Post on 18-Apr-2018
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Hitting the SlopesDetermining Rate of Change from a Graph
VocabularyMatch each definition to its corresponding term.
1. rate which has a second term of 1 unit
e. unit rate
a. rate of change
2. ratio in which the units of the quantities being compared are different
d. rate
b. per
3. the vertical change from one point to another point on a graph
g. rise
c. run
4. phrase used when a rate is used to describe a rate of increase
(or decrease) in a real-life situation
a. rate of change
d. rate
5. the rate of change on a graph written as the vertical change from one
point to another over the horizontal change of the same two points
f. rise ____ run
e. unit rate
6. means “for each” or “for every”
b. per
f. rise ____ run
7. the horizontal change from one point to another point on a graph
c. run
g. rise
Lesson 3.1 Skills Practice
Name ________________________________________________________ Date _________________________
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Lesson 3.1 Skills Practice page 2
Problem SetWrite a rate for each situation. State whether the rate is a rate of increase or a rate of decrease.
1. Use the graph to write a rate that compares the track length to the change in time at point A.
Time (in seconds)
Trac
k le
ngth
(in
met
ers)
x86
180
10 12 14 16 18 20420
y
270
360
450
540
600
Length of Track Remaining in Bobsled Run
90
A
B
C
Point A is at (4, 480).
Because the bobsled started at 600 meters, the change in the track length remaining is
120 meters. The change in time is 4 seconds. So, the rate of decrease is 120 ____ 4
.
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Lesson 3.1 Skills Practice page 3
Name ________________________________________________________ Date _________________________
2. Use the graph to write a rate that compares the track length to the change in time at point B.
Time (in seconds)
Trac
k le
ngth
(in
met
ers)
x86
180
10 12 14 16 18 20420
y
270
360
450
540
600
Length of Track Remaining in Bobsled Run
90
A
B
C
Point B is at (10, 300).
Because the bobsled started at 600 meters, the change in the track length remaining is
300 meters. The change in time is 10 seconds. So, the rate of decrease is 300 ____ 10
.
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Lesson 3.1 Skills Practice page 4
3. Use the graph to write a rate that compares the track length to the change in time at point C.
Time (in seconds)
Trac
k le
ngth
(in
met
ers)
x86
180
10 12 14 16 18 20420
y
270
360
450
540
600
Length of Track Remaining in Bobsled Run
90
A
B
C
Point C is at (15, 150).
Because the bobsled started at 600 meters, the change in the track length remaining is
450 meters. The change in time is 15 seconds. So, the rate of decrease is 450 ____ 15
.
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Lesson 3.1 Skills Practice page 5
Name ________________________________________________________ Date _________________________
4. Use the graph to write a rate that compares the change in elevation to the change in time
at point D.
Time (in minutes)
Dis
tanc
e (in
feet
)
x86 10 12 14 16 18 20420
y Distance Ascended by a Mountain Climber
20
40
60
80
100
120
140
160
180
200
D
E
F
Point D is at (5, 50).
Because the mountain climber started at 0 feet, the change in elevation is 50 feet. The change
in time is 5 minutes. So, the rate of increase is 50 ___ 5 .
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Lesson 3.1 Skills Practice page 6
5. Use the graph to write a rate that compares the change in elevation to the change in time
at point E.
Time (in minutes)
Dis
tanc
e (in
feet
)
x86 10 12 14 16 18 20420
y Distance Ascended by a Mountain Climber
20
40
60
80
100
120
140
160
180
200
D
E
F
Point E is at (12, 120).
Because the mountain climber started at 0 feet, the change in elevation is 120 feet. The change
in time is 12 minutes. So, the rate of increase is 120 ____ 12
.
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Lesson 3.1 Skills Practice page 7
Name ________________________________________________________ Date _________________________
6. Use the graph to write a rate that compares the change in elevation to the change in time
at point F.
Time (in minutes)
Dis
tanc
e (in
feet
)
x86 10 12 14 16 18 20420
y Distance Ascended by a Mountain Climber
20
40
60
80
100
120
140
160
180
200
D
E
F
Point F is at (18, 180).
Because the mountain climber started at 0 feet, the change in elevation is 180 feet. The change
in time is 18 minutes. So, the rate of increase is 180 ____ 18
.
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Lesson 3.1 Skills Practice page 8
Label two points on each graph to determine the rate of change for the graph. Write the rate in the
format rise ____ run and as a unit rate.
7.
Time (in seconds)
Ele
va
tio
n (
in m
ete
rs)
x86 10 12 14 16 18 20420
y
20
40
60
80
100
120
140
160
180
200
B
A
Sample answer:
Point A: (4, 90)
Point B: (6, 50)
rise ____ run 5 240 meters ___________ 2 seconds
unit rate 5 220 meters ___________ 1 second
8.
Time (in seconds)
Ele
va
tio
n (
in m
ete
rs)
x43 5 6 7 8 9 10210
y
5
10
15
20
25
30
35
40
45
50
B
A
Sample answer:
Point A: (2, 35)
Point B: (4, 25)
rise ____ run 5 210 meters ___________ 2 seconds
unit rate 5 25 meters __________ 1 second
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Lesson 3.1 Skills Practice page 9
Name ________________________________________________________ Date _________________________
9.
Time (in minutes)
Dis
tan
ce
(in
fe
et)
x86 10 12 14 16 18 20420
y
6
12
18
24
30
36
42
48
54
60
A
B
Sample answer:
Point A: (4, 24)
Point B: (6, 36)
rise ____ run 5 12 feet __________ 2 minutes
unit rate 5 6 feet _________ 1 minute
10.
