students says that the slope of a line for the equation

20 15y x= − is 20 and the y-intercept is 15. Find and correct the

error.

5. Graph the lines 3y x= , 3 3y x= − , and 3 3y x= + . What do

you notice about the three lines? Make a conjecture based on your observation.

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

6. 2 1y x= − 7. 1

32

y x= + 8. 2y x= − −

m

b

0

25

50

75

100

125

150

175

200

225

250

0 25 50 75 100 125 150 175 200 225 250

Co

st (

$)

Visits


Activity 7-14: Graphing Non-Linear Proportional Relationships (4.3) Name:

1. The equation 15 10y x= + gives your score on a math quiz, where x

is number of questions you answered correctly.

a. Graph the equation.

b. Interpret the slope and y-intercept of the line.

___________________________________________________________

___________________________________________________________

c. What is your score if you answered 5 questions correctly?

___________________________________________________________

Proportional and Nonproportional Situations (4.4)

2. Determine if each of the following graphs represents a proportional or nonproportional relationship.

If it is a linear equation it can be written in the form y mx b= + . If it is also proportional, then 0b = and

it can also be written as y kx= .

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16

18

20

0 4 8 12 16 20 24 28 32 36 40

IS THE GRAPH A STRAIGHT LINE?

Non-Linear, Non-Proportional DOES IT GO THROUGH THE ORIGIN?

Linear, Proportional Linear, Non-proportional

YES

YES NO

NO


Activity 7-15: Proportional and Nonproportional Situations (4.4) Name:

1. Determine if each of the following equations represents a proportional or nonproportional relationship.

65d t= 0.1 2000p s= + 450 3n p= − 36 12d=

A linear relationship is a proportional relationship when y

x is constant. Otherwise, the linear relationship

is nonproportional. 2. Determine if the linear relationship represented by each table is a proportional or nonproportional relationship.

x y x y

2 30 5 1

8 90 40 8

14 150 65 13

3. Determine which situation is a proportional relationship and which situation is a nonproportional relationship.

The cost for Test Prep Center A is $20 times the number of hours that you attend.

The cost for Test Prep Center B is $25 an hour, but you have a $100 coupon that you can use to reduce the cost.

4.-9. Determine if each relationship is a proportional or nonproportional situation.

1

22

q p= + 1

10v u= 15t d= 0.75 2m d= −

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

0

3

6

9

12

15

18

21

24

27

30

0 3 6 9 12 15 18 21 24 27 30


Activity 7-16: Proportional and Nonproportional Situations (4.4) Name:

1.-2. Determine if each relationship is a proportional or nonproportional situation.

x y x y

3 12 22 4

9 16 46 8

21 84 58 10

3. Describe a real-world situation where the relationship is linear and nonproportional.

4.-9. Determine if each relationship is a proportional or nonproportional situation. Explain your reasoning. Assume all tables represent linear relationships.

y x=

2 1r b= +

x y

2 11

5 26

12 61

x y

4 36

10 90

13 117

10. This year, Andrea celebrated her 12th birthday, and her brother Carlos celebrated his 6th birthday. Andrea noted that she was now twice as old as her brother was. Is the relationship between their ages proportional? Support your answer.

11.

Lily is considering buying books on display three different tables. Each table has one of the following signs. What will be the total cost if Lily buys 6 books from the table whose sign indicates a non-proportional relationship?

12.

Ed and Riley are cycling in the same direction on the same stright road. Ed’s distance from a roadside rest area is given by d=6t. Riley’s distance from the same rest area is iven by d= 4.5t+12. In each equation d is distance in miles and t is time in hours.

a. Determine if each equation is proportional or non-proportional. b. When are Ed and Riley the same distance from the rest area? Show your work.

Each Book $10 Each Book $8 for Club Members (One-time Membership Fee: $15)

Each Book 50% off


Activity 7-17: Proportional vs. Non-Proportional (4.4) Name:

1. Complete the tables below and then graph each set of data. Proportional Relationship

x y y

x

1 2

2 4

3 6

5 10

Rate of change (slope) =

Non-Proportional Relationship

x y y

x

1 1

2 3

3 5

4 7

Rate of change (slope) =

2. What did you notice about the rate of change for each relationship?

3. What did you notice about the graphs for the proportional and non-proportional relationship?

4. For which relationship is the rate of change the same as the ratio y

x ?

