FLORIDA
CHAPTER
8
Linear Functions
Name
Class
Date
Lesson
Worktext
Student
Textbook
MA.8.A.1.6
8-1 Graphing Linear Equations
269 – 276
346 – 350
MA.8.A.1.2
8-2 Slope of a Line
277 – 284
351 – 355
MA.8.A.1.2
8-3 Using Slopes and Intercepts
285 – 294
356 – 360
MA.8.A.1.2
8-4 Linear Functions
295 – 302
361 – 364
MA.8.A.1.1
8-5 Point-Slope Form
305 – 312
365 – 368
MA.8.A.1.1
8-6 Direct Variation
313 – 320
369 – 373
MA.8.A.1.4
8-7 Using Graphs and Tables to
Solve Linear Systems
321 – 328
376 – 380
MA.8.A.1.3
8-8 Using Algebra to Solve Linear
Systems
329 – 336
381 – 385
Study It!
339 – 341
Write About It!
342
Chapter 8 Linear Functions 267
CHAPTER
Copyright © by Holt McDougal. All rights reserved.
Benchmark
8
Chapter at a Glance
Vocabulary Connections
LA.8.1.6.5 The student will relate new vocabulary to familiar words.
Key Vocabulary
Vocabulario
Vokabilè
constant of variation
constante de variación
constant
direct variation
variación directa
varyasyon dirèk
linear equation
ecuación lineal
ekwasyon lineye
slope
pendiente
pant
slope-intercept form
forma de pendienteintersección
solucion de un sistema
de ecuaciones
sistema de ecuaciones
fòm kanonik
intersección con el
eje x
intersección con el
eje y
absis a lorijin
solution of a system of
equations
system of equations
x-intercept
CHAPTER
y-intercept
solisyon yon system
ekwasyon
systèm ekwasyon
òdone a lorijin
To become familiar with some of the vocabulary terms in the chapter, consider the
following. You may refer to the chapter, the glossary, or a dictionary if you like.
1. The word linear means “relating to a line.” What do you think the graph of a linear
equation looks like?
3. The adjective direct can mean “passing in a straight line.” What do you suppose the
graph of an equation with direct variation looks like?
4. A system is a group of related objects. What do you think a system of equations is?
268 Chapter 8 Linear Functions
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8
2. The word intercept can mean “to interrupt a course or path.” Where on a graph do
you think you should look to find the y-intercept of a line?
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8-1
Date
MA.8.A.1.6 Compare the graphs of
linear and non-linear functions for realworld situations.
Graphing Linear Equations
Explore Linear and Non-linear Equations
One way to graph equations is to create a table of values, and graph the
corresponding ordered pairs using the x- and y-values in the table.
Activity
1 The equation y = 2x is a linear equation. From the word “linear,” what do
you think the graph of a linear equation looks like?
y
2 Substitute the given values of x to find the corresponding
value of y. Then graph the corresponding ordered pairs in the
coordinate plane at the right.
4
2
x
–2
–1
0
1
x
2
-4
y = 2x
O
-2
4
-2
-4
3 Connect the points, and describe the shape of the graph.
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2
4 The equation y = x2 is a non-linear equation. From the word “non-linear,”
what do you think the graph of a non-linear equation looks like?
y
5 Substitute the given values of x to find the corresponding
values of y. Then graph the corresponding ordered pairs in
the coordinate plane at the right.
x
-2
-1
0
1
8
6
2
4
y = x2
2
x
6 Connect the points, and describe the shape of the graph.
-6
-4
-2
O
2
4
6
-2
8-1 Graphing Linear Equations 269
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Try This
Make a table of values, and graph each equation. Describe the shape of
the graph.
1. y = 2.5x
6
x
-3
-2
-1
0
1
2
y
4
3
2
y = 2.5x
x
2. y = 0.5x
2
-6
-4
O
-2
2
4
6
-2
x
-3
-2
-1
0
1
2
3
-4
y = 0.5x 2
Draw Conclusions
Compare the exponent of x the linear equations with the exponent of x in the
non-linear equations.
3. Make a conjecture: How are the x-terms of linear equations alike?
4. Make a conjecture: How are the x-terms of non-linear equations
different from the x-terms of linear equations?
y = 3x + 5
y = –4x – 4
y = 2x 2 – 4
y = _12 x + 3
y = –x + 2
y = –x 2 + 2
6
4
x
2
y
O
x
-6
-4
-2
-2
-4
-6
270 8-1 Graphing Linear Equations
y
2
4
6
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5. Use your conjecture: Select one of the following equations that you
believe is a linear equation. Circle it. Make a table of values and graph the
equation to support or disprove your conjecture.
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8-1
Date
MA.8.A.1.6 Compare the graphs of
linear and non-linear functions for realworld situations.
Graphing Linear Equations (Student Textbook pp. 346–350)
Lesson Objective
Identify and graph linear equations
Vocabulary
linear equation
rate of change
Example 1
Graph each e
equation and tell whether it is linear.
y
A. y = 3x - 1
Find the differences between consecutive data points.
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+1
+1
4
+1
x
-1
0
1
2
y
-4
-1
2
5
2
x
-4
-2
O
2
4
-2
+3
+3
+3
y = 3x - 1
-4
a linear equation. Its graph is a
Each time x increases by a constant
.
, the change in y is
.
y
B. y = 2
Find the differences between consecutive data points.
+1
+1
+1
x
-1
0
1
2
y
2
2
2
2
+0
y=2
+0
4
2
x
-4
-2
4
-4
a linear equation. Its graph is a
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2
-2
+0
Each time x increases by a constant
O
.
, the change in y is
.
8-1 Graphing Linear Equations 271
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Graph each equation and tell whether it is linear.
G
1b. y = x 2 + 1
1a. y = 2x + 1
x
Practice It!
y
y
6
y
x
y
4
4
2
2
x
x
-4
O
-2
2
4
-4
O
-2
6
2
4
-2
-2
-4
-4
Example 2
Determine whether the rates of change are constant or variable.
