CHAPTER
9
Linear Relations
GET READY
474
Math Link
476
9.1 Warm Up
477
9.1 Analysing Graphs of Linear Relations
478
9.2 Warm Up
487
9.2 Patterns in a Table of Values
488
9.3 Warm Up
498
9.3 Linear Relationships
499
Chapter Review
507
Practice Test
512
Wrap It Up!
515
Key Word Builder
516
Math Games
517
Challenge in Real Life
518
Answers
519
Name: _____________________________________________________
Date: ______________
Describe Patterns in Words
To describe a pattern in words, say:
• what it is (number, letter, shape)
• where it starts
• how it changes (skips, decreases)
Pattern
6, 9, 12, 15, …
Description in Words
• whole numbers
• begins with 6
• increases by 3 each time
1. Describe each pattern in words.
a) b, e, h, …
b) 9, 4, –1, …
• _______________________________
• _______________________________
• _______________________________
• _______________________________
• _______________________________
• _______________________________
Show Patterns in a Table
A café has small tables that seat 4 people.
Two tables put together seat 6 people.
Three tables put together seat 8 people.
Describe the pattern using a table of values.
1
4
Number of Tables
Number of People
2
6
Describe the pattern using words:
• The number of people begins at 4 and
increases by 2 each time a table is added.
3
8
2. Complete the table of values and describe each pattern in words.
a)
Figure 1 Figure 2
b)
Figure 3
Figure 3
Figure 2
Figure 1
Figure Number
1
2
3
Number of Squares
474
Figure Number
1
2
3
Number of Cubes
Describe in words:
Describe in words:
____________________________________
_________________________________
____________________________________
_________________________________
MHR ● Chapter 9: Linear Relations
Name: _____________________________________________________
Date: ______________
Describe Patterns Using an Expression
expression
● a number or letter or combination of numbers and letters connected by
+, – , ×, or ÷
● examples: 5, r, 8t, x + 9, 2y – 5
variable
● a letter that represents an unknown number or amount
A fish tank holds 3 algae eaters and some guppies.
How many fish are there in total?
The number of
guppies is unknown.
Number of guppies: g
Number of algae eaters: 3
Total number of fish in the tank: g + 3
3. Write an expression. Tell what your variable describes.
b) Shay has 5 packages of pencils. Each
package has the same number of pencils.
How many pencils are in all 5 packages?
a) Simon has many shirts.
He gives 2 shirts away.
How many shirts does he still have?
Variable: s = the number of
Variable:
Expression: s –
Expression:
Use a Coordinate Grid
You can describe points on a coordinate grid using ordered pairs: (x, y).
●
●
The x-coordinate tells you how many units to move
left or right starting at the origin (0, 0).
The y-coordinate tells you how many units to move
up or down starting at the x-axis.
y
2
A
D
B
E
C
F
0
–2
–2
To plot the point E (3, 1), start at (0, 0).
Move 3 units right and 1 unit up.
x
2
G
4. Use the grid above. Write the coordinates of each point in the table of values.
Point
A
B
C
D
E
x
–1
3
y
0
1
F
G
Get Ready ● MHR 475
Name: _____________________________________________________
Date: ______________
Adventure Travel
Have you ever wanted to climb a mountain?
Via Ferrata is a steep mountain trail in Whistler, British Columbia.
Paulette decided to climb the Via Ferrata trail.
She recorded how long it took her to climb each 100-m section.
Her times were 10 min, 9 min, 10 min, 11 min, 10 min, and 30 min.
a) Complete the table with Paulette’s data.
100-m Section
Time to Climb
That Section
1
2
3
10
9
10
4
5
6
b) Describe the pattern of the time it took Paulette to climb sections 1 to 5.
_________________________________________________________________________
c) Give 2 reasons why you think the time for section 6 was so different from the others.
• _______________________________________________________________________
• _______________________________________________________________________
d) Using the table, draw a graph of Paulette’s total distance climbed compared to her total time.
Total Time
(min)
10
19
29
40
50
80
y
80
70
Total Time (min)
Total Distance
(m)
100
200
300
400
500
600
60
50
40
30
20
10
0
e) Describe the patterns you see on your graph.
100 200 300 400 500 600 x
Total Distance (m)
_________________________________________________________________________
_________________________________________________________________________
476
MHR ● Chapter 9: Linear Relations
Name: _____________________________________________________
Date: ______________
9.1 Warm Up
Ask yourself: “Do I
add or subtract?”
1. For each chart, describe a pattern to go from the input to the output.
Input
2
4
3
8
Output
5
7
6
11
Pattern:
Input
6
3
7
10
Output
5
2
6
9
Input
1
3
5
6
Pattern:
Pattern:
Ask yourself: “Do I
multiply or divide?”
2. For each chart, describe a pattern to go from the input to the output.
Input
4
6
3
5
Output
8
12
6
10
Pattern:
Input
6
12
3
15
Output
7
9
11
12
Output
2
4
1
5
Input
3
1
4
8
Pattern:
Output
30
10
40
80
Pattern:
3. Write the coordinates of each point on the grid.
A (3, 4)
y
10
B(
,
)
F
9
8
C(
,
)
7
C
6
D(
,
)
5
B
A
4
E(
,
)
3
D
2
F(
,
)
1
E
0
1
2
3
4
5
6
7
8
9
10 x
9.1 Warm Up ● MHR 477
Name: _____________________________________________________
Date: ______________
9.1 Analysing Graphs of Linear Relations
linear relation
● a pattern made by a set of points that lie in a straight line
● example: y
linear = straight line
4
3
2
1
0
1
2
3
4 x
table of values
● shows 2 sets of related numbers
Working Example 1: Make a Table of Values From a Graph
The graph shows that the total cost depends on the number of baseballs you buy.
Total cost and number of baseballs are related to each other.
C Cost of Baseballs
14
12
The variable b shows the number of baseballs.
The variable C shows the total cost in dollars.
Total Cost ($)
10
8
6
4
2
0
1
2
3
4 b
Number of Baseballs
a) Describe the patterns you see on the graph.
Solution
●
The graph shows data on the cost of baseballs.
One ball costs $3, 2 balls cost
●
The total cost increases by $
, 3 balls cost
,….
each time you buy a baseball.
To move from 1 point to the next, go 1 unit horizontally (↔) and
vertically (↕).
●
478
The points lie in a
The graph shows a linear relation.
MHR ● Chapter 9: Linear Relations
units
.
Name: _____________________________________________________
Date: ______________
b) Make a table of values from the graph.
Solution
Read each point as an ordered pair to make a table of values. Complete each table.
In a horizontal table of values, the top row
shows the x-coordinates.
The bottom row shows the y-coordinates.
Number of
1
Baseballs (b)
2
Total Cost
(C )
6
3
3
In a vertical table of values, the first column
shows the x-coordinates.
The second column shows the y-coordinates.
Number of
Baseballs (b)
1
2
4
Total Cost
(C )
3
6
3
4
c) Write an expression that represents the cost of buying b baseballs.
Solution
Let b describe the number of baseballs.
The expression 3b describes the cost of buying
b baseballs.
Number of
Baseballs (b)
1
2
3
4
Total Cost
(C )
3
6
9
12
×3
d) If the relationship in the graph continues, how much will it cost to buy 14 baseballs?
Solution
Use the expression 3b to find the cost of 14 baseballs.
Cost of 14 baseballs = 3(
)
← Substitute 14 for b.
3(14) = 3 × 14
=
The cost of 14 baseballs is $
.
9.1 Analysing Graphs of Linear Relations ● MHR 479
Name: _____________________________________________________
Date: ______________
a) Describe 2 patterns you see in the graph.