Time (in hours)
Dis
tan
ce
(in
mile
s)
x43 5 6 7 8 9 10210
y
200
150
100
50
A
B
Sample answer:
Point A: (6, 90)
Point B: (8, 120)
rise ____ run 5 30 miles ________ 2 hours
unit rate 5 15 miles ________ 1 hour
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Lesson 3.1 Skills Practice page 10
11.
Time (in seconds)
Ele
va
tio
n (
in m
ete
rs)
x86 10 12 14 16 18 20420
y
6
12
18
24
30
36
42
48
54
60
B
A
Sample answer:
Point A: (2, 36)
Point B: (4, 12)
rise ____ run 5 224 meters ___________ 2 seconds
unit rate 5 212 meters ___________ 1 second
12.
Time (in hours)D
ista
nce
(in
mile
s)
x43 5210
y
30
60
90
120
150
180
210
240
270
300B
A
Sample answer:
Point A: (2, 150)
Point B: (4, 300)
rise ____ run 5 150 miles _________ 2 hours
unit rate 5 75 miles ________ 1 hour
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Lesson 3.1 Skills Practice page 11
Name ________________________________________________________ Date _________________________
Determine the unit rate of change for lines A and B on each graph.
13.
Number of items
Cos
t (in
dol
lars
)
x86 1042 75 9310
y
1
2
3
4
5
6
7
8
9
10
C
D
F
E
AB
Line A sample answer:
Point C: (0, 0)
Point D: (1, 1.5)
rise ____ run 5 1.5 dollars __________ 1 item
unit rate 5 1.5 dollars __________ 1 item
Line B sample answer:
Point E: (0, 0)
Point F: (1, 3)
rise ____ run 5 3 dollars _________ 1 item
unit rate 5 3 dollars _________ 1 item
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Lesson 3.1 Skills Practice page 12
14.
Number of items
Cos
t (in
dol
lars
)
x86 1042 75 9310
y
2
4
6
8
10
12
14
16
18
20
E
F
AB
C
D
Line A sample answer:
Point C: (1, 7)
Point D: (2, 9)
rise ____ run 5 2 dollars _________ 1 item
unit rate 5 2 dollars _________ 1 item
Line B sample answer:
Point E: (1, 9)
Point F: (2, 13)
rise ____ run 5 4 dollars _________ 1 item
unit rate 5 4 dollars _________ 1 item
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Lesson 3.1 Skills Practice page 13
Name ________________________________________________________ Date _________________________
15.
Time (in seconds)
Ele
vatio
n (in
met
ers)
x1612 2084 1410 18620
y
10
20
30
40
E
AB
F
D
C
Line A sample answer:
Point C: (2, 30)
Point D: (4, 20)
rise ____ run 5 210 meters ___________ 2 seconds
unit rate 5 25 meters __________ 1 second
Line B sample answer:
Point E: (2, 24)
Point F: (4, 8)
rise ____ run 5 216 meters ___________ 2 seconds
unit rate 5 28 meters __________ 1 second
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Lesson 3.1 Skills Practice page 14
16.
Time (in seconds)
Ele
vatio
n (in
met
ers)
x86 1042 75 9310
y
10
20
30
40
50
60
70
80
90
100
E
AB
F
D
C
Line A sample answer:
Point C: (1, 70)
Point D: (3, 50)
rise ____ run 5 220 meters ___________ 2 seconds
unit rate 5 210 meters ___________ 1 second
Line B sample answer:
Point E: (1, 60)
Point F: (3, 20)
rise ____ run 5 240 meters ___________ 2 seconds
unit rate 5 220 meters ___________ 1 second
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Lesson 3.1 Skills Practice page 15
Name ________________________________________________________ Date _________________________
17.
Number of items
Cos
t (in
dol
lars
)
x43 5210
y
10
20
30
40A
B
E
F
D
C
Line A sample answer:
Point C: (0, 0)
Point D: (2, 18)
rise ____ run 5 18 dollars __________ 2 items
unit rate 5 9 dollars _________ 1 item
Line B sample answer:
Point E: (0, 0)
Point F: (2, 6)
rise ____ run 5 6 dollars _________ 2 items
unit rate 5 3 dollars _________ 1 item
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Lesson 3.1 Skills Practice page 16
18.
Time (in minutes)
Ele
vatio
n (in
feet
)
x202 4 6 8 10 12 14 16 180
y
2
4
6
8
10
12
14
16
18
20
B
AF
E
C
D
Line A sample answer:
Point C: (4, 16)
Point D: (8, 14)
rise ____ run 5 22 feet __________ 4 minutes
unit rate 5 20.5 feet _________ 1 minute
Line B sample answer:
Point E: (2, 13)
Point F: (4, 8)
rise ____ run 5 25 feet __________ 2 minutes
unit rate 5 22.5 feet _________ 1 minute
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At the ArcadeDetermining Rate of Change from a Table
VocabularyDefine the term in your own words.
1. first differences
First differences are the values determined by subtracting consecutive y-values in
a table when the x-values are consecutive integers.
Problem SetUse the informal method to determine the rate of change for the data in each table. The rate of change
is constant for the data in each table. Write the rate as a unit rate.
1.