A.
x
0
1
3
5
8
y
0
2
6
10
16
B.
Find the
between
consecutive data points.
3
4
6
9
0
2
6
6
3
Find the
between
consecutive data points.
+2
+2
x
0
1
3
5
8
x
1
3
4
6
9
y
0
2
6
10
16
y
0
2
6
6
3
+2
+4
+4
Find each
change in x.
2=2
__
1
1
y
_____ = 2
of change in y to
4=2
__
2
The table shows
The rates of change are
_____ = 2
Find each
to change in x.
_____ = 1
2
.
272 8-1 Graphing Linear Equations
of change in y
_____ =
1
The table shows
data.
-3
The rates of change are
data.
.
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+1
x
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2a.
x
0
1
y
1
3
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D
Determine
whether the rates of change are constant or
vvariable.
2b. x
2
3
4
0
-2 -1
5
7
9
y
-3
0
-4
1
2
-3
0
Example 3
A lift on a ski
ki slope rises according to the equation
a = 130t + 6250, where a is the altitude in feet and t is the
number of minutes that a skier has been on the lift. Five
friends are on the lift. What is the altitude of each person
if they have been on the ski lift for the times listed in the
table? Draw a graph that represents the relationship
between the time on the lift and the altitude.
a = 130(
) + 6250
(
)
3 min
a = 130(
) + 6250
(
)
2 min
a = 130(
) + 6250
(
)
1.5 min
a = 130(
1 min
a = 130(
The approximate altitudes are Anna,
Kwani,
ft; Tony,
(
) + 6250
4 minutes
Tracy
3 minutes
Kwani
2 minutes
Tony
1.5 minutes
George
1 minute
7000
4 min
) + 6250
Anna
(t, a)
)
(
ft; Tracy
ft; and George,
6800
6600
6400
6200
)
0
0 1 2 3 4 5
Time (min)
ft;
ft. This is a
equation because when t increases by 1 unit, a increases by
units.
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Ah
home iimprovement store charges a base fee of $150 ,
plus $25 for each hour of machinery rental. The cost C
for h hours is given by C = 25h + 150. Find the cost for
1, 2, 3, 4, and 5 hours. Is this a linear equation? Draw a
graph that represents the relationship between the
cost and the number of hours of rental.
400
Rental Cost ($)
Copyright © by Holt McDougal. All rights reserved.
a
Time on Lift
Altitude (ft)
a = 130t + 6250
t
Skier
300
200
100
1
2
3
4
5
Time (h)
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8-1 Graphing Linear Equations 273
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8-1
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Date
LA.8.2.2.3 The student will organize
information to show understanding or
relationships…
Graphing Linear Equations
Think and Discuss
1. Explain whether an equation is linear if three ordered-pair solutions lie
on a straight line but a fourth does not.
2. Compare the equations y = 3x + 2 and y = 3x2 + 2x. Without graphing,
explain why one of the equations is not linear.
3. Describe why neither number in the ordered pair can be negative in
Example 3.
Definition
Examples
274 8-1 Graphing Linear Equations
Property
Linear
Equation
Nonexample
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4. Get Organized Complete the graphic organizer. Write the definition of a
linear equation and a property of linear equations. Then give examples and
nonexamples of linear equations.
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8-1
Date
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MA.8.A.1.6 Compare the graphs of
linear and non-linear functions for
real-world situations.
Graphing Linear Equations
Graph each equation and tell whether it is linear.
y
1. y = -3x - 4
2
-3x - 4
x
y
(x, y)
x
-4
-2
O
-2
2
4
-2
-1
-4
0
-6
1
2
2. y = x2 + 2
x2 + 2
x
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y
y
6
(x, y)
-2
4
-1
2
x
0
-4
O
-2
2
4
-2
1
2
Determine whether the rates of change are constant or variable.
3.
4.
x
-2
-1
0
1
2
x
-2
-1
0
1
2
y
6
3
2
3
6
y
7
4
1
-2
-5
5. A pharmaceutical representative earns 5.5% commission on all her sales plus
a base salary of $400 a month. Write an equation to represent this situation
and then explain whether it is a linear equation.
8-1 Graphing Linear Equations 275
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Date
MA.8.A.1.6 Compare the graphs of
linear and non-linear functions for
real-world situations.
Graphing Linear Equations
The table shows the price of a wooden frame
for a square poster with various side lengths.
Use the table for 1−8.
Side
Length (ft)
1
2
Price of
Frame ($)
3
4
18
24
5
5. The price g of the glass for a square poster
with side length l is given by g = 2l 2. Graph
this equation on the same coordinate plane
you used for Problem 4. Be sure to label the
two graphs so you can tell which is which.
6. Describe the two graphs that you drew.
How are they different?
1. The rates of change in the table are
constant. Complete the table.
2. Anne wants to buy a wooden frame for a
square poster that is 7 feet long. How much
should she expect to pay?
7. For what size poster is the price of the frame
the same as the price of the glass?
4. Graph the equation that gives the price of
the frame. Use the coordinate plane below.
8. Gridded Response
Jerome has a square
poster that is 2.5 feet
long on each side. How
much should he expect
to pay, in dollars,
altogether for the frame
and the glass?
50
Price ($)
40
30
20
10
1
2
3
4
5
Side Length (ft)
276 8-1 Graphing Linear Equations
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3. Write an equation that gives the price f of a
wooden frame for a square poster with side
length ℓ.
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8-2
Date
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MA.8.A.1.2 Interpret the slope…
when graphing a linear equation for a
real-world problem.
Slope of a Line
Explore the Slant of a Line
The slant of a line may be steep or relatively flate. The line may slant upward or
downward as you move from left to right along the graph. In the activity, you will
investigate how the coefficient of x in a linear equation affects the slant of its
graph, a line.
Activity
y
8
1 As you move from left to right along each
line in Figure 1, the direction of the slant
is the same. Is it upward or downward?