Number of Squares
●
s
The points appear in a
.
●
To move from 1 point to the next:
Squares in
a Pattern
25
20
15
10
5
0
__________________________________________
1
2
3
4
Figure Number
b) The graph shows the number of squares in relation to the figure number.
These figures match the graph. Draw Figure 4.
Figure 1
Figure 2
Figure 3
Figure 4
c) Complete the table of values.
Figure Number, f
1
2
3
4
Number of Squares, s
d) Is this a linear relation? Circle YES or NO.
e) Let f describe the figure number.
Circle the expression that describes the number of squares in any figure: 5f or f + 5
f ) If the pattern continues, how many squares will there be in Figure 8?
Sentence: ______________________________________________________________
480
MHR ● Chapter 9: Linear Relations
f
Name: _____________________________________________________
Date: ______________
Working Example 2: Analyse Data on a Graph of a Linear Relation
Nicole has a part-time job.
The graph shows her pay related to the number of hours she works.
P
Nicole’s Rate of Pay
50
40
Pay ($)
a) Describe the patterns you see in the graph.
Solution
●
20
10
2
3
4
5
6
t
Time (h)
, the pay for 2 h is
the pay for 3 h is
1
0
The graph shows data on the pay Nicole receives for each
hour of work.
The pay for 1 h is
●
30
,
,….
The points lie in a straight
, so the graph is a linear relation.
To move from 1 point to the next, move
and 10 units vertically (↕).
The Pay axis
counts
by 10s.
b) Make a table of values.
unit horizontally (↔)
●
Solution
Time (h)
1
Pay ($)
10
2
3
4
40
c) How much does Nicole make per hour?
Solution
Look at the table of values.
In 1 h of work, Nicole makes $
.
For each hour she works, Nicole makes $10 more, so she makes $10/hr.
d) Is it possible to have points between the points shown on the graph?
Explain why or why not.
Solution
Yes, it is possible to have points between the points on the graph.
Nicole could work full hours and half hours.
If she worked 2.5 h, she would be paid 2.5 × $10 = $
On the graph, this point would be (2.5, 25).
.
9.1 Analysing Graphs of Linear Relations ● MHR 481
Name: _____________________________________________________
Chad is buying notebooks at Bob’s Bargain Store.
The graph shows the cost of notebooks.
Date: ______________
C
Cost of Notebooks
Cost ($)
8
a) Describe the 2 patterns you see in the graph.
6
4
2
●
The points appear to lie in a
0
2
3
4
5
n
Number of Notebooks
.
●
1
To move from 1 point to the next:
_____________________________________________________________________
b) Complete the table of values.
Number of Notebooks, n
1
2
3
4
Cost, C ($)
c) Use the table to find the cost of 1 notebook.
d) Write an expression to show how much n notebooks cost:
e) How much would it cost to buy 52 notebooks?
Expression →
Substitute →
Solve →
Sentence: _____________________________________________________________
f) Is it possible to have points between the ones on the graph? Explain why or why not.
_____________________________________________________________________
_____________________________________________________________________
482
MHR ● Chapter 9: Linear Relations
Name: _____________________________________________________
Date: ______________
1. For each graph, is it possible to have points between the ones on the graph? Explain.
C
Graph A
Cost ($)
Cost ($)
6
4
12
_____________________________________
8
4
2
0
Graph A: _____________________________
Graph B
16
8
1
2
3
4 d
Distance (km)
0
Graph B: _____________________________
1
2
3
4 n
_____________________________________
Number of Cartons
of Ice Cream
2. The graph shows how much higher you get each time you go up
a step of a staircase.
a) Describe the 2 patterns you see in the graph.
●
The pattern lies in a
Total Height of Stairs
80
60
40
20
0
.
●
h
100
Total Height (cm)
C
1
2
3
4
5
6 s
Number of Steps
To move from 1 point to the next:
_______________________________________________________________________
b) Use the graph to complete the table of values.
Number of Steps
1
2
3
4
Total Height (cm)
c) Describe the pattern in the table of values.
The total height starts at
cm and increases by
cm.
d) Write an expression for the total height after climbing s stairs:
e) If the relationship in the graph continues, what is the total height on step 10?
9.1 Analysing Graphs of Linear Relations ● MHR 483
Name: _____________________________________________________
Date: ______________
3. Tessa and Vince go shopping at Bulk Bin.
The graph shows the cost of banana chips.
C
360
Cost of Banana Chips
at Bulk Bin
a) Does the graph show a linear relation? Explain why or why not.
______________________________________________
Cost (¢)
300
240
180
120
60
______________________________________________
0
100 200 300 400 500 600 q
Quantity (g)
b) Describe 2 patterns shown on this graph.
●
The pattern of the points: ___________________________________________________
●
To move from 1 point to the next: ____________________________________________
c) Complete the table of values for this graph.
Quantity (g)
0
100
200
300
400
Cost ($)
d) Can the graph show the cost of 250 g of banana chips? Explain your answer.
_________________________________________________________________________
4. a) Complete the table of values for the ordered pairs on the graph.
y
7
x
1
y
6
5
4
3
b) Describe the 2 patterns you see in the graph.
●
The pattern of the points:
____________________________________________
●
To move from 1 point to the next:
____________________________________________
c) Extend your table of values so the x-column goes to 9.
d) If this pattern continues, what is the value of y when x = 9?
484
MHR ● Chapter 9: Linear Relations
2
1
0
1
2
3 x
Name: _____________________________________________________
Date: ______________
5. The graph shows the rate of pay based on the number of hours worked.
P
Rate of Pay
80
Rate of pay means how
much money you are paid
for 1 h of work.
70
Pay ($)
60
50
40
30
20
10
1
0
2
3
4
5
6
t
Time Worked (h)
a) Make a table of values for the ordered pairs on the graph.
Time Worked (h)
Pay ($)
1
Look at the graph.
b) What is the hourly rate of pay?
c) If the time worked is 4.5 h, how much pay is earned?
6. The graph shows part of a linear relation that describes the cost
of cake flower decorations.
C
Cost of Flowers
Cost (¢)
120
90
60
30
0
1
2
3
4
f
Number of Flowers
Ask yourself,
“Can I buy 2 flowers?”
Is it reasonable to have points between the ones on the graph? Explain your answer.
____________________________________________________________________________
9.1 Analysing Graphs of Linear Relations ● MHR 485
Name: _____________________________________________________
Date: ______________
Do you like adventures?
There are many different adventure tours in Canada.
a) You are going on a polar bear tour.
The graph shows the cost of the trip.
Describe 2 patterns on the graph.
2400
The overall pattern:
___________________________________________
●
Polar Bear Adventure
2800
Cost ($)
●
C
3200
2000
1600
1200
To move from 1 point to the next:
800
400
___________________________________________
0
b) Complete the table of values for this graph.
Number of Days
0
1
2
1
2
3
4
5
6
7 d
Number of Days
3
4
5
6
7
Cost ($)
c) What would the cost be for 8 days on the tour?
d) The tour company offers a deluxe tour that includes food and clothing supplies.
They charge $300 for signing up plus $400 for each day on the tour.
Complete the table of values for the deluxe tour.
Number of Days
Cost ($)
0
1
2
3
4
5
6
7
300
+ 400
e) Compare the data in the 2 tables of values.
How are they alike? _______________________________________________________
________________________________________________________________________
How are they different? ____________________________________________________
________________________________________________________________________
486
MHR ● Chapter 9: Linear Relations
Name: _____________________________________________________
Date: ______________
9.2 Warm Up
1. Graph each set of points.
Input
2
4
3
8
Output
5
7
6
11
y
12
10
Output
a)
8
6
4
2
0
2
4
6
8 x
+, –, ×, or ÷
Input
To go from the input to the output, the pattern is
b)
x
6
3
7
10
y
5
2
6
9
.
y
10
8
6
4
2
0
2
4
6
8
10 x
To go from x to y, the pattern is
.