5. What did you notice about the ratio y

x for the each relationship?

Proportional Linear Functions Both Non-Proportional Linear

Functions

Graphs will not pass

through the origin.

Graphs will pass through the

origin. Graphs have a constant rate of

change (slope).

There is a constant of proportionality.

Lines are in the form of

y=kx.

Lines are in the form of y=mx+b.

There is only one and exactly one y-value that corresponds

(maps) to any given x-value.


Activity 7-18: Solving Systems of Linear Equations by Graphing (4.5) Name:

You have learned several ways to graph a linear equation in slope-intercept form. For example, you can use the slope and y-intercept or you can find two points that satisfy the equation and connect them with a line.

Graph the pair of equations together on the same graph: 3 2y x= − and 2 3y x= − +

What is the point of intersection of the two lines? Will that point of intersection be a solution to the first equation? Check by substituting the point into the equation. Will that point of intersection be a solution to the second equation? Check by substituting the point into the equation. An ordered pair (x,y) is a solution of an equation in two variables if substituting the x- and y- values into the equation results in a true statement. A system of equations is a set of equations that have the same variables. An ordered pair is a solution of a system of equations if it is a solution of every equation in the set.

Solve the system by graphing. 4y x= − + and 3y x=

Solve the system by graphing. 2y x= − + and 4 1y x= − −

Solve the system by graphing. 2 5y x= − + and 3y x=


Activity 7-19: Solving Systems of Linear Equations (4.5) Name:

Show all graphs on graph paper. Show all work on notebook or graph paper.

1.

Bowl-o-Rama charges a shoe rental fee of $2.00 and a cost per game of $2.50. Bowling Pinz charges a rental fee of $4.00 and a cost per game of $2.00.

Let x represent the number of games played and let y represent the total cost.

Write an equation to represent the total cost at Bowl-O-Rama.

Write an equation to represent the total cost at Bowling Pinz.

Graph each equation and to find the solution. What does the solution represent?

2.

Mr. Underwood runs 7 miles per week and increases his distance by 1 mile each week. Mr. Mangham runs 3 miles per week and increases his distance by 2 miles each week. In how many weeks will Mr. Underwood and Mr. Mangham be running the same distance? What will that distance be?

Write a system of equations and graph to find the solution.

3.

Write an interesting, creative real-world problem that could be represented by the system of equations shown below.

4 10 3 15y x y x= + = +

4.

You have two options for your internet service: Option 1: $50 set-up fee plus $30 per month Option 2: No set-up fee plus $40 per month

In how many months will the total cost of both options be the same?

Write a system of equations and graph to find the solution. If you plan to cancel your internet service after 9 months, which is the cheaper option?

5.

Eight friends wanted to start a business. They will wear either a baseball cap or a shirt imprinted with their logo while working. They want to spend exactly $36 on the shirts and caps. Shirts cost $6 each and caps cost $3 each.

Let x represent the number of shirts and let y represent the number of caps.

Write an equation to represent the total number of caps and shirts they will purchase.

Write an equation to represent the total cost of all the shirts and caps.

Rewrite both of your equations in y mx b= + form.

Graph each equation and to find the solution. What does the solution represent?



Solve each linear system by graphing. Mentally check your answer.

1. 1

2 7

y

y x

= −

= −

_________________ 2. 5

2 4

y x

y x

= −

= − +

_________________

3. 4

3 6

x

y x

=

= − +

__________________ 4.

4 3

32

2

y x

y x

= −

= +

__________________

5. Two skaters are racing toward the finish line of a

race. The first skater has a 50 meter lead and is traveling at a rate of 5 meters per second. The second skater is traveling at a rate of 15 meters per second. How long will it take for the second skater to pass the first skater?



Solve each linear system by graphing. Check your answer.

1.

32

4

5

y x

y x

= − +

= −

2. 6

2 3

y x

y x

= − −

= +

________________________________________ _______________________________________

3.

0

23

5

x

y x

=

= − −

__________________ 4.

3 6

33

2

y x

y x

= −

= − +

__________________

5. Two skaters are racing toward the finish line of a

race. The first skater has a 100 meter lead and is traveling at a rate of 10 meters per second. The second skater is traveling at a rate of 15 meters per second. How long will it take for the second skater to pass the first skater?

Chapter 7 Linear - Mangham Math

Documents

Transcript of Chapter 7 Linear - Mangham Math