6
4
2
2 Examine the lines graphed at the right.
Describe what happens to the steepness
of the line as the value of the coefficient
of x increases from _14 to 3.
x
-8
-6
-4
O
-2
y=
1
4
x
-2
y=
1
2
x
-4
2
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-6
y=x
3 As you move from left to right along each
line in Figure 2, the direction of the slant
is the same. Is it upward or downward?
y = -x
4
6
8
Figure 1
-8
y = 2x
y = 3x
y = -3x
y = -2x
8
y
6
4 Examine the lines graphed at the right.
Describe what happens to the steepness
of the line as the absolute value of the
coefficient of x increases from _14 to 3.
y =-
1
2
x
4
y =-
1
4
x
2
x
-8
-6
-4
-2
O
2
4
6
8
-2
-4
Figure 2
-6
-8
8-2 Slope of a Line 277
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The equation of a line and two points on that line are given. Use the coordinate
plane to plot the points and draw the line. Then circle “upward” or “downward”
to indicate the line’s slant from left to right.
y
1. y = 2x - 5; (1, -3), (4, 3)
upward OR downward
6
2. y = -__12x + 3; (4, 1), (8, -1)
4
upward OR downward
2
3. y = __25 x - 2; (5, 0), (-5, -4)
x
-6
-4
-2
O
upward OR downward
-2
4. y = -3x + 2; (0, 2), (3, -7)
-4
upward OR downward
-6
2
4
6
Draw Conclusions
Compare the slant of a line from left to right with the sign of the x-term in its
equation. Then answer the questions.
5. Make a conjecture : How is the slant of a line affected by a negative x-term in
its equation?
Compare the steepness of a line with the absolute value of the coefficient of x in
its equation. Then answer the questions.
7. Make a conjecture : How is the steepness of a line affected as its equation
changes from y = - _13 x to y = -3x?
8. Make a conjecture : How is the steepness of a line affected as its equation
changes from y = 5x to y = _25 x ?
278 8-2 Slope of a Line
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6. Make a conjecture : How is the slant of a line affected by a positive x-term in
its equation?
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8-2
Date
MA.8.A.1.2 Interpret the slope…
when graphing a linear equation for a
real-world problem..
Slope of a Line (Student Textbook pp. 351–355)
Lesson Objective
Find the slope of a line and use slope to understand and draw graphs
Vocabulary
rise
run
slope
Example 1
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Find
Fi
d th
the slope
l
of the line.
y
8
6
(7, 5)
4
2
(1, 2)
x
O
2
4
6
8
to find the rise.
Begin at one point and count
Then count
to the second point to find the run.
rise = _____ = _____
slope = ____
run
The slope of the line is
.
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8-2 Slope of a Line 279
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y
1. Find
d the
h slope
l
off the
h line.
6
4
2
x
6
-4
-2
O
2
4
6
-2
-4
-6
Example 2
Fi
d the
h slope
l
Find
of the line that passes through (-2, -3) and (4, 6).
Let (x1, y1) be (-2, -3) and (x2, y2) be (4, 6).
y2 - y1
____________ =
______
=
x2 - x1
-
The slope of the line is
=
.
2. Find the slope of the line that passes through (-4, -6) and (2, 3).
280 8-2 Slope of a Line
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Example 3
The table shows the total cost of fruit per pound purchased at
the grocery store. Use the data to make a graph. Find the slope
of the line and explain what it shows.
on the x-axis and
Cost ($)
Graph the data. Put
50
45
40
35
30
25
20
15
10
5
0
Cost of Fruit
on the y-axis.
Pounds
Cost ($)
0
0
5
15
10
30
15
45
Find the slope of the line.
y -y
15
2
1
_________ = __
m = ______
=
x -x =
5
2
1
0 2 4 6 8 10 12 14 16
Pounds
The slope of the line is
.
This means for every pound of fruit, you will pay another $
.
3. The table shows how much money Andre and Marla
made working at the concession stand at a baseball
game one weekend. Use the data to make a graph.
Find the slope of the line and explain what it shows.
Time Worked (h) Amount Earned ($)
2
15
4
30
6
45
8
60
120
Amount ($)
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90
60
30
2
4
6
8
Time (h)
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8-2 Slope of a Line 281
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Date
LA.8.2.2.3 The student will organize
information to show understanding or
relationships…
Slope of a Line
Think and Discuss
1. Explain why it does not matter which point you choose as
(x1, y1) and which point you choose as (x2, y2) when finding slope.
2. Give an example of two pairs of points from each of two parallel lines.
3. Get Organized Complete the graphic organizer. Fill in the boxes by writing
the definition of rise and run. Then write the definition of the slope of a line.
Slope of a Line
Run
Slope
282 8-2 Slope of a Line
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Rise
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8-2
Date
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MA.8.A.1.2 Interpret the slope…
when graphing a linear equation for a
real-world problem.
Slope of a Line
Find the slope of each line.
y
1.
y
2.
4
4
2
2
x
x
-4
O
-2
2
-4
4
O
-2
-2
-2
-4
-4
2
4
Find the slope of the line that passes through each pair of points.
4. (2, 8), (8, 2)
5. (-2, 6), (-4, -4)
7. The table shows the distance in feet from a motion
detector that a toy car travels over a period of time.
Use the data to make a graph. Find the slope of the
line and explain what it shows.
Time (s)
1
2
3
4
5
Distance (ft)
9
13
17
21
25
Distance (ft)
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3. (5, 7), (6, 9)
6. (-8, -7), (-9, -1)
50
45
40
35
30
25
20
15
10
5
0
0 1 2 3 4 5 6 7 8 9 10
Time (s)
8-2 Slope of a Line 283
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MA.8.A.1.2 Interpret the slope…
when graphing a linear equation for a
real-world problem..
Slope of a Line
1. The roof of a house rises vertically 3 feet for
every 12 feet of horizontal distance. What is
the slope, or pitch, of the roof ?
2. The table shows the altitude of a hot-air
balloon over a period of time.