2. Solve.
a) 12 ÷ 4 =
b) 6 × 5 =
c) 13 – 9 =
d) 12 + 5 =
3. Fill in the boxes.
a) 6 ×
= 12
b) 5 ×
= 15
c) 2 ×
= 18
d) 2 ×
–1=5
e) 3 ×
+ 5 = 11
f) 5 ×
–4=6
Do multiplication
first, and then add
or subtract.
9.2 Warm Up ● MHR 487
Name: _____________________________________________________
Date: ______________
9.2 Patterns in a Table of Values
Working Example 1: Identify the Relationship in a Table of Values
relationship
• a pattern formed by 2 sets of numbers
The pattern in this table of values describes a linear relation.
A
0
1
2
3
4
a) Graph the ordered pairs.
Solution
The ordered pairs are (0, 0), (1, 3), (
(
,
,
), and (
B
,
),
B
0
3
6
9
12
).
Plot the last 3 ordered pairs.
12
10
8
6
4
Consecutive numbers go in
order from smallest to largest
(e.g: 6, 7, 8, 9 …).
2
0
1
2
3
4 A
b) What is the difference between consecutive A-values?
Solution
3–2=
2–1=
1–0=
Consecutive A-values have a difference of
.
c) What is the difference between consecutive B-values?
Solution
9–6=
6–3=
3–0=
Consecutive B-values have a difference of
.
d) Look at the graph. Describe how to move from (0, 0) to the next point.
Solution
Starting at (0, 0), move
vertically (↕).
488
MHR ● Chapter 9: Linear Relations
unit horizontally (↔) and
units
Name: _____________________________________________________
Date: ______________
e) Write an expression for B in terms of A.
Solution
Look at the table of values: To get from A to B, multiply by
Look at the graph: When A increases by
There are 3 ways to write B in terms of A:
.
, B increases by 3.
Words
Ordered Pair
(x, y)
Expression
B is 3 times A
(A, 3A)
3 × A or 3A
This table of values represents a linear relation.
a) Complete the table. Then, graph each ordered pair.
x
y
1
5
2
10
3
15
4
20
y
Ordered Pair
(x, y)
25
20
15
10
5
O
1
2
3
4
5 x
×
b) What is the difference in consecutive values of x and y?
x
y
4–3=
20 – 15 =
3–2=
15 – 10 =
When x increases by
y increases by
,
.
c) Write an expression for y in terms of x.
To get from x to y, multiply x by
An expression for y in terms of x is y =
.
× x.
9.2 Patterns in a Table of Values ● MHR 489
Name: _____________________________________________________
Date: ______________
Working Example 2: Use a Table to Determine a Linear Relation
Table 1
4
7
2
3
x
y
6
11
8
15
m
n
1
1
Table 2
2
4
3
7
4
8
a) Complete the chart to show the pattern in the values for each variable.
Solution
Table 1
Difference Between
Consecutive First
Variables
Difference Between
Consecutive Second
Variables
Table 2
2
7–3=
4–1=
11 – 7 =
7–4=
15 – 11 =
The y-values have a difference
8–7=
The n-values have differences
of
of 3, 3, and
.
.
b) Graph each table of values. Which relation is linear?
Solution
Finish graphing the points.
y
Graph of Table 1
n
18
10
15
8
12
6
9
4
6
2
3
0
0
2
4
6
Graph of Table 2
1
2
3
The graph of Table
linear relation.
shows a
4 m
8 x
c) Look at the graphs.
Table 1 is a linear relation and Table 2 is not. Explain how you know.
Solution
Table 1: The difference in the y-coordinates is
. The relation is linear.
(the same or not the same)
Table 2: The difference in the n-coordinates is
. The relation is not linear.
(the same or not the same)
490
MHR ● Chapter 9: Linear Relations
Name: _____________________________________________________
Table 1:
1
1
x
y
4
10
p
q
7
19
0
0
Date: ______________
Table 2:
1
1
2
3
3
6
a) Complete the chart to find the difference between the consecutive values of each variable.
Table 1
Difference
Between
Consecutive
First
Variables
Table 2
4–1=
–
=
7–4=
Consecutive x-values have a
–
=
difference of
.
–
=
Consecutive p-values have a
difference of
Difference
10 – 1 =
Between
–
=
Consecutive
Consecutive y-values have a
Second
Variables
difference of
.
.
1–0=
–
=
–
The q-values differ by
,
=
,
.
b) Is Table 1 linear? Explain.
_______________________________________________________________________
Is Table 2 linear? Explain.
_______________________________________________________________________
c) Check each answer by graphing.
y
Graph of Table 1
q
20
6
18
5
16
4
14
3
12
2
10
1
8
0
Graph of Table 2
1
2
3
4
5
6 p
6
4
2
0
2
4
6
8 x
9.2 Patterns in a Table of Values ● MHR 491
Name: _____________________________________________________
Date: ______________
Working Example 3: Use a Table of Values in Solving a Problem
Sam is paid $7 for every hour of babysitting.
a) The table of values shows how much she is paid for 1 h, 2 h, and 3 h of babysitting.
Complete the table.
Solution
Number of
Hours, n
1
2
3
Sam’s Pay,
P
7
14
21
4
5
b) Is this a linear relation? Explain how you know.
Solution
Look at the table of values.
Consecutive n-values have a difference of
, or they increase by
.
Consecutive P-values have a difference of
, or they increase by
.
This relation is linear because:
• the number of hours (n) changes by the same amount
• Sam’s pay (P) changes by the same amount
c) Graph this relation.
Solution
Plot the numbers in the table of values on the grid.
P
42
Sam’s Pay
35
28
21
14
7
0
492
1
2
3
4
5
Number of Hours
6 n
MHR ● Chapter 9: Linear Relations
Name: _____________________________________________________
Date: ______________
d) Write an expression to show Sam’s pay.
Solution
Look at the table of values. To get from n to P, multiply by
Look at the graph. When n increases by
.
, P increases by
.
There are 3 ways to write Sam’s pay in terms of the number of hours she babysits:
Words
P is ________ times n
Ordered Pair
Expression
n)
n
(n,
e) How much will Sam be paid for 9 h of babysitting?
Solution
Sam’s pay = 7 × n
=7×
Substitute 9 for n.
=
Sam will be paid $
.
Sky sells magazine subscriptions. She gets $20 for every 5 subscriptions she sells.
a) Complete the table.
Number of Subscriptions, n
0
5
10
Pay, P
0
20
40
15
20
25
b) Is this a linear relation? Explain how you know.
• difference of consecutive n-values =
The relation is
• difference of consecutive P-values =
.
(linear or not linear)
c) Let the number of subscriptions be n.
An expression for Sam’s pay is
.
d) How much will Sky get paid if
she sells 40 subscriptions?
Sky’s pay = 4n
9.2 Patterns in a Table of Values ● MHR 493
Name: _____________________________________________________
Date: ______________
1. Giselle and Tim are discussing the table of values.
3
1
m
a
5
3
7
5
9
7
a) Who is correct? Circle GISELLE or TIM.
b) How do you know?
2.
x
1
2
3
4
a) The difference between consecutive x-values is
y
5
8
11
14
(4 – 3 =
, 3 – 2 =,
, and 2 – 1 =
b) The difference between consecutive y-values is
(14 – 11 =
.
, 11 – 8 =
, and
)
.