Number of Balloons
Total Cost of Balloons
(in Dollars)
2 6
4 12
6 18
8 24
Sample answer:
Number of balloons 5 4 2 2
5 2
Cost of balloons 5 12 2 6
5 6
Rate of change 5 6 dollars __________ 2 balloons
Unit rate 5 3 dollars _________ 1 balloon
Lesson 3.2 Skills Practice
Name ________________________________________________________ Date _________________________
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Lesson 3.2 Skills Practice page 2
2.Number of
LawnsTotal Earned (in Dollars)
3 25.50
5 42.50
7 59.50
9 76.50
Sample answer:
Number of lawns 5 5 2 3
5 2
Total earned 5 42.50 2 25.50
5 17
Rate of change 5 17 dollars __________ 2 lawns
Unit rate 5 8.5 dollars __________ 1 lawn
3.Number of
TouchdownsTotal Points
Scored
2 12
3 18
4 24
5 30
Sample answer:
Number of touchdowns 5 3 2 2
5 1
Total points scored 5 18 2 12
5 6
Rate of change 5 6 points _____________
1 touchdown
Unit rate 5 6 points _____________
1 touchdown
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Lesson 3.2 Skills Practice page 3
Name ________________________________________________________ Date _________________________
4.Number of
Minutes on an Exercise Bike
Total Number of Calories Burned
15 180
30 360
45 540
60 720
Sample answer:
Number of minutes 5 30 2 15
5 15
Number of calories 5 360 2 180
5 180
Rate of change 5 180 calories ____________ 15 minutes
Unit rate 5 12 calories ___________ 1 minute
5.
Number of HoursTotal Number
of Miles Traveled
2 130
5 325
8 520
11 715
Sample answer:
Number of hours 5 5 2 2
5 3
Number of miles 5 325 2 130
5 195
Rate of change 5 195 miles _________ 3 hours
Unit rate 5 65 miles ________ 1 hour
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Lesson 3.2 Skills Practice page 4
6.Number of Songs
Downloaded
Total Cost of Songs
(in Dollars)
10 9.90
20 19.80
30 29.70
40 39.60
Sample answer:
Number of songs 5 20 2 10
5 10
Cost of songs 5 19.80 2 9.90
5 9.9
Rate of change 5 $9.90 _________ 10 songs
Unit rate 5 $0.99 _______ 1 song
Use the formula y2 2 y1 _______ x2 2 x1
to calculate the unit rate of change for the data in each table. The rate of
change is constant for the data in each table.
7.Number of
Raffle Tickets
Total Cost of Raffle Tickets
(in Dollars)
2 1
4 2
8 4
10 5
Sample answer:
( x1, y1 ) 5 (2, 1)
( x2, y2 ) 5 (4, 2)
y2 2 y1 ________
x2 2 x1
5 2 2 1 ______ 4 2 2
5 1 __ 2
5 0.5 ___ 1
Unit rate 5 $0.50 _______ 1 ticket
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Lesson 3.2 Skills Practice page 5
Name ________________________________________________________ Date _________________________
8.x y
22 8
0 0
2 28
4 216
Sample answer:
( x1, y1 ) 5 (22, 8)
( x2, y2 ) 5 (2, 28)
y2 2 y1 ________
x2 2 x1
5 28 2 8 ________ 2 2 (22)
5 216 _____ 4
5 24 ___ 1
Unit rate 5 24 ___ 1
9.
Number of Photos Printed
Total Cost of Photos
(in Dollars)
10 2
20 4
30 6
40 8
Sample answer:
( x1, y1 ) 5 (10, 2)
( x2, y2 ) 5 (20, 4)
y2 2 y1 ________
x2 2 x1
5 4 2 2 ________ 20 2 10
5 2 ___ 10
5 0.2 ___ 1
Unit rate 5 $0.20 ________ 1 photo
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Lesson 3.2 Skills Practice page 6
10.x y
3 27
5 45
7 63
9 81
Sample answer:
( x1, y1 ) 5 (3, 27)
( x2, y2 ) 5 (5, 45)
y2 2 y1 ________
x2 2 x1
5 45 2 27 ________ 5 2 3
5 18 ___ 2
5 9 __ 1
Unit rate 5 9 __ 1
11.
Number of Greeting Cards
Total Cost of Greeting Cards
(in Dollars)
2 6.50
3 9.75
6 19.50
8 26.00
Sample answer:
( x1, y1 ) 5 (2, 6.50)
( x2, y2 ) 5 (3, 9.75)
y2 2 y1 ________
x2 2 x1
5 9.75 2 6.50 ___________ 3 2 2
5 3.25 _____ 1
Unit rate 5 $3.25 ______ 1 card
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Lesson 3.2 Skills Practice page 7
Name ________________________________________________________ Date _________________________
12.x y
23 54
1 218
3 254
7 2126
Sample answer:
( x1, y1 ) 5 (23, 54)
( x2, y2 ) 5 (1, 218)
y2 2 y1 ________
x2 2 x1
5 218 2 54 __________ 1 2 (23)
5 272 _____ 4
5 218 _____ 1
Unit rate 5 218 _____ 1
Calculate the rate of change between the points listed in each table. Determine if the table represents a
linear relationship.
13.x y
2 14
5 35
7 49
10 70
Yes, the table represents a linear
relationship.
(2, 14) and (5, 35) (5, 35) and (7, 49)
35 2 14 ________ 5 2 2
5 21 ___ 3
49 2 35 ________ 7 2 5
5 14 ___ 2
5 7 __ 1
5 7 __ 1
(7, 49) and (10, 70)
70 2 49 ________ 10 2 7
5 21 ___ 3
5 7 __ 1
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Lesson 3.2 Skills Practice page 8
14.x y
210 50
22 10
4 220
14 270
Yes, the table represents a linear
relationship.
(210, 50) and (22, 10) (22, 10) and (4, 220)
10 2 50 ___________ 22 2 (210)
5 240 _____ 8
220 2 10 __________ 4 2 (22)
5 230 _____ 6
5 25 ___ 1
5 25 ___ 1
(4, 220) and (14, 270)
270 2 (220)
____________ 14 2 4
5 250 _____ 10
5 25 ___ 1
15.x y
21 224
2 48
4 90
8 192
No, the table does not represent a linear
relationship.