Time (min)
3. The state of Kansas has a fairly steady
slope from the west side of the state to the
east side of the state. At the western edge
of the state, the elevation is 4039 ft. At the
eastern edge, 413 miles across the state, the
elevation is 771 ft. What is the approximate
slope of Kansas? (Hint:1 mile = 5280 feet)
Altitude (ft)
4
1000
6
1400
8
1800
10
2200
4. Jason types an essay at a constant rate. After
3 minutes, he has typed 135 words. After 8
minutes, he has typed 360 words. At what
rate does Jason type? How long will it take
him to type an essay that is 900 words long?
2400
2200
2000
1800
1600
1400
1200
1000
800
600
400
200
1 2 3 4 5 6 7 8 9 10 11 12
Time (min)
b. Find the slope of the line and explain
what it shows.
Height (ft)
5. Short Response The graph shows the
height of an elevator over a period of time.
How far does the elevator travel in
15 seconds? Explain how you know.
300
270
240
210
180
150
120
90
60
30
1 2 3 4 5 6 7 8 9 10
Time (s)
c. Assuming the balloon continues at this
rate, what is its altitude after 14 minutes?
284 8-2 Slope of a Line
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Altitude (ft)
a. Use the data to make a graph.
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8-3
Date
MA.8.A.1.2 Interpret the slope and
the x- and y-intercepts when graphing
a linear equation for a real-world
problem.
Using Slopes and Intercepts
Explore Intercepts
You have seen that a line can be described in part by its slope. A line can
also be described by where it crosses the axes of a coordinate plane.
y
Activity 1
1 Graph the following equations together on the same
coordinate plane at the right.
4
2
y=x+4
y=x+0
y=x-1
y=x+2
x
-4
y=x-3
2
-2
4
-2
2 How are the lines alike?
-4
3 How are the lines different?
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4 Look at the point on the y-axis where each line crosses. How is the number added
to x, or constant, in the equation related to this point?
5 What happens to the graph as the constant increases? decreases?
Try This
y
Graph each equation, and match it with the location
where its graph crosses the y-axis.
4
1. y = x - 4
a. crosses the y-axis at 1
2. y = x + 4
b. crosses the y-axis at -2
3. y = x + 1
c. crosses the y-axis at 4
-2
4. y = x - 2
d. crosses the y-axis at -4
-4
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2
x
-4
2
-2
4
8-3 Using Slopes and Intercepts 285
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Activity 2
6
1 For each equation, identify the coordinates of the point
where the line crosses the y-axis.
The graph of y - 1 = 3x crosses at
.
The graph of 2x - y = 2 crosses at
.
The graph of 4 + y = x crosses at
.
y
4
2
x
-6
-4
-2
6
4
2
-2
-4
-6
2 What is the x-coordinate of any point on the y-axis?
.
3 Substitute 0 for x in each equation, and solve for y. Then write the coordinates
from Step 1.
y - 1 = 3x : When x = 0, y =
2x - y = 2: When x = 0, y =
4 + y = x : When x = 0, y =
. The graph crosses the y-axis at
.
. The graph crosses the y-axis at
.
. The graph crosses the y-axis at
.
4 Look for patterns: For each equation in Step 3, how are the y-values
when x = 0 and the y-coordinates where the line crosses the y-axis related?
2
y
5. Graph the equation y = x - 2, and complete the statements.
• The graph crosses the y-axis at
.
x
-4
O
-2
2
4
-2
• When x = 0, y =
.
-4
-6
Draw Conclusions
6. Make a conjecture: Without graphing, how do you think it is possible to find
out where the graph of a linear equation will cross the y-axis? Justify your answer.
286 8-3 Using Slopes and Intercepts
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Try This
Learn It!
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Summarize It!
Name
Practice It!
Apply It!
Class
8-3
Date
MA.8.A.1.2 Interpret the slope and
the x- and y-intercepts when graphing
a linear equation for a real-world
problem.
Learn It!
Using Slopes and Intercepts (Student Textbook pp. 356–360)
Lesson Objective
Use slopes and intercepts to graph linear equations
Vocabulary
x-intercept
y-intercept
slope-intercept form
Example 1
Copyright © by Holt McDougal. All rights reserved.
Fi d the
Find
h x-intercept
i
and y-intercept of the line 4x - 3y = 12. Use the intercepts to
graph the equation.
Find the x-intercept ( y =
).
Find the y-intercept (x =
4x - 3y = 12
4x - 3(
).
4x - 3y = 12
) = 12
4(
) -3y = 12
4x = 12
-3y = 12
12
4x = __
__
4
4
-3y ___
12
____
= -3
-3
y
y=
x=
2
The x-intercept is
.
The y-intercept is
The graph of 4x - 3y = 12 is the line that crosses
.
x
-4
-2
O
2
4
-2
the x-axis at the point (
y-axis at the point (
) and the
-4
).
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8-3 Using Slopes and Intercepts 287
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Check It Out!
1. Find the x-intercept and y-intercept of the line 8x - 6y = 24. Use the intercepts to
graph the equation.
2
y
x
-2
O
2
4
6
-2
-4
-6
Example 2
Write each equation in slope-intercept form, and then find the slope and
y-intercept.
A. 2x + y = 3
2x + y = 3
-
-
Subtract
from both sides.
y = 3 - 2x
Rewrite to match slope-intercept form.
The equation is in
m=
-intercept form.
b=
The slope of the line 2x + y = 3 is
, and the y-intercept is
.
B. 5y = 3x
5y
3x
______
=______
y=
Divide both sides by
x
Rewrite the equation in
y=
m=
.
form.
x+
b=
The slope of the line 5y = 3x is
288 8-3 Using Slopes and Intercepts
and the y-intercept is
.
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y=
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Write each equation in slope-intercept form, and then find the slope
and y-intercept.
C. 4x + 3y = 9
4x + 3y = 9
-
-
Subtract
from both sides.
3y = -4x + 9
3y
+9
______
_______
=-4x
Divide both sides by
y = -__43 x + 3
y=
m=
The equation is in
form.
x+
b=
The slope of the line 4x + 3y = 9 is
Check It Out!