)
c) Does this table of values describe a linear relation? Circle YES or NO.
Give 1 reason for your answer.
_________________________________________________________________________
d) Graph the table of values.
e) Look at the graph. Describe the movement from
(1, 5) to the next point.
y
14
Starting at (1, 5), move
12
10
unit horizontally and
8
6
units vertically.
4
2
0
494
1
2
3
4 x
MHR ● Chapter 9: Linear Relations
Name: _____________________________________________________
Date: ______________
3. The table of values describes a linear relation.
0
0
x
a
1
4
2
8
3
12
4
16
5
20
a) The difference between consecutive x-values is
.
b) The difference between consecutive a-values is
.
c) Graph the ordered pairs.
d) Look at the graph. Describe in words how to
move from (0, 0) to the next point.
a
20
___________________________________
16
12
___________________________________
8
4
0
___________________________________
1
2
3
4
5 x
e) Write a in terms of x.
Words
a is
4.
x
2
3
4
5
times x
Ordered Pair
Expression
x)
x
(x,
y
7
10
13
16
a) What is the difference between consecutive x-values?
b) What is the difference between consecutive y-values?
Is the difference the same for consecutive values? Circle YES or NO.
c) Is the relationship in the table of values a
linear relation? Circle YES or NO.
Give 1 reason for your answer.
d) Check by graphing.
y
16
_________________________________
14
_________________________________
10
12
8
_________________________________
_________________________________
6
4
2
0
1
2
3
4
5 x
9.2 Patterns in a Table of Values ● MHR
495
Name: _____________________________________________________
Date: ______________
5. Mara reads 90 words per minute.
a) Complete the table of values.
Number of Minutes, m
1
2
3
4
5
6
Number of Words, w
b) Explain how you can find out if this is a linear relation.
________________________________________________________________________
________________________________________________________________________
c) If the number of minutes is m, then the expression for the number of words is
× m.
d) How many words can Mara read in 15 min?
Sentence: ___________________________________________________________________
6. A community centre has a new banquet hall.
The centre charges $5 per person to rent the hall.
a) Complete the table of values.
Number of People, p
1
Rental Cost, C ($)
5
20
40
60
80
100
b) If the number of people is p, then the expression for the rental cost is
c) How much will it cost for 150 people?
Sentence: __________________________________________________________________
496
MHR ● Chapter 9: Linear Relations
.
Name: _____________________________________________________
Date: ______________
Plan a canoe trip.
You can rent canoes at many national parks.
One canoe costs $40 a day to rent.
A park pass costs $36 for 1 week.
a) Complete the table.
The first row has been done for you.
Number
of Days, d
Cost, C ($)
b) Graph the ordered pairs in your table.
Label the axes.
y
Ordered
Pair
320
300
36 + (40 × 1)
1
280
260
(1, 76)
= 36 + 40
240
= 76
220
200
36 + (40 × 2)
2
= 36 + ______
180
(2, _____)
160
= ______
140
120
100
3
80
60
4
40
20
0
5
1
2
3
4
5
6
7
8 x
6
7
c) Is this a linear relation? Circle YES or NO.
Give 1 reason for your answer.
________________________________________________________________________
d) Write an expression for the cost, C, based on the number of days, d.
C=
+
×d
9.2 Math Link ● MHR 497
Name: _____________________________________________________
Date: ______________
9.3 Warm Up
1. Complete the patterns.
a) 3, 8, 13,
,
c) 10, 16, 22,
,
,
b) 9, 7, 5,
,
d) 38, 35, 32,
,
2. Whole numbers start at 0 and increase by 1 each time.
a) List the first 10 even whole numbers.
________________________________
b) List the first 10 odd whole numbers.
________________________________
3. Integers are … –3, –2, –1, 0, 1, 2, 3 , …
Zero is not positive or negative.
a) List the first 5 positive integers.
________________________________
b) List the first 5 negative integers.
________________________________
4. Use substitution to evaluate.
b) 8 + 2x, when x = 3
a) 3x – 4, when x = 5
3(
=
)–4
–4
Multiply.
=8+
=
=
d) –7x – 1, when x = 0
c) –5x + 6, when x = 2
–5(
=
)+6
+6
=
5. Let n describe Nancy’s age.
The expression n – 5 represents Jane’s age.
If Nancy is 12 years old, how old is Jane?
Jane is
498
8+2(
Substitute.
years old.
MHR ● Chapter 9: Linear Relations
)
Substitute.
Multiply.
Name: _____________________________________________________
Date: ______________
9.3 Linear Relationships
Working Example 1: Graph From a Linear Formula
formula
● an equation that shows how 1 variable is related to another
● example: P = 2l + 2w shows how the perimeter of a rectangle is related to its
length and width
A car travels at 60 km/h.
Use the formula d = 60t to express this relationship, where d is the distance travelled, in kilometres,
and t is the time, in hours.
a) Make a table of values.
Time cannot be negative in
this question. So, t cannot be
a negative integer.
Solution
0
Time, t
1
2
3
Distance, d
To find d, substitute each value of t from the table into the formula d = 60t.
d = 60t
d = 60(0)
d = 60t
d = 60(1)
d = 60t
d = 60t
d = 60(
)
d = 60(
)
d=
d=
d=
d=
b) Graph the ordered pairs in your table of values.
Solution
d
Distance (km)
180
Speed of Car
150
120
90
60
30
O
1
2
3
4 t
Time (h)
c) Is it possible to have points between the ones on this graph?
Solution
105 km is halfway between
90 km and 120 km.
It is possible to have points between the ones on the graph.
The distance at 2.5 h is
km. The time at 105 km is
h.
9.3 Linear Relationships ● MHR 499
Name: _____________________________________________________
Date: ______________
d) How far will the car travel in 3.5 h?
Solution
d = 60t
d = 60 × (
)
Substitute 3.5 for t.
d=
The car will travel 210 km in 3.5 h.
Paul rents a lawn mower for $8 per hour.
The formula C = 8t describes the relationship between the rental cost and the time.
C describes the rental cost and t describes time, in hours.
a) Complete the table of values.
t
C
What are possible
values for t ?
b) Graph your ordered pairs.
To draw a graph:
Label the x-axis “t” and the y-axis “C.”
Write a description along each axis.
Mark the intervals on the x- and y- axes.
Give the graph a title.
Plot the points.
c) Find the rental cost for 12 h.
500
MHR ● Chapter 9: Linear Relations
Name: _____________________________________________________
Date: ______________
Working Example 2: Graph From a Linear Equation Using Integers
Integers are
… –3, –2, –1, 0, 1, 2, 3, … .
equation
● 2 expressions that have the same value and are joined with an equal sign
● examples: 2x + 3 = 10 and y = x – 5
y = –3x + 4 is a linear equation.
a) Use the linear equation to make a table of values.
Solution
You do not know what x describes, so, use integer values for x.
To find y, substitute each of the values for x into the equation y = –3x + 4.
x
–2
–1
0
y=
1
For x = 1:
y = –3x + 4
y = –3(1) + 4
2
3
y=
y=
For x = 0:
y = –3x + 4
y = –3(0) + 4
For x = –1:
y = –3x + 4
y = –3(–1) + 4
For x = –2:
y = –3x + 4
y = –3(–2) + 4
y=6+4
y = 10
y
10
+4
y=
+4
+4
y=
y=
For x = 2:
y = –3x + 4
y = –3(2) + 4
For x = 3:
y = –3x + 4
y = –3(3) + 4
y=
y=
+4
+4
y=
y=
b) Graph the ordered pairs.
c) Find the y-value in the ordered pair (11, y).