(21, 224) and (2, 48) (2, 48) and (4, 90)
48 2 (224)
___________ 2 2 (21)
5 72 ___ 3
90 2 48 ________ 4 2 2
5 42 ___ 2
5 24 ___ 1
5 21 ___ 1
(4, 90) and (8, 192)
192 2 90 _________ 8 2 4
5 102 ____ 4
5 25.5 _____ 1
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Lesson 3.2 Skills Practice page 9
Name ________________________________________________________ Date _________________________
16.x y
26 12
23 6
3 26
6 210
No, the table does not represent a linear
relationship.
(26, 12) and (23, 6) (23, 6) and (3, 26)
6 2 12 __________ 23 2 (26)
5 26 ___ 3
26 2 6 ________ 3 2 (23)
5 212 _____ 6
5 22 ___ 1
5 22 ___ 1
(3, 26) and (6, 210)
210 2 (26)
___________ 6 2 3
5 24 ___ 3
17.x y
2 13.5
5 33.75
10 67.5
15 101.25
Yes, the table represents a linear
relationship.
(2, 13.5) and (5, 33.75)
33.75 2 13.5 ____________ 5 2 2
5 20.25 ______ 3
5 6.75 _____ 1
(5, 33.75) and (10, 67.5)
67.5 2 33.75 ____________ 10 2 5
5 33.75 ______ 5
5 6.75 _____ 1
(10, 67.5) and (15, 101.25)
101.25 2 67.5 _____________ 15 2 10
5 33.75 ______ 5
5 6.75 _____ 1
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Lesson 3.2 Skills Practice page 10
18.x y
24 238
21 29.5
2 19
3 27
No, the table does not represent a linear
relationship.
(24, 238) and (21, 29.5)
29.5 2 (238)
_____________ 21 2 (24)
5 28.5 _____ 3
5 9.5 ___ 1
(21, 29.5) and (2, 19)
19 2 (29.5)
___________ 2 2 (21)
5 28.5 _____ 3
5 9.5 ___ 1
(2, 19) and (3, 27)
27 2 19 ________ 3 22
5 8 __ 1
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To Put It in ContextDetermining Rate of Change from a Context
Problem SetDetermine the rate of change for each situation.
1. Lashawna is making jewelry to sell at a craft fair. On Monday, she makes 12 bracelets. On Tuesday,
she works an additional 2.5 hours and has a total of 22 bracelets. What is the unit rate of the time
it takes her to make each bracelet?
22 bracelets 2 12 bracelets 5 10 bracelets made on Tuesday
2.5 hours ____________ 10 bracelets
5 0.25 hour __________ 1 bracelet
The unit rate is 0.25 hour __________ 1 bracelet
or 15 minutes per bracelet.
2. Nina and her friends are going to the downtown rib festival. The festival organizers expect 10,000
people to attend the four-day festival. At the end of the festival the organizers say that they have
exceeded their expected attendance by 2000 people. What was the average number of people to
attend the festival per day?
10,000 people 1 2000 people 5 12,000 people
12,000 people
______________ 4 days
5 3000 people
____________ 1 day
The unit rate is 3000 people
____________ 1 day
.
Lesson 3.3 Skills Practice
Name ________________________________________________________ Date _________________________
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Lesson 3.3 Skills Practice page 2
3. Rosa is ordering a submarine sandwich from the corner deli. The deli charges $6.25 for a 7-inch
sub. Some additional toppings cost extra. Rosa’s sandwich with two extra toppings costs $7.75.
What is the cost per additional topping?
$7.75 2 $6.25 5 $1.50 for extra toppings
$1.50 __________ 2 toppings
5 $0.75 _________ 1 topping
The unit rate is $0.75 _________ 1 topping
.
4. Aiko spends 2.5 hours baking croissants for a community center bake sale. Aiko bakes the
90 croissants in 5 batches. What is the unit rate of the number of batches baked per hour?
5 batches __________ 2.5 hours
5 2 batches __________ 1 hour
The unit rate is 2 batches __________ 1 hour
.
5. Nelson is selling his photographs at an art festival. The festival is open for 6 hours each day for
3 days. At the conclusion of the festival, Nelson has sold 54 photographs. What is the unit rate of
the number of photographs sold per hour?
3 days 3 6 hours per day 5 18 hours total
54 photographs
_______________ 18 hours
5 3 photographs
______________ 1 hour
The unit rate is 3 photographs
______________ 1 hour
.
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Lesson 3.3 Skills Practice page 3
Name ________________________________________________________ Date _________________________
6. Clayton wants to purchase tickets for the rides at a carnival. He can choose to purchase tickets
individually or he can purchase a ticket package. The package includes 25 tickets for $18.75.
What is the cost per ticket if he purchases the package?
Package 5 $18.75 __________ 25 tickets
5 $0.75 ________ 1 ticket
The unit rate is $0.75 _______ 1 ticket
.
7. Carmen is selling pies at the cherry festival to raise money for her local volunteer fire department.
She sells 85 pies for $12 each. The supplies to make the pies cost Carmen $340. What is the unit
rate of the profit made for each pie?
85 pies 3 $12 per pie 5 $1020 earned
$1020 2 $340 5 $680 profit
$680 _______ 85 pies
5 $8 ______ 1 pie
The unit rate is $8 ______ 1 pie
.
8. Tameca is planning a hiking trip. The trail she would like to follow is 7.5 miles long. She plans to
start her hike at 10:00 am. She hopes to reach the end of the trail at 3:00 pm. What is the unit rate
of the number of miles per hour that Tameca plans to hike?
10:00 am to 3:00 pm is 5 hours.
7.5 miles _________ 5 hours
5 1.5 miles _________ 1 hour
The unit rate is 1.5 miles _________ 1 hour
.
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Lesson 3.3 Skills Practice page 4
9. Jamal is shopping with a gift card he received for his birthday. After he purchases two T-shirts the
gift card balance has dropped from $50 to $20.02. What is the unit rate of the cost per T-shirt?