.
and the y-intercept is
.
W
Write
each equation in slope-intercept form, and then find the
sslope and y-intercept.
Copyright © by Holt McDougal. All rights reserved.
2a. 3y = 5x
2b. 5x − y = 8
2c. 6y + 7 = 2x
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8-3 Using Slopes and Intercepts 289
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Example 3
A video club charges $8 to join, and $1.25 for each DVD that is rented. The
linear equation y = 1.25x + 8 represents the amount of money y spent after
renting x DVDs. Graph the equation, and explain the meaning of the slope
and y-intercept.
y = 1.25x + 8
The equation is in slope-intercept form.
m=
b=
The slope of the line is
, and the y-intercept is
Total Charge ($)
y-axis at (0, 8) and moves
25
24
22
20
18
16
14
12
10
8
6
4
2
0
. The line crosses the
1.25 units for every 1 unit it moves
.
The y-intercept represents the charge for
The slope represents the charge for
.
.
0 2 4 6 8 10 12 13
DVDs Rented
3. A home improvement store charges a base fee of $150, plus $25 for each hour
of machinery rental. The cost y for x hours is given by y = 25x + 150. Graph the
equation, and explain the meaning of the slope and y-intercept.
290 8-3 Using Slopes and Intercepts
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Example 4
W i the
Write
h equation of the line that passes through (3, -4) and (-1, 4) in
slope-intercept form.
Find the slope.
y2 - y1
_____________
______
=
x2 - x1
-
=
=
The slope is
.
Substitute either point and the slope into the slope-intercept form,
y = mx + b and solve for b.
=
+b
=
-
+b
-
Substitute -1 for x, 4 for y, and -2 for m.
Simplify.
Subtract
from both sides.
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=b
Write the equation of the line, using
for m and
for b.
y=
Check It Out!
4. Write the equation of the line that passes through (5, 1) and (–1, –5) in slopeintercept form.
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8-3 Using Slopes and Intercepts 291
8-3
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Summarize It!
Name
Practice It!
Class
Apply It!
Date
Summarize It!
LA.8.2.2.3 The student will organize
information to show understanding or
relationships…
Using Slopes and Intercepts
Think and Discuss
1. Describe the line represented by the equation y = -5x + 3.
2. Give a real-life example with a graph that has a slope of 5 and a
y-intercept of 30.
3. Get Organized Complete the graphic organizer. Fill in the boxes by
writing an advantage and a disadvantage of using each of the given
forms of a linear equation.
Standard Form
Ax + By = C
Slope-Intercept Form
Advantage
Advantage
Disadvantage
Disadvantage
292 8-3 Using Slopes and Intercepts
y = mx + b
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Linear
Equations
Explore It!
Learn It!
Practice It!
Summarize It!
Name
Apply It!
Class
8-3
Date
MA.8.A.1.2 Interpret the slope
and the x- and y-intercepts when
graphing a linear equation for a realworld problem.
Practice It!
Using Slopes and Intercepts
Find the x-intercept and y-intercept of each line. Use the intercepts to graph
the equation.
1. x = y - 5
2. 3x - 2y = 6
y
2
x
4
-4
2
-4
-2
-2
O
2
4
-2
x
O
y
2
-4
4
-2
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Write each equation in slope-intercept form, and then find the slope and
y-intercept.
3. 2x = y - 4
4. 4x - 3y = -9
5. x - 4y = 8
Write the equation of the line that passes through each pair of points in
slope-intercept form.
6. (-1, 7), (4, -3)
7. (2, 5), (-8, 15)
8. (8, 4), (-10, -5)
9. Thad’s father gives him $10 for a passing report card plus $5 for every
grade of A. Write an equation of a line in slope-intercept form to
express y, the amount received with x grades of A. State the slope,
and y-intercept of the equation.
8-3 Using Slopes and Intercepts 293
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8-3
Learn It!
Name
Summarize It!
Class
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Practice It!
Date
MA.8.A.1.2 Interpret the slope
and the x- and y-intercepts when
graphing a linear equation for a
real-world problem.
Apply It!
Using Slopes and Intercepts
Vanessa buys a $20 bus pass. Each time she
rides the bus, $1.25 is deducted from the
pass. The equation y = -1.25x + 20 gives the
amount of money y on the pass after x rides.
Use this information for 1−4.
4. A different town also offers a $20 bus pass.
For this pass, the equation y = -0.8x + 20
gives the amount of money y on the pass
after x rides. Without graphing, explain how
the graph of this equation compares to that
of the equation you graphed in Problem 1.
Amount Remaining on Pass ($)
1. Graph the equation on the coordinate
plane.
20
18
16
14
12
10
8
6
4
2
Alfredo drives at a constant speed. The table
shows the time he drives and his distance
from his home. Use the table for 5−6.
Distance from Home (mi)
2 4 6 8 10 12 14 16 18 20
2
149
Number of Rides
5
335
2. Find the slope and the y-intercept and
explain the meaning of each.
5. Write an equation that gives Alfredo’s
distance from home y after he drives for
x hours.
6. Short Response Did Alfredo start his
drive from home? Justify your answer.
3. What is the maximum number of bus rides
Vanessa can take using the card? Explain
how your answer to this question is shown
in the graph.
294 8-3 Using Slopes and Intercepts
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Time (h)
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Class
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8-4
Date
MA.8.A.1.2 Interpret the slope
and the x- and y-intercepts when
graphing a linear equation for a
real-world problem.
Linear Functions
Model Linear Relationships
The perimeter, P, of a rectangle with length ℓ and width w is given by the formula
P = 2ℓ + 2w. In this activity you will find a linear relationship using that formula.
Activity 1
1 Find the perimeter of a piece of paper that is
8.5 inches wide and 11 inches long. Record this
perimeter in the first row of the table.
Width of fold
(w) (inches)
2 Fold the paper to remove 1 inch from the length
of the paper, and find the new perimeter.
Record it in the table.