Solution
Solution
The point (11, y) tells you that the x-value is 11.
Substitute 11 into the formula y = –3x + 4.
y
10
8
y = –3x + 4
6
4
y = –3(
2
–2
–1 0
–2
–4
–6
1
2
3
4 x
y=
)+4
+4
Substitute.
Multiply.
y=
The value of the y-coordinate is
The ordered pair is (11,
.
).
9.3 Linear Relationships ● MHR 501
Name: _____________________________________________________
Date: ______________
Use y = 2x + 3 to answer the questions.
a) Make a table of values using integers.
x
–2
–1
0
1
2
3
b) Graph the ordered pairs in your table
of values.
y
y
10
9
8
7
6
5
4
3
2
1
Let x = –2
y = 2x + 3
y = 2(–2) + 3
y = –4 + 3
y = –1
Let x = –1
y = 2x + 3
Let x = 0
-5 -4 -3 -2 -1 O
-1
2
3
4
5 x
-2
-3
-4
y = 2( ___ ) + 3
-5
-6
-7
y = _____ + 3
-8
-9
y = ______
Let x = 1
1
Let x = 2
-10
Let x = 3
c) What are the coordinates for the point that lies on the y-axis?
x = 0 for all points that lie on the y-axis.
y = 2x + 3
y = 2(
y=
)+3
+
y=
The coordinates for the point that lies on the y-axis are (0,
502
MHR ● Chapter 9: Linear Relations
).
Name: _____________________________________________________
Date: ______________
1. A store sells hats for $10 each. Let h describe the number of hats.
The formula S = 10h shows the amount of money collected in sales, S.
a) Can negative integers be used for the values of h? Give 1 reason for your answer.
_______________________________________________________________________
b) Can whole numbers be used for the values of h? Give 1 reason for your answer.
_______________________________________________________________________
2. a) Name the point that is 3 units to the right of the origin. (
,
)
(0, 0)
b) Name the point that is 4 units to the left of the origin. (
,
)
c) What do you notice about these 2 points? ___________________________________
d) Is this true for all points that lie on the x-axis? Give 1 reason for your answer.
_______________________________________________________________________
3. Find the value of each equation.
a) y = 5x – 3 when x = 6
y = 5(
y=
y=
c) y = x – 8 when x = 5
b) y = 3x + 2 when x = –4
)–3
–3
← Substitute →
y = 3(
← Multiply →
y=
)+2
+2
y=
d) y = –5x when x = –2
9.3 Linear Relationships ● MHR 503
Name: _____________________________________________________
4. Use the formula C = 6t to describe a long-distance
telephone plan, where C is the cost in cents and t is
the time in minutes.
a) Make a table of values.
Use at least 4 whole number values for t.
t
1
1
2
3
4
5
6
7
8
9
*
0
#
Date: ______________
Only 6¢ per minute
anytime for calls
across Canada!
CallCanada
Let t = 1
C = 6t
C = 6(1)
C
C=
b) Graph the ordered pairs from your
table of values.
y
c) If you round part minutes up to the next
whole minute, is it possible to have points
between the ones on your graph? Explain.
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
x
5. Complete each table of values.
a) y = 3x + 2
x
–2
0
2
4
y
Let y = –2
y = 3(–2) + 2
y=
y=
b) y = –4x
x
–2
0
2
4
504
y
MHR ● Chapter 9: Linear Relations
+2
Let y = 0
Let y = 2
Let y = 4
Name: _____________________________________________________
Date: ______________
6. An animal shelter pays you $5 for each dog you walk.
Use the formula M = 5d to relate the money you make to the number of dogs you walk.
M is the money you make and d is the number of dogs you walk.
b) Graph the ordered pairs.
a) Make a table of values.
y
d
M
x
7. For each equation, find the value of y in the ordered pair (2.5, y).
a) y = 3x + 2
b) y = x – 5
y = 3(2.5) + 2
y=
← Substitute →
+2
← Multiply →
y=
8. The graph shows Nigel’s monthly pay.
e
Nigel’s Monthly Earnings
Monthly pay ($)
1600
A break in the y-axis means the
length of the axis has been
shortened. It can be shown as
1500
1400
1300
1200
0
1000 2000 3000 4000 5000 6000 7000 s
Sales ($)
a) If Nigel does not make any sales, what is his monthly pay?
b) Nigel has sales of $4000 in 1 month. How much does he make?
c) Nigel earns $1300 in 1 month. What are his sales?
9.3 Linear Relationships ● MHR 505
Name: _____________________________________________________
Date: ______________
9. You are given part of a table of values for a linear relation.
–3
x
–2
–1
y
0
1
2
6
8
10
a) How could you find the missing y-coordinates?
_________________________________________________________________________
b) Complete the table.
10. You can buy work gloves from a web site.
Use the formula C = 5g + 2 to find the price.
C is the cost in dollars and g is the number of pairs of gloves.
a) Complete the table of values using whole numbers.
g
C
b) Graph the ordered pairs.
To draw a graph:
Label each axis using g and C.
Describe each axis.
Mark the intervals on each axis.
Give the graph a title.
Plot the points.
c) Is this a linear relation? Circle YES or NO.
Give 1 reason for your answer.
___________________________________
___________________________________
___________________________________
d) Are other points possible between the ones on your graph? Circle YES or NO.
Give 1 reason for your answer.
_________________________________________________________________________
_________________________________________________________________________
506
MHR ● Chapter 9: Linear Relations
Name: _____________________________________________________
Date: ______________
9 Chapter Review
Key Words
For #1 to #5, fill in the blanks.
Unscramble the letters for each term to complete the sentence.
1. A pattern that creates points that lie in a straight line is called a
relation. EILRAN
2. A chart showing the relationship between 2 sets of numbers is called a
.
LEBTA FO SUAVLE
3. In the expression 5g – 2, g is called a
. ABEVIRAL
4. When 2 expressions are joined with an equal sign, you have an
QONATUIE
.
5. An equation that shows how 1 variable is related to another is called a
. MALUROF
9.1 Analysing Graphs of Linear Relations, pages 478–486
6. Klaus works after school. The graph shows his rate of pay.
P
45
a) Fill in the table of values.
1
2
y
36
Pay ($)
x
0
Klaus’s Rate
of Pay
27
18
9
0
1
2
3
4
t
Time (h)
3
4
b) Does the graph represent a linear relation? Circle YES or NO. Give 1 reason for your answer.
_________________________________________________________________________
c) Is it possible to have other points between the ones on the graph? Circle YES or NO.
Give 1 reason for your answer.
_________________________________________________________________________
Chapter Review ● MHR 507
Name: _____________________________________________________
7. The graph shows the amount of money a grade 8 class made while
doing a car wash fundraiser.
I
Money Collected ($)
1
2
3
Grade 8 Car Wash
50
a) Using the graph, fill in the table of values.
Number of Cars
Date: ______________
4
40
30
20
10
Money Collected ($)
0
1
2
3
4
5 c
Number of Cars
b) Describe the 2 patterns you see in the graph.
• The overall pattern: ________________________________________________________
• To move from 1 point to the next: _____________________________________________
c) Write an expression that describes the amount of money collected after washing c cars.
d) If the students wash 15 cars, how much money will they collect?
Amount collected = 10c
10c = 10 × c
= 10 ×
=
9.2 Patterns in a Table of Values, pages 488–497
8. a) Is this a linear relation?
A
0
1
2
3
4
5
B
0
7
14
21
28
35
The difference between consecutive A-values is
(1 – 0 =
,2–1=
.
,3–2=
)
The difference between consecutive B-values is
(7 – 0 =
, 14 – 7 =
.