$50 2 $20.02 5 $29.98 spent
$29.98 _________ 2 T-shirts
5 $14.99 ________ 1 T-shirt
The unit rate is $14.99 ________ 1 T-shirt
.
10. Olivia is printing photos for a scrapbook project. She prints 150 photos in 1 hour and 15 minutes.
What is the unit rate of the number of minutes it takes to print each photo?
1 hour 5 60 minutes
60 minutes 1 15 minutes 5 75 minutes
75 minutes ___________ 150 photos
5 0.5 minute __________ 1 photo
The unit rate is 0.5 minute __________ 1 photo
.
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Lesson 3.3 Skills Practice page 5
Name ________________________________________________________ Date _________________________
11. Franco is traveling to a vacation destination 715 miles from home. On the first day of his trip he
travels 390 miles in 6 hours. On the second day of his trip he leaves at 8:00 am and arrives at his
destination at 1:00 pm. What is the unit rate of the total number of miles per hour traveled? Convert
the information to coordinate points and use the formula y2 2 y1 _______ x2 2 x1
to answer the question.
8:00 am to 1:00 pm is 5 hours
6 hours 1 5 hours 5 11 hours
(x1, y1) 5 (6, 390)
(x2, y2) 5 (11, 715)
y2 2 y1 _______ x2 2 x1
5 715 2 390 __________ 11 2 6
5 325 miles _________ 5 hours
5 65 miles ________ 1 hour
The unit rate is 65 miles ________ 1 hour
.
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Lesson 3.3 Skills Practice page 6
12. Rakesha loves reading and is participating in a read-a-thon to raise money for a charity. She plans
to read 15 books during the 90-day read-a-thon. During the first 30 days she reads 7 books. What
is the unit rate of the number of days she has to read each book to meet her goal? Convert the
information to coordinate points and use the formula y2 2 y1 _______ x2 2 x1
to answer the question.
(x1, y1) 5 (7, 30)
(x2, y2) 5 (15, 90)
y2 2 y1 _______ x2 2 x1
5 90 2 30 ________ 15 2 7
5 60 days
________ 8 books
5 7.5 days
________ 1 book
The unit rate is 7.5 days
________ 1 book
.
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All Together Now!Determining Rate of Change from an Equation
VocabularyGive an example of each term.
1. slope
Sample answer: The slope of the line represented by the equation y 5 3x 1 4 is 3.
2. slope-intercept form
Sample answer: The equation y 5 3x 1 4 is written in slope-intercept form.
Problem SetDetermine the slope of the line represented by each equation.
Lesson 3.4 Skills Practice
Name ________________________________________________________ Date _________________________
1. y 5 4x 1 5 1 2x
y 5 4x 1 5 1 2x
y 5 6x 1 5
slope 5 6
3. 2x 1 3y 5 21
2x 1 3y 5 21
3y 5 22x 1 21
y 5 22 __ 3 x 1 7
slope 5 22 __ 3
2. y 5 3x 1 8 2 12x
y 5 3x 1 8 2 12x
y 5 29x 1 8
slope 5 29
4. 8y 2 2x 5 24
8y 2 2x 5 24
8y 5 2x 1 24
y 5 2 __ 8
x 1 3
y 5 1 __ 4
x 1 3
slope 5 1 __ 4
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Lesson 3.4 Skills Practice page 2
5. y 5 8
y 5 8
y 5 0x 1 8
slope 5 0
7. y 5 25(2x 2 3)
y 5 25(2x 2 3)
y 5 210x 1 15
slope 5 210
9. x 5 1 __ 2
This equation represents a vertical line
at x 5 1 __ 2
. The slope is undefined.
6. y 5 6x 2 2 1 x
y 5 6x 2 2 1 x
y 5 7x 2 2
slope 5 7
8. 10y 2 6x 5 290
10y 2 6x 5 290
10y 5 6x 2 90
y 5 6 ___ 10
x 2 9
y 5 3 __ 5
x 2 9
slope 5 3 __ 5
10. 12y 2 4x 5 224
12y 2 4x 5 224
12y 5 4x 2 24
y 5 4 ___ 12
x 2 2
y 5 1 __ 3
x 2 2
slope 5 1 __ 3
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Lesson 3.4 Skills Practice page 3
Name ________________________________________________________ Date _________________________
Graph each equation with your graphing calculator and sketch its graph on the given grid.
Determine the slope of the line.
11. y 5 4x 1 2
slope 5 4
x
y
–16
–12
–8
–4
4
8
12
16
–2–4 2 4 6–6–8 8
12. y 5 1 __ 2
x 1 1
slope 5 1 __ 2
x
y
–8
–6
–4
–2
2
4
6
8
–2–4 2 4 6–6–8 8
13. y 5 21 __ 3 x 2 5
slope 5 21 __ 3
x
y
–8
–6
–4
–2
2
4
6
8
–2–4 2 4 6–6–8 8
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Lesson 3.4 Skills Practice page 4
14. y 5 22x 2 3
slope 5 22
x
y
–8
–6
–4
–2
2
4
6
8
–2–4 2 4 6–6–8 8
15. y 5 3 __ 4
x 2 7
slope 5 3 __ 4
x
y
–8
–6
–4
–2
2
4
6
8
–2–4 2 4 6–6–8 8
16. y 5 22 __ 3 x 1 8
slope 5 22 __ 3
x
y
–8
–6
–4
–2
2
4
6
8
–2–4 2 4 6–6–8 8
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Lesson 3.5 Skills Practice
Name ________________________________________________________ Date _________________________
Where it CrossesDetermining y-Intercepts from Various Representations
VocabularyDefine each term in your own words.
1. y-intercept
the y-coordinate of the point where a graph crosses the y-axis, written in the form (0, y).