3 Repeat. Fold the paper to remove 2 inches, 3
inches, and then 4 inches from the length of
the paper, and find the new perimeter each
time. Record your results in the table.
0
1
2
3
4
40
Perimeter (in.)
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4 Plot the ordered pairs (w, P) from your table on
the coordinate plane at the right.
5 Describe a reasonable domain for w and range for P. Do
negative values make sense? Justify your answer.
Perimeter (P)
(inches)
P
30
20
10
w
1 2 3 4 5 6 7
Width (in.)
6 Describe the pattern of the points on the coordinate plane.
Do they form a line or a curve?
7 Is this relation continous or discrete? Do rational numbers like 2.5 or __5
2
make sense?
8-4 Linear Functions 295
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Try This
1. Use the values in the table for Activity 1 to predict the perimeter after removing
5 inches from the length. Justify your answer.
2. Look for a pattern: Explain how you could find the perimeter after
removing any given amount from the length.
Activity 2
1 A square has side length s and perimeter P. Complete
the table for at least 5 different values of s, and plot the
ordered pairs (s, P).
Side length (s)
Perimeter (P)
P
40
35
30
25
20
15
10
5
s
1 2 3 4 5 6 7 8 9 10
2 Describe all the possible values of s and P.
3 Describe the pattern of the points on the coordinate plane.
Try This
3. Look for a pattern: Explain how to find P given any value of s.
Draw Conclusions
4. Do you think the following relationship is linear? Justify your answer.
the distance in miles and the time in hours that a car
travels while driving at a constant 60 miles per hour
296 8-4 Linear Functions
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4 Does it make sense to connect the points? Justify your answer.
Learn It!
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Summarize It!
Name
Practice It!
Apply It!
Class
8-4
Date
MA.8.A.1.2 Interpret the slope and
the x- and y-intercepts when graphing
a linear equation for a real-world
problem.
Learn It!
Linear Functions (Student Textbook pp. 361–364)
Lesson Objective
Identify and write linear functions
Vocabulary
linear function
function notation
Example 1
D t
Determine
i whether each function is linear. If so, give the slope
and y-intercept of the function’s graph.
A. f (x) = 2x 3
Copyright © by Holt McDougal. All rights reserved.
f (x) = 2x 3
represent a linear function because x has an
other than 1.
B. f (x) = 3x + 3x + 3
f (x) =
x+3
Write the equation in
The function is
f (x) = mx + b. The slope m is
form.
because it can be written in the form
and the y-intercept b is
.
Check It Out!
D
Determine
whether each function is linear. If so, give the slope
and y-intercept of the function’s graph.
a
1a. f (x) = -2x + 4
1b. f (x) = -_1x + 4
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8-4 Linear Functions 297
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Example 2
Write a rule for each linear function.
A.
B.
y
4
2
x
-3
-1
1
3
y
-8
-2
4
10
Step 1: Locate two points.
x
-4
O
-2
(–1,
4
2
-2
)(1,
)
Step 2: Find the slope m.
-4
y -y
2
1
4________
m = ______
= _____ = 3
x -x =
2
1
1-
Step 1: Identify the y-intercept b.
Step 3: Substitute the slope and one pair of
x- and y-values into the equation
y = mx + b, and solve for b.
b=
Step 2: Locate another point (x, y).
y = mx + b
(1, 4)
=
Step 3: Substitute 2 for b and the x- and
y-values into the equation
y = mx + b, and solve for m.
= m(
(1) + b
1=b
The rule is
)+
.
The rule is f (x) =
Check It Out!
x+
W
Write a rule for each linear function.
y
2a.
.
2b.
4
x
-2
-1
0
1
2
y
-5
-3
-1
1
3
2
x
-4
-2
O
2
4
-2
-4
298 8-4 Linear Functions
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=m
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Example 3
A video game club costs $15 to join. Each game that is rented costs $1.50.
Find a rule for the linear function that describes the total cost of renting
games as a member of the club, and find the total cost of renting 12 games.
The y-intercept is the cost for joining the video game club, $15.
b=
The rate of change in cost is $1.50 per game.
m=
The rule for the function is
f (12) = 1.5(
=
.
) + 15
+ 15
=
The cost renting 12 games is $
.
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Check It Out!
3. A book club has a membership fee of $20. Each book purchased costs $2.
Find a rule for the linear function that describes the total cost of buying
books as a member of the club, and find the total cost of buying 10 books.
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8-4 Linear Functions 299
Explore It!
8-4
Learn It!
Summarize It!
Name
Practice It!
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Class
Date
Summarize It!
LA.8.2.2.3 The student will organize
information to show understanding or
relationships…
Linear Functions
Think and Discuss
1. Describe how to use a graph to find the equation of a linear function.
2. Get Organized Complete the graphic organizer. Fill in the boxes by writing the
rule for a linear function. Then provide a table and graph for the function.
Rule
Linear
Function
Copyright © by Holt McDougal. All rights reserved.
Table
Graph
y
5
x
-5
O
-5
300 8-4 Linear Functions
5
Explore It!
Learn It!
Practice It!
Summarize It!
Name
Apply It!
Class
8-4
Date
MA.8.A.1.2 Interpret the slope
and the x- and y-intercepts when
graphing a linear equation for a
real-world problem.
Practice It!
Linear Functions
Determine whether each function is linear. If so, give the slope and y-intercept
of the function’s graph.
1. 4y = 2x
3. 4y = 10 - __1x
2. 5 = 8 - 3y
Write a rule for each linear function.
4.
6
y
5.
4
2
4
x
2
-6
-4
x
Copyright © by Holt McDougal. All rights reserved.
-6
-4
-2
O
y
2
-2
O
2
4
6
-2
4
-2
-4
-4
-6
-6
6.
x
-4
-2
3
5
f (x)
13
7
-8
-14
7.
x
f (x)
-6
-10
-4
-9
2
8
-6
-3
8. A salesperson is paid a base salary of $300
plus 8% commission.
a. Write a function rule for the salary. Graph the
function.
b. If the salesperson has $900 in sales, what is the
salesperson’s salary?