, 21 – 14 =
)
The relation is
, since the A-values change by the same amount
and the B-values change by the same amount.
b) Write B in terms of A.
Words
B is ________ times A
508
MHR ● Chapter 9: Linear Relations
Ordered Pair
(A,
A)
Expression
A
Name: _____________________________________________________
Date: ______________
9. The table of values shows a relation.
p
1
2
3
4
5
a) Is this a linear relation?
Q
5
8
10
13
15
The difference between consecutive p-values is
.
The difference between consecutive Q-values is
.
The relation is
, since
______________________________________________________.
b) Graph the ordered pairs to check your answer.
q
16
14
12
10
8
6
4
2
O
1
2
3
4
5 p
10. A recreation centre charges $5 per person to use the gym.
a) Complete the table of values.
Number of People
3
5
7
9
Gym Charge
b) Without graphing, explain if this is a linear relation or not.
__________________________________________________________________________
__________________________________________________________________________
c) Let n describe the number of people. Write an expression for the cost to use the gym.
d) How much will it cost for 25 people to use the gym?
Sentence: ____________________________________________________________________
Chapter Review ● MHR 509
Name: _____________________________________________________
Date: ______________
9.3 Linear Relationships, pages 499–506
11. Craig travels at a constant speed of 15 km/h.
The formula d = 15t describes the relationship.
His speed does
not change.
a) What does each variable describe?
d describes the
t describes the
b) Make a table of values. Use 5 consecutive whole number values for t.
t
2
d
Let t = 2
d = 15t
d = 15(2)
d=
c) Graph your ordered pairs.
To draw a graph:
Label each of the axes using t and d.
Describe each axis.
Mark the intervals on both axes.
Give the graph a title.
Plot the points.
d) Is it reasonable to have points between the ones on the graph? Circle YES or NO. Explain.
__________________________________________________________________________
__________________________________________________________________________
e) How far would Craig travel in 8 h?
Sentence: _________________________________________________________________
510
MHR ● Chapter 9: Linear Relations
Name: _____________________________________________________
Date: ______________
12. y = 2x + 4
a) Use 5 integers for x to complete this table of values (use positive and negative integers).
x
Let y = −2
y = 2x + 4
y = 2(−2) + 4
y
–2
y=
+4
y=
b) Graph the ordered pairs.
y
10
9
8
7
6
5
4
3
2
1
-2 -1 O
1
2
3
4
5
6
7
8
9
10 x
c) Find the value of y in the ordered pair (2.5, y).
y = 2x + 4
y = 2(
y=
)+4
+4
Substitute.
Add.
y=
d) Find the value of y in the ordered pair (–6, y).
Chapter Review ● MHR 511
Name: _____________________________________________________
Date: ______________
9 Practice Test
For #1 to #5, choose the best answer.
1. You can describe 2x – 1 as a(n)
.
A constant
C expression
B equation
D variable
2. The table shows the number of toothpicks in the base of a triangle in relation to its perimeter.
Toothpicks
in Base, b
1
2
3
Toothpicks in
Perimeter, p
3
6
9
Which expression describes the number of toothpicks in the perimeter of any triangle in
this pattern?
A b+3
b
C
3
B 3b
D b–3
3. Which table of values describes this linear relation?
y
8
6
4
2
0
512
1
2
3
4 x
A
x
y
0
8
1
6
2
6
3
5
4
4
B
x
y
0
8
1
7
2
6
3
4
4
2
C
x
y
0
8
1
7
2
6
3
5
4
4
D
x
y
0
8
1
6
2
4
3
3
4
2
MHR ● Chapter 9: Linear Relations
Name: _____________________________________________________
4. Which table of values describes the linear equation y = 3x – 2?
A
x
1
2
3
y
1
4
8
B
x
0
2
4
y
2
8
1
C
x
2
3
4
y
4
7
10
D
x
3
5
7
y
9
15
21
Date: ______________
Substitute values for
x into the equation.
5. Which graph describes this statement:
“A banquet room rents for $50 a night plus $2 per person”?
The formula for the total cost is C = 50 + 2p, where p is the number of people.
10 20 30 40 n
Number of People
10 20 30 40 n
Number of People
C
110
90
70
50
Cost ($)
Cost ($)
0
D
C
90
70
50
0
C
110
90
70
50
Cost ($)
90
70
50
0
C
B
C
Cost ($)
A
10 20 30 40 n
Number of People
0
10 20 30 40 n
Number of People
For #6 and #7, complete the statements.
6. If the equation is y = 4x + 2, the value for y in (1, y) is
7. For this the graph, when the x-coordinate increases by 1,
the y-coordinate
.
y
8
6
by
.
4
2
0
1
2
3
4 x
Practice Test ● MHR 513
Name: _____________________________________________________
Date: ______________
Short Answer
8. The graph shows the cost of a new drink called Zap.
C
a) What is the cost of 1 can of Zap?
Cost of Zap
18
15
b) Describe 2 patterns in the graph.
Cost ($)
12
9
• The pattern of the points: ____________________
6
• To move from 1 point to the next: _____________
3
0
1
2
3
4
5
6 n
_____________________________________
Number of Cans
c) Complete the table of values.
Number
of Cans, n
1
2
3
d) Describe 2 patterns in the table of values.
The difference between consecutive values of
Cost ($),
C
3
6
9
n is
.
The difference between consecutive values of
C is
4
.
5
9. The formula b = 4f represents the pattern in the table.
The number of black dots is b and the figure number is f.
Figure 1
Figure 2
Figure 3
a) Complete the table of values.
Figure
Number, f
Number of
Black Dots, b
1
2
3
4
5
514
MHR ● Chapter 9: Linear Relations
b) Use the formula to find the number of black dots in
Figure 10.
Name: _____________________________________________________
Date: ______________
You decide to go on an adventure tour!
Choose your adventure:
• hang gliding
• white-water rafting
• any other adventure
• hiking
• dogsledding
• canoeing
• cycling
a) Describe 3 different adventure tours you find on the Internet:
1. _____________________________________________________________________
2. _____________________________________________________________________
3. _____________________________________________________________________
b) Choose 1 of the adventures from part a):
c) Using travel brochures, the Internet, or
other resources, find or create data for a
linear relation.
d) Graph the ordered pairs in your table of
values.
Examples:
• cost related to the number of hours
• cost related to the rental charges
• distance travelled over time
• height climbed over time
Record your findings in a table of values.
e) Is it possible to have points between the ones on your graph? Explain why or why not.
________________________________________________________________________
Practice Test ● MHR 515
Name: _____________________________________________________
Date: ______________
Draw a line from the term in column A to the example in column B.
Then, find each term in column A in the word search below.
A
B
A. equation
1. P = 2l + 2w, where P is the perimeter, l is the length of a
rectangle, and w is the width of the rectangle
B. expression
2.
C. formula
D. linear relation
3.
0
20
Height (m)
Temperature (°C)
15
30
45
y
15
12
E. table of values
9
6
F. variable
3
1
0
2
3
4
5
6
x
4. y = 4x
5. An unknown number, c
6. 6n – 2
E
J
O Y N M P
N
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516
T
MHR ● Chapter 9: Linear Relations
I
60
75
Name: _____________________________________________________
Date: ______________
Math Games
−4
Friends and Relations
−3
Play the Friend and Relations game with a partner.
3
4
2
1
−2
−1
0
• spinner BLM
• paper clip
• set of 20 Friends and Relations
game cards
• Friends and Relations chart
for each student
Rules:
• Each player spins the spinner once to decide who will deal
the cards.
If there is a tie, spin again.
• The dealer shuffles and deals 10 Friends and Relations game
cards to each player.