2. direct variation
the relationship between two quantities x and y having a constant ratio such that when
one increases or decreases by a specific amount the other value increases or decreases
by a constant amount.
Problem SetExamine each linear graph and determine the y-intercept. Write the y-intercept in coordinate form.
Show your work.
1.
x109876543210
y
10
9
8
7
6
5
4
3
2
1
(3, 9)
(4, 7)
Since the graph is a line, it will keep
increasing by 2 for every point going
backward.
The next points to the left would be
(2, 11), then (1, 13), then (0, 15).
The y-intercept is (0, 15).
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Lesson 3.5 Skills Practice page 2
2.
x543210
y
5
4
3
2
1
(2, 5)
(4, 4)
Since the graph is a line, it will keep
increasing by 1 for every 2 points going
backward.
The next point to the left would be (0, 6).
The y-intercept is (0, 6).
3.
x100
y
10
10987654321
10
9
8
7
6
5
4
3
2
1
(3, 8)
(2, 6)
Since the graph is a line, it will keep
decreasing by 2 for every point going
backward.
The next points to the left would be (1, 4),
and then (0, 2).
The y-intercept is (0, 2).
4.
x20181614121086420
y
20
18
16
14
12
10
8
6
4
2
(4, 14)
(2, 2)
Since the graph is a line, it will keep
decreasing by 12 for every 2 points
going backward.
The next point to the left would be (0, 210).
The y-intercept is (0, 210).
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Lesson 3.5 Skills Practice page 3
Name ________________________________________________________ Date _________________________
5.
x100
y
10
987654321
9
8
7
6
5
4
3
2
1
(4, 5)
(3, 10)
Since the graph is a line, it will keep
increasing by 5 for every point going
backward.
The next points to the left would be
(2, 15), then (1, 20), then (0, 25).
The y-intercept is (0, 25).
6.
x543210
y
5
4
3
2
1
(3, 5)
(2, 2)
Since the graph is a line, it will keep
decreasing by 3 for every point going
backward.
The next points to the left would
be (1, 21), and then (0, 24).
The y-intercept is (0, 24).
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Lesson 3.5 Skills Practice page 4
Determine the y-intercept for the linear relation represented in each table. Write the y-intercept as a
coordinate pair.
7.x y
6 25
9 34
12 43
15 52
Counting backward in the table the x-values would
be 3, 0.
Skip counting backward in the table the y-values
would be 16, 7.
The y-intercept is (0, 7).
8.x y
4 12
6 13
8 14
10 15
Counting backward in the table the x-values would
be 2, 0.
Skip counting backward in the table the y-values
would be 11, 10.
The y-intercept is (0, 10).
9. x y
10 238
15 258
20 278
25 298
Counting backward in the table the x-values would
be 5, 0.
Skip counting backward in the table the y-values
would be 218, 2.
The y-intercept is (0, 2).
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Lesson 3.5 Skills Practice page 5
Name ________________________________________________________ Date _________________________
10.x y
8 29
12 212
16 215
20 218
Counting backward in the table the x-values would
be 4, 0.
Skip counting backward in the table the y-values
would be 26, 23.
The y-intercept is (0, 23).
11. x y
6 35
9 53
12 71
15 89
Counting backward in the table the x-values would
be 3, 0.
Skip counting backward in the table the y-values
would be 17, 21.
The y-intercept is (0, 21).
12. x y
10 295
15 2145
20 2195
25 2245
Counting backward in the table the x-values would
be 5, 0.
Skip counting backward in the table the y-values
would be 245, 5.
The y-intercept is (0, 5).
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Lesson 3.5 Skills Practice page 6
Determine the y-intercept for the linear relation represented in each equation. Write the
y-intercept as a coordinate pair.
13. 3x 1 7y 5 63
3x 1 7y 5 63
3(0) 1 7y 5 63
7y 5 63
y 5 9
The y-intercept is (0, 9).
14. 5x 2 4y 5 120
5x 2 4y 5 120
5(0) 2 4y 5 120
24y 5 120
y 5 230
The y-intercept is (0, 230).
15. 2x 2 8y 5 248
2x 2 8y 5 248
2(0) 2 8y 5 248
28y 5 248
y 5 6
The y-intercept is (0, 6).
16. 20x 1 15y 5 105
20x 1 15y 5 105
20(0) 1 15y 5 105
15y 5 105
y 5 7
The y-intercept is (0, 7).
17. 216x 1 23y 5 253
216x 1 23y 5 253
216(0) 1 23y 5 253
23y 5 253
y 5 11
The y-intercept is (0, 11).
18. 19x 1 17y 5 251
19x 1 17y 5 251
19(0) 1 17y 5 251
17y 5 251
y 5 23
The y-intercept is (0, 23).
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Lesson 3.5 Skills Practice page 7
Name ________________________________________________________ Date _________________________
Determine the y-intercept for the linear relation represented in each context. Write the y-intercept as a
coordinate pair. Explain what the y-intercept represents in the problem situation.
19.Carmen estimated that she would pay $19 to park in a downtown parking garage for a 3-hour
event. After spending 5 hours downtown she paid $25 for parking.
The difference between the prices is $6. The difference between the times parked is 2 hours.
The rate paid is $3 per hour. The estimate for 3 hours of parking is $19, and $9 is due to the per
hour rate. So, $10 must be an initial flat fee. The y-intercept is (0, 10).
20.Pedro is traveling on a toll road. He plans to exit the road 5 miles ahead at First Avenue and pay
$1.75. He changes his plans and travels 9 miles to Butler Street and pays $2.75.
The difference in the tolls is $1.00. The difference in the miles traveled is 4 miles.