8-4 Linear Functions 301
Explore It!
8-4
Learn It!
Name
Summarize It!
Practice It!
Class
Apply It!
Date
MA.8.A.1.2 Interpret the slope
and the x- and y-intercepts when
graphing a linear equation for a
real-world problem.
Apply It!
Linear Functions
The table shows the relationship between the
number of times a cricket chirps in a minute
and the temperature. Use the table for 1−6.
Number of Chirps per
Minute
Temperature (°F)
80
60
100
65
120
70
140
75
1. Write a rule for the linear function that
describes the temperature based on the
number of cricket chirps x in a minute.
4. A cricket chirps 150 times in a minute. What
is the temperature? Describe two different
ways to solve this problem.
5. Suppose the number of chirps increases
by 12 chirps per minute. What can you say
about the temperature? Explain.
2. Graph the function.
302 8-4 Linear Functions
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3. Find the slope of the line and explain what
it represents.
6. Gridded Response
Suppose the
temperature is 85 °F.
How many times does
a cricket chirp in 10
minutes?
8-1
Name
Class
THROUGH
Date
8-4
Got It?
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Quiz for Lessons 8-1 through 8-4
Graphing Linear Equations (Student Textbook pp. 346–350)
At Maggi’s Music, the equation
u = _34 n + 1 represents the price for a
used CD u with a selling price n when
the CD was new.
New
Price
Used
Price
$8
1. How much will a used CD cost for
each of the listed new prices?
Used Price ($)
8-1
$12
$14
2. Graph the equation and tell
whether it is linear.
$20
20
18
16
14
12
10
8
6
4
2
2 4 6 8 10 12 14 16 18 20
8-2
New Price ($)
Slope of a Line (Student Textbook pp. 351–355)
Find the slope of the line that passes through each pair of points.
3. (6, 3) and (2, 4)
4. (1, 4) and (-1, -3)
5. (0, -3) and (-4, 0)
7.
8.
Find the slope of each line.
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6.
y
6
5
4
3
2
1
y
2
x
O
x
-2
x
-2
y
O
2
-2
2
-4
-2
1 2 3 4 5 6
8-3
Using Slopes and Intercepts (Student Textbook pp. 356–360)
Write the equation of the line that passes through each pair of points in
slope-intercept form.
9. (-4, 3) and (-2, 1)
8-4
10. (2, 7) and (5, 2)
11. (4, 2) and (2, -5)
Linear Functions (Student Textbook pp. 361–364)
Determine whether each function is linear. If so, give the slope and
y-intercept of the function’s graph.
12. f (x) = 2x 3
13. f (x) = 6x - 3x +1
14. f (x) = 2 ( __31 x - 1 )
Chapter 8 Linear Functions 303
8-1
THROUGH
8-4
Name
Class
Date
Connect It!
MA.8.A.1.1; MA.8.A.1.2;
MA.8.A.1.5
Connect the concepts of Lessons 8-1 through 8-4
Manatee Weigh-In
A manatee is a marine mammal that is found along the Florida coast.
The table shows the weight of a typical manatee at various ages.
1. Is the rate of change constant
Age (months)
Weight (pounds)
or variable?
9
10
11
12
222
240
258
276
2. Write a rule for the function. Tell what the variables represent.
3. What is the slope of the function’s graph? What does it represent?
4. What is the y-intercept of the function’s graph? What does it represent?
5. Predict a manatee’s weight at 4 years of age. Justify your answer.
An Amusing Museum
x-intercept: 8
A museum in Sanibel, Florida, is devoted to several
million objects that are not often seen in museums.
Find out what they are by solving the puzzle!
5x = 10y
1
Slope: 2
1. Each box contains a linear equation and
information about the slope or intercepts. Cross
out all the boxes in which the information is false.
3y = 6x -1
2. Arrange the letters in the remaining boxes to find
out what you can see at the museum.
y =7
Slope: 0
3. How can you change the equation and information
in the box with the letter H without changing the
puzzle’s solution?
304 Chapter 8 Linear Functions
Slope: 2
8x + y = 5
O
Slope: 8
L
T
y-intercept: -1
y = 2x - 5
Think About The Puzzler
S
S
2x + 5y = -10
E
y-intercept: -2
2x + 5y = 20
2
Slope: 5
3x = 15y
R
L
y-intercept: 0
H
y = -x + 3
x-intercept: 3
A
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3x + 4y = 24
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Date
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MA.8.A.1.1 Create…models to
represent, analyze, and solve problems
related to linear equations…
Point-Slope Form
Explore Another Form of Linear Equations
Slope-intercept form is one way to write a linear
equation. In this activity you will explore another form
of a linear equation.
Activity
At the top of each column there is an equation and three ordered pairs. Answer the
questions in each row for each equation.
y - 1 = 1 (x - 2)
(2, 1), (0, -1), (-3, -3)
1 (x - 2)
y - 1 = -__
2
(2, 1), (0, 5), (-4, 4)
y - 1 = -2 (x - 2)
(2, 1), (0, 5), (-4, 4)
Copyright © by Holt McDougal. All rights reserved.
1 Which two ordered
pairs are solutions
of the equation?
2 Plot each pair
of points from
Step 1, and use
a straightedge to
draw a line through
each pair.
y
6
4
2
x
-6
-4
-2
O
2
4
6
-2
-4
-6
3 Write the equation
of each line in
slope-intercept
form.
8-5 Point-Slope Form 305
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1. Use Step 3 in the Activity to write the slope of each of the original equations
from the Activity.
y - 1 = 1(x - 2)
slope:
y - 1 = -2(x - 2)
slope:
y - 1 = -_12(x - 2)
slope:
2. Make a conjecture : Compare the slopes with the equations. Do you see a
number in the equation that could represent the slope? Explain.
3. Look at Step 1 in the Activity. What point was a solution of all three equations?
How do the constants in each equation compare with the coordinates of
that point?
Draw Conclusions
5. Without graphing the equation, can you find a point that the line
y – 3 = 2(x - 4) goes through? Justify your answer.