The other player takes the first turn.
For each turn:
• Flip over the top card in your stack. Copy the linear relation from the
card into the Linear Relation column of your chart.
• Spin the spinner and record the result in the x-column of your chart.
• Calculate the y-value by substituting the x-value into the linear
relation. Have your partner check your calculation.
• Record the y-value in the y-column of your chart.
• After each turn, add your y-values. Compare the totals.
My y-value was 14 in
my first turn and –8
in my second turn.
My total score after
2 turns is 14 + (–8),
which equals 6.
How to win:
• The player with the highest total score is the winner.
• If there is a tie after 10 turns, shuffle the deck again and deal the cards.
Take more turns until 1 player wins.
Linear Relation
x
y
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
TOTAL of y-values
Math Games ● MHR 517
Name: _____________________________________________________
Date: ______________
Challenge in Real Life
Comparing Wages
Five people work at Moy’s Food Mart.
The table shows a weekly time sheet.
• grid paper
• calculator
Total
Hours
Hourly
Wage
$12.50
Employee
Mr. Moy
Mon
8
Tues
8
Wed
8
Thurs
8
Fri
9
Sat
—
Sun
—
Ms. Wong
—
8
8
7
—
9
9
$9.50
Maria
8
5
—
—
6
9
5
$7.50
Tom
4
4
4
4
4
8
8
$7.50
Jacob
—
—
5
7
7
5
5
$7.50
Total
Wage
TOTAL
1. For each employee, find the “Total Hours” and the “Total Wage.”
Total wage
= total hours × hourly wage
2. Who made the most money?
3. Who worked the most hours?
and
4. What is the total wages paid out for the week? _______________
5. Complete the tables of values to find the rates of pay.
Mr. Moy
Hours Wage
0
5
10
15
20
Ms. Wong
Hours Wage
0
5
10
15
20
Maria
Hours Wage
0
5
10
15
20
Mr. Moy’s hourly rate
= $12.50
Wage for 0 h = 0 × 12.50
=
Ms. Wong’s hourly rate
Maria’s hourly rate
=
=
Wage for 5 h = 5 × 12.50
=
6. Graph each table of values on a grid. Use a different colour for each set of points.
518
MHR ● Chapter 9: Linear Relations
Answers
3. a) Yes, the graph is linear because the points lie on a straight line.
Get Ready, pages 474–475
b) Answers may vary. Example: The points lie in a straight line; move
100 units right and 60 units up.
1. a) letters of the alphabet; begins with b; skips 2 letters
b) integers; begins with 9; decreases by 5
2. a)
1
4
Figure Number
Number of Squares
2
7
3
10
c)
1
4
Figure Number
Number of Cubes
2
6
3
8
x
1
2
3
4
5
6
7
8
9
y
2
4
6
8
10
12
14
16
18
b) Answers may vary. Example: The points lie in a straight line; move
1 unit right and 2 units up d) 18
x
–1
0
–1
3
3
3
2
y
2
1
0
2
1
0
–1
5. a)
1
2
3
4
5
6
10
9
10
11
10
30
b) Answers may vary. Example: It took Paulette approximately the same
time for the first 5 sections, creating a consistent pattern.
c) Answers may vary. Example: The terrain could have been more difficult,
or Paulette could have taken a break.
Paulette’s Climb
t
Total Time
(min)
100
80
60
40
20
0
e) Answers may vary. Example: The first
5 dots appear to lie along a straight
line, but the last dot does not.
200 400 600 d
Total Distance
(m)
b) $15 c) $67.50
Math Link
a) Answers may vary. Example: The points lie in a straight line; move 1 unit
right and 400 units up.
b)
d)
2. multiply by 2; divide by 3; multiply by 10
Number
of Days
0
1
2
3
4
5
6
7
Cost ($)
0
400
800
1200
1600
2000
2400
2800
Number
of Days
0
1
2
3
4
5
6
7
Cost ($)
300
700
1100
1500
1900
2300
2700
3100
e) Answers may vary. Example: They are similar because the rate of increase
is $400.00 per day. They are different because the starting value for the
first table is (0, 0) and the starting value for the second table is (0, 300).
3. B (1, 5), C (0, 7), D (7, 2), E (6, 0), F (8, 9)
9.1 Analysing Graphs of Linear Relations, pages 478–486
Working Example 1: Show You Know
1
5
2
10
9.2 Warm Up, page 487
Figure 4
3
15
1. a) add 3
4
20
10
8
6
4
2
0
Working Example 2: Show You Know
1
2
2
4
3
6
4
8
c) $2.00 d) 2n e) $104.00 f) No; you cannot buy part of a book.
2 4 6 8 10
x
x
1. Answers may vary. Example: Graph A: Yes, you can travel part of a
kilometre. Graph B: No, you cannot buy part of a carton of ice cream.
Practise
2. a) Answers may vary. Example: straight line; move 1 unit right and 20 units up
3
60
3. a) 2 b) 3 c) 9 d) 3 e) 2 f) 2
9.2 Patterns in a Table of Values, pages 488–497
Working Example 1: Show You Know
a)
Communicate the Ideas
2
40
2 4 6 8
Input
2. a) 3 b) 30 c) 4 d) 17
a) straight line; move 1 unit right and 2 units up
1
b) Number of Steps
20
Total Height (cm)
c) 20 cm; 20 cm d) 20 s e) 200 cm
y
12
10
8
6
4
2
0
d) YES e) 5f f) 40 squares
Number of Notebooks, n
Cost, C ($)
b) subtract 1
y
Output
a) straight line; move 1 unit right and 5 units up b)
b)
Pay
($)
15
30
45
60
75
6. Yes, you can buy 2 flowers.
1. add 3; subtract 1; add 6
Figure Number, f
Number of Squares, s
Time Worked
(h)
1
2
3
4
5
c) $3200
9.1 Warm Up, page 477
c)
400
240
G
Math Link
d)
300
180
4. a) and c)
3. a) shirts; s – 2 b) Let p stand for the number of pencils; 5p
4.
Point
A
B
C
D
E
F
100-m Section
Time to Climb
That Section
200
120
Apply
Start with 4 cubes and increase by 2.
a)
100
60
d) Answers may vary. Example: Yes, you can have a point between the
ones on the graph because you can buy between 100 and 200 grams.
Start with 4 squares and increase by 3.
b)
0
0
Quantity (g)
Cost ($)
4
80
x
1
2
3
4
y
5
10
15
20
Ordered Pair
(x, y)
1, 5
2, 10
3, 15
4, 20
y
35
30
25
20
15
10
5
0
1 2 3 4
x
b) 1, 5 c) y = 5x
Answers ● MHR 519
Apply
Working Example 2: Show You Know
a)
Difference Between
Consecutive First
Variables
Difference Between
Consecutive Second
Variables
Table 1
4−1=3
7−4=3
Consecutive x-values
have a difference of 3.
Table 2
1−0=1
2−1=1
3−2=1
Consecutive p-values
have a difference of 1.
10 − 1 = 9
19 − 10 = 9
Consecutive y-values
have a difference of 9.
1−0=1
3−1=2
6−3=3
The q-values differ
by 1, 2, 3.
b) Yes, Table 1 is linear because the difference in consecutive values are the
same; No, Table 2 is not linear because the difference in consecutive
values of q is not the same.
of Table 1
c) Graph
y
5. a) Number of
Minutes, m
Number of
Words, w
6. a)
1 2 3 4 5 6p
5
10
15
20
25
0
20
40
60
80
100
b) linear c) 4n d) $160
1. a) TIM b) Answers may vary. Example: The relation is linear because all
consecutive values of m differ by 2 and values of a differ by 2.