The rate paid is $0.25 per mile. The toll for traveling 5 miles is $1.75, and $1.25 is due to the per
mile rate. So, $0.50 must be an initial flat fee. The y-intercept is (0, 0.50).
21.Alberto is saving for a new video game. After adding 2 weeks of his allowance to a savings
account he has $105. After adding 3 more weeks of his allowance to his savings he has $150.
Three weeks of allowance is $45 ($150 2 $105). Alberto’s allowance each week is $15.
Counting backward by $15 per week from $105 for the first two weeks of allowance added
is $75. The y-intercept is (0, 75). $75 is the initial balance of Alberto’s savings account before
adding his allowance.
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Lesson 3.5 Skills Practice page 8
22. Noah is renewing a magazine subscription. One package offers to renew the magazine for 3 years
for $26. A second package offers to renew the magazine for 5 years for $38.
The difference in the cost is $12. The difference in the length of the subscription is 2 years.
The rate paid is $6 per year. The offer to renew for 3 years is $26, and $18 is due to the per year
rate. So, $8 must be an initial flat fee. The y-intercept is (0, 8).
23. Serena received a gift card to the local movie theater. After going to 2 movies, the balance
of her gift card dropped to $64. After going to 3 more movies, the balance of her gift card
dropped to $40.
Three movies cost $24 ($64 2 $40). The cost is $8 per movie. Counting forward by $8 per
movie from $64 for the first two movies is $80. The y-intercept is (0, 80). $80 is the initial value
of Serena’s gift card.
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Slope-Intercept FormDetermining the Rate of Change and y-Intercept
VocabularyMatch each definition to its corresponding term.
1. Ax 1 By 5 C, where A, B, and C are constants
and A and B are not both zero.
b. standard form
a. point-slope form
2. m(x 2 x1) 5 (y 2 y1), a linear equation that passes
through the point (x1, y1) and has slope m
a. point-slope form
b. standard form
Problem SetSketch the graph of each line.
1. y 5 3x 1 2
y
x
2. y 5 2 2 __ 3
x 1 5
y
x
Lesson 3.6 Skills Practice
Name ________________________________________________________ Date _________________________
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Lesson 3.6 Skills Practice page 2
3. y 5 1 __ 2
x 2 6
y
x
4. y 524x 2 3
y
x
5. y 5 2 3 __ 4
x 2 4
y
x
6. y 5 3 __ 8
x 1 1
y
x
Determine the y-intercept of each line given the slope and a point that lies on the line.
7. m 5 4 and (2, 13)
y 5 mx 1 b
13 5 4(2) 1 b
13 5 8 1 b
5 5 b
8. m 5 3 __ 2
and (4, 26)
y 5 mx 1 b
26 5 3 __ 2 (4) 1 b
26 5 6 1 b
212 5 b
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Lesson 3.6 Skills Practice page 3
Name ________________________________________________________ Date _________________________
9. m 5 2 2 __ 5
and (5, 12)
y 5 mx 1 b
12 5 2 2 __ 5
(5) 1 b
12 5 22 1 b
14 5 b
10. m 5 2 1 __ 4
and (3, 8)
y 5 mx 1 b
8 5 2 1 __ 4
(3) 1 b
8 5 2 3 __ 4
1 b
8 3 __ 4 5 b
11. m 5 7 and ( 2, 13 1 __ 2 )
y 5 mx 1 b
13 1 __ 2
5 7(2) 1 b
13 1 __ 2
5 14 1 b
2 1 __ 2
5 b
12. m 5 3 __ 4
and (8, 21)
y 5 mx 1 b
21 5 3 __ 4
(8) 1 b
21 5 6 1 b
15 5 b
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Lesson 3.6 Skills Practice page 4
Write the equation of each line given two points that lie on the line.
13. (3, 25) and (4, 31)
Calculate the slope, m.
m 5 y2 2 y1 _______ x2 2 x1
5 31 2 25 ________ 4 2 3
5 6 __ 1
Substitute m and b.
y 5 mx 1 b
y 5 6x 1 7
Calculate the y-intercept, b.
y 5 mx 1 b
25 5 6(3) 1 b
25 5 18 1 b
7 5 b
14. (10, 22) and (15, 24)
Calculate the slope, m.
m 5 y2 2 y1 _______ x2 2 x1
5 24 2 22 ________ 15 2 10
5 2 __ 5
Substitute m and b.
y 5 mx 1 b
y 5 2 __ 5
x 1 18
Calculate the y-intercept, b.
y 5 mx 1 b
22 5 2 __ 5
(10) 1 b
22 5 4 1 b
18 5 b
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Lesson 3.6 Skills Practice page 5
Name ________________________________________________________ Date _________________________
15. (6, 1) and (18, 21)
Calculate the slope, m.
m 5 y2 2 y1 _______ x2 2 x1
5 21 2 1 _______ 18 2 6
5 22 ___ 12
5 2 1 __ 6
Substitute m and b.
y 5 mx 1 b
y 5 2 1 __ 6
x 1 2
Calculate the y-intercept, b.
y 5 mx 1 b
1 5 2 1 __ 6
(6) 1 b
1 5 21 1 b
2 5 b
16. ( 2, 15 3 __ 4
) and ( 5, 39 3 __ 4
)
Calculate the slope, m.
m 5 y2 2 y1 _______ x2 2 x1
5 39 3 __
4 2 15 3 __
4 __________
5 2 2
5 24 ___ 3
5 8
Substitute m and b.
y 5 mx 1 b
y 5 8x 2 1 __ 4
Calculate the y-intercept, b.
y 5 mx 1 b
15 3 __ 4 5 8(2) 1 b
15 3 __ 4 5 16 1 b
2 1 __ 4
5 b
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Lesson 3.6 Skills Practice page 6
17. (7, 1) and (21, 11)
Calculate the slope, m.