Practise
2. a) 1 b) 3 c) YES. The relation is linear because all consecutive values of
x differ by 1, and all consecutive values of y differ by 3.
e) 1 unit, 3
d
14
12
10
8
6
4
2
450
540
Number of
People, p
Rental Cost,
C ($)
1
20
40
60
80
100
5
100
200
300
400
500
Cost, C ($)
76
116
156
196
236
276
316
c) YES. This relationship is linear because
there is a common difference between the
consecutive values for both variables.
d) C = 36 + 40 × d
320
300
280
260
240
220
200
180
160
140
120
100
80
60
40
20
0
1 2 3 4 5 6 7
Number of Days
Ordered Pair
(# of Days, Cost)
(1, 76)
(2, 116)
(3, 156)
(4, 196)
(5, 236)
(6, 276)
(7, 316)
n
9.3 Warm Up, page 498
3. a) 1, 2, 3, 4, 5 b) –1, –2, –3, –4, –5
4. a) 11 b) 14 c) 4 d) –1
1 2 3 4 5a
5. 7
d) move 1 unit right, and 4 units up
e) four, 4, 4
9.3 Linear Relationships, pages 499–506
Working Example 1: Show You Know
a)
1 2 3 4 5x
4. a) 1 b) 3; YES c) YES. The relation is linear because all consecutive
values of x differ by 1, and all consecutive values of y differ by 3.
d
t
1
2
3
4
C
8
16
24
32
Cost of
Lawnmover Rental
b)
C
Rental Cost ($)
A
20
16
12
8
4
0
520
360
1. a) 18, 23, 28 b) 3, 1 c) 28, 34 d) 29, 26
3. a) 1 b) 4 c)
40
32
24
16
8
0
16
14
12
10
8
6
4
2
0
270
2. a) 0, 2, 4, 6, 8, 10, 12, 14, 16, 18 b) 1, 3, 5, 7, 9, 11, 13, 15, 17, 19
0
d)
180
Cost of Canoe Trip
Communicate the Ideas
d)
90
C
Total Cost ($)
0
6
Number of Days, d
1
2
3
4
5
6
7
b)
Number of
Subscriptions, n
Pay, P
5
a)
Working Example 3: Show You Know
a)
4
b) 5p c) $750
2 4 6 8x
0
3
Math Link
Graph of Table 2
0
2
b) Answers may vary. Example: You can find out if this is a linear
relationship by checking the consecutive values for m and w. c) 90
d) 1350
q
6
5
4
3
2
1
20
18
16
14
12
10
8
6
4
2
1
1 2 3 4 5n
MHR ● Chapter 9: Linear Relations
1 2 3 4 5 t
Rental Time (h)
c) $96
Working Example 2: Show You Know
a)
x
–2
–1
0
1
2
3
Chapter Review, pages 507–511
b)
y
–1
1
3
5
7
9
c) (0, 3)
y
10
9
8
7
6
5
4
3
2
1
1. linear 2. table of values 3. variable 4. equation 5. formula
6. a)
b) YES. The points on the graph are
in a straight line. c) YES. Klaus
could work part of his shift.
y
0
9
18
27
36
1
10
7. a) Number of Cars
Money Collected ($)
1 2 3 4 5x
-2 0
-2
x
0
1
2
3
4
2
20
3
30
4
40
b) The points lie in a straight line. Move 1 unit right, and
10 units up. c) 10c d) $150
Communicate the Ideas
8. a) linear b) seven; 7; 7
1. a) No, because you cannot sell a negative number of hats. b) Yes, because
you can sell any number of hats, beginning at 0 and going up.
9. a) not linear; the changes in consecutive values of Q are not the same.
b)
2. a) (3, 0) b) (–4, 0) c) The second coordinate is 0. d) Yes, because they
lie in a straight line.
Practise
3. a) y = 27 b) y = –10 c) y = –3 d) y = 10
C
6
12
18
24
b)
Cost of Long
Distance
30 Phone Plan
C
0
24
18
12
6
10. a)
c) No, because any part minutes are rounded up to the next nearest minute.
6. a)
x
–2
0
2
4
y
–4
2
8
14
1
5
d
M
2
10
b)
3
15
x
–2
0
2
4
4
20
M Money Earned
35
30
25
20
15
10
5
for Dog Walking
0
1 2 3 4 5 6d
Number of Dogs
g
1
2
3
4
C
7
12
17
22
–2
2
b)
–1
4
0
6
1
8
2
10
Cost of Work
Total Cost ($)
10. a)
–3
0
9
45
c)
d
30
45
60
75
90
Distance Travelled
by a Cyclist
d
90
75
60
45
30
15
2
4
Time (h)
6 t
d) YES. Craig can travel for a length of time that is not a whole number.
e) 120 km
8. a) $1200 b) $1400 c) $2000
x
y
t
2
3
4
5
6
0
Apply
9. a) Use a pattern. b)
7
35
c) 5n d) $125
b)
b)
7. a) y = 9.5 b) y = –2.5
5
25
11. a) distance travelled in kilometres; time in hours
y
8
0
–8
–16
Money ($)
5. a)
3
15
Number of People
Gym Charge
b) Yes, this is a linear equation because the consecutive values of both
variables have a common difference.
1 2 3 4 t
Time (min)
0
1 2 3 4 5p
Distance (km)
t
1
2
3
4
Total Cost (¢)
4. a)
q
16
14
12
10
8
6
4
2
12. a) Answers may vary. Example:
x
–2
–1
0
1
2
y
0
2
4
6
8
b) Answers may vary. Example:
y
10
8
6
4
–2 0
2 4 6 8 10 x
c) y = 9 d) y = –8
C Gloves
20
10
0
2
4 g
Pairs of Gloves
c) YES. The points are in a straight line. d) NO. The values of g must be
whole numbers because they represent the number of pairs of gloves.
Answers ● MHR 521
Practice Test, pages 512–514
2. Mr. Moy
1. C 2. B 3. C 4. C 5. D
3. Mr. Moy; Ms. Wong
6. 6
4. $1637.00
7. decreases; 1
5.
8. a) $3.00 b) Answers may vary. Example: The points appear to lie in a
straight line; The values for both variables have a constant difference;
move 1 unit right and 3 units up.
c)
Number of Cans, n
1
2
3
4
5
9. a)
Cost ($), C
3
6
9
12
15
Number of Black
Dots, b
4
8
12
16
20
Figure Number
1
2
3
4
5
Mr. Moy
Hours
0
5
10
15
20
d) 1; 3
Maria
Hours
0
5
10
15
20
b) 40
6.
Wrap It Up!, page 515
Wages ($)
a)–e) Answers will vary.
Key Word Builder, page 516
1. C 2. E 3. D 4. A 5. F 6. B
Challenge in Real Life, page 518
1.
Employee
Mr. Moy
Ms. Wong
Maria
Tom
Jacob
TOTAL
522
Total
Hours
41
41
33
36
29
180
Hourly
Wage
$12.50
$9.50
$7.50
$7.50
$7.50
Wage
$0
$62.50
$125.00
$187.50
$250.00
Total Wage
$512.50
$389.50
$247.50
$270.00
$217.50
$1637.00
MHR ● Chapter 9: Linear Relations
w
260
240
220
200
180
160
140
120
100
80
60
40
20
0
Wage
$0
$37.50
$75.00
$112.50
$150.00
Wages at Mr. Moy’s
Food Mart
5
10 15
Hours Worked
20 h
Mr. Moy
Ms. Wong
Maria
Ms. Wong
Hours
Wage
0
$0
5
$47.50
10
$95.00
15
$142.50
20
$190.00