m 5 y2 2 y1 _______ x2 2 x1
5 11 2 1 _______ 21 2 7
5 10 ___ 14
5 5 __ 7
Substitute m and b.
y 5 mx 1 b
y 5 5 __ 7
x 2 4
Calculate the y-intercept, b.
y 5 mx 1 b
1 5 5 __ 7
(7) 1 b
1 5 5 1 b
24 5 b
18. (2, 27) and (4, 210)
Calculate the slope, m.
m 5 y2 2 y1 _______ x2 2 x1
5 210 2 (27)
___________ 4 2 2
5 23 ___ 2
Substitute m and b.
y 5 mx 1 b
y 5 2 3 __ 2
x 2 4
Calculate the y-intercept, b.
y 5 mx 1 b
27 5 2 3 __ 2 (2) 1 b
27 5 23 1 b
24 5 b
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Lesson 3.6 Skills Practice page 7
Name ________________________________________________________ Date _________________________
Write the equation of each line given the slope and a point that lies on the line. Write the equation in
slope-intercept form.
19. slope 5 9 and (2, 21)
m(x 2 x1) 5 (y 2 y1)
9(x 2 2) 5 (y 2 21)
9x 2 18 5 y 2 21
9x 1 3 5 y
y 5 9x 1 3
20. slope 5 22 and (4, 7)
m(x 2 x1) 5 (y 2 y1)
22(x 2 4) 5 (y 2 7)
22x 1 8 5 y 2 7
22x 1 15 5 y
y 5 22x 1 15
21. slope 5 2 __ 5
and (10, 13)
m(x 2 x1) 5 (y 2 y1)
2 __ 5
(x 2 10) 5 (y 2 13)
2 __ 5
x 2 4 5 y 2 13
2 __ 5
x 1 9 5 y
y 5 2 __ 5
x 1 9
22. slope 5 2 4 __ 3
and (6, 216)
m(x 2 x1) 5 (y 2 y1)
2 4 __ 3
(x 2 6) 5 (y 2 (216))
2 4 __ 3
x 1 8 5 y 1 16
2 4 __ 3 x 2 8 5 y
y 5 2 4 __ 3
x 2 8
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Lesson 3.6 Skills Practice page 8
23. slope 5 1 __ 3
and (9, 212)
m(x 2 x1) 5 (y 2 y1)
1 __ 3 (x 2 9) 5 (y 2 (212))
1 __ 3 x 2 3 5 y 1 12
1 __ 3
x 2 15 5 y
y 5 1 __ 3 x 2 15
24. slope 5 2 5 ___ 12
and (24, 26)
m(x 2x1) 5 (y 2 y1)
2 5 ___ 12
(x 2 24) 5 (y 2 (26))
2 5 ___ 12
x 1 10 5 y 1 6
2 5 ___ 12
x 1 4 5 y
y 5 2 5 ___ 12
x 1 4
Calculate the y-intercept and x-intercept for each linear equation in standard form. Sketch the graph
of the line.
25. 4x 1 6y 5 48
4x 16y 5 48
4(0) 1 6y 5 48
6y 5 48
y 5 8
The y-intercept is (0, 8).
4x 1 6y 5 48
4x 1 6(0) 5 48
4x 5 48
x 5 12
The x-intercept is (12, 0).
x
y
–16
–12
–8
–4
4
8
12
16
–4–8 4 8 12–12–16 16
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Lesson 3.6 Skills Practice page 9
Name ________________________________________________________ Date _________________________
26. 3x 115y 5 135
3x 1 15y 5 135
3(0) 1 15y 5 135
15y 5 135
y 5 9
The y-intercept is (0, 9).
3x 1 15y 5 135
3x 1 15(0) 5 135
3x 5 135
x 5 45
The x-intercept is (45, 0).
x
y
–8
–6
–4
–2
2
4
6
8
–10–20 10 20 30–30–40 40
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Lesson 3.6 Skills Practice page 10
27. 22x 1 8y 5 56
22x 1 8y 5 56
22(0) 1 8y 5 56
8y 5 56
y 5 7
The y-intercept is (0, 7).
22x 1 8y 5 56
22x 1 8(0) 5 56
22x 5 56
x 5 228
The x-intercept is (228, 0).
x
y
–8
–6
–4
–2
2
4
6
8
–8–16 8 16 24–24–32 32
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Lesson 3.6 Skills Practice page 11
Name ________________________________________________________ Date _________________________
28. 5x 16y 5 290
5x 1 6y 5 290
5(0) 1 6y 5 290
6y 5 290
y 5 215
The y-intercept is (0, 215).
5x 1 6y 5 290
5x 1 6(0) 5 290
5x 5 290
x 5 218
The x-intercept is (218, 0).
x
y
–16
–12
–8
–4
4
8
12
16
–4–8 4 8 12–12–16 16
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Lesson 3.6 Skills Practice page 12
29. 2x 1 8y 5 224
2x 1 8y 5 224
2(0) 1 8y 5 224
8y 5 224
y 5 23
The y-intercept is (0, 23).
2x 1 8y 5 224
2x 1 8(0) 5 224
2x 5 224
x 5 212
The x-intercept is (212, 0).
x
y
–16
–12
–8
–4
4
8
12
16
–4–8 4 8 12–12–16 16
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Lesson 3.6 Skills Practice page 13
Name ________________________________________________________ Date _________________________
30. 7x 2 3y 5 242
7x 2 3y 5 242
7(0) 2 3y 5 242
23y 5 242
y 5 14
The y-intercept is (0, 14).
7x 2 3y 5 242
7x 2 3(0) 5 242
7x 5 242
x 5 26
The x-intercept is (26, 0).
x
y
–16
–12
–8
–4
4
8
12
16
–4–8 4 8 12–12–16 16
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