306 8-5 Point-Slope Form
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4. Use your conjecture: Without graphing the equation or writing it in
slope-intercept form, name the slope of the graph of y - 7 = -5(x + 9).
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MA.8.A.1.1 Create…models to
represent, analyze, and solve problems
related to linear equations…
Point-Slope Form (Student Textbook pp. 365–368)
Lesson Objective
Find the equation of a line given one point and the slope
Vocabulary
point-slope form
Example 1
U th
Use
the point-slope
i
form of each equation to identify a point the line passes
through and the slope of the line.
A.
y - 7 = 3(x - 4)
y - y1 = m(x - x1)
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y-
=
(x -
)
The line defined by y - 7 = 3(x - 4) has
slope
, and passes through the
point
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=
y-
(x1, y1) =
m=3
y - 1 = __13 (x + 6)
B.
(x -
)
(x1, y1) =
m=
The line defined by y - 1 = __13 (x + 6) has
slope
, and passes through the
.
point
.
Use the point-slope form of the equation to identify a point
the line passes through and the slope of the line.
1a. y - 2 = __23(x + 3)
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1b. y + 5 = 2(x - 1)
8-5 Point-Slope Form 307
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Example 2
W
Write
it th
the point-slope
i
form of the equation with the given slope that passes
through the indicated point.
A. the line with slope 4 passing through (5, -2)
y - y1 = m(x - x1)
y-
=
y+
=
(x -
)
(x -
Substitute
)
and
for x1,
for y1,
for m.
The equation of the line with slope 4 that passes through (5, -2) in point-slope
form is
.
B. the line with slope -5 passing through (-3, 7)
y - y1 = m(x - x1)
y-
=
(x -
)
y-
=
(x +
)
Substitute
for x1,
for y1, and
for m.
The equation of the line with slope -5 that passes through (-3, 7) in point-slope
form is
Write the point-slope form of the equation with the given
slope that passes through the indicated point.
2a. the line with slope 2 passing
through (2, -2)
308 8-5 Point-Slope Form
2b. the line with slope __14 passing
through (-3, 2)
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.
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Example 3
A roller coaster starts by ascending 20 feet for every 30 feet it moves forward.
The coaster starts at a point 18 feet above the ground. Write the equation of
the line that the roller coaster travels along in point-slope form, and use it to
determine the height of the coaster after traveling 150 feet forward. Assume
that the roller coaster travels in a straight line for the first 150 feet.
, y increases by
As x increases by
so the slope of the line is
=
,
.
The line must pass through the point (0,
).
y - y1 = m(x - x1)
=
y-
(x - 0)
Substitute 0 for x1,
for y1, and
for m.
The equation of the line the roller coaster travels along is
.
for x to find the height after traveling 150 ft forward.
Substitute
y-
= __23 (
y-
=
)
y=
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The height of the coaster after traveling 150 feet forward is
.
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3. At sea level, the boiling point of water is 212°F. The boiling point decreases
1°F for every 500 ft of increase in altitude. Write an equation for the boiling
point of water in point-slope form, and use it to find the boiling point of water
at 6000 ft above sea level.
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8-5 Point-Slope Form 309
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LA.8.2.2.3 The student will organize
information to show understanding or
relationships…
Point-Slope Form
Think and Discuss
1. Describe the line, using the point-slope equation, that has a slope of 2 and
passes through (-3, 4).
2. Tell how you find the point-slope form of the line when you know the
coordinates of two points.
3. Get Organized Complete the graphic organizer. Fill in the boxes by writing
the slope-intercept form of a linear equation and the point-slope form of a
linear equation. Then give an example of each.
Slope-Intercept Form
Point-Slope Form
Example
Example
310 8-5 Point-Slope Form
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Linear
Equations
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8-5
Date
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MA.8.A.1.1 Create…models to
represent, analyze, and solve problems
related to linear equations…
Point-Slope Form
Use the point-slope form of the equation to identify a point the line passes
through and the slope of the line.
1. y - 6 = 3(x - 2)
2. y + 4 = -2(x + 1)
3. y - 5 = -1(x - 8)
4. y + 3 = 4(x + 2)
5. y - 7 = -6(x - 1)
6. y - 4 = 9(x + 6)
7. y + 5 = -7(x - 9)
8. y - 7 = __13(x + 9)
9. y - 3 = 3.2(x + 8)
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Write the point-slope form of the equation with the given slope that passes
through the indicated point.
10. the line with slope -3 passing through
(4, 3)
11. the line with slope -1 passing through
(6, -2)
12. the line with slope 5 passing through
(-7, 1)
13. the line with slope 7 passing through
(-8, -4)
14. the line with slope 2 passing through
(-9, -6)
15. the line with slope -8 passing through
(7, -3)
16. An oil tank is being filled at a constant rate. The depth of the oil begins at 3 feet.
Every 5 minutes, the depth increases by 1 foot. Write the equation for the depth of
the oil in point-slope form. Use the equation to find out how long it would take for
the depth of oil to reach 25 feet.
8-5 Point-Slope Form 311
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Date
MA.8.A.1.1 Create…models to
represent, analyze, and solve problems
related to linear equations…
Point-Slope Form
1. A 1600 square foot house in Orlando sells
for about $215,000. The price increases by
about $132 per additional square foot.
5. Describe in words the relationship between
a person’s weight on Earth and the person’s
weight on Mars.
a. Write an equation in point-slope form
that describes the price y of a house in
Orlando based on the number of square
feet x.
b. Write the equation in slope-intercept
form.
c. How much would you expect to pay for a
2000 square foot house in Orlando?
Weight on Earth (lb)
Weight on Mars (lb)
150
57
200
76
2. There is a linear relationship between
a person’s weight on Earth and their
weight on Mars. What is the slope for this
relationship?
3. Write an equation in point-slope form that
describes a person’s weight on Mars y based
on their weight on Earth x
4. Write the equation in slope-intercept form.
312 8-5 Point-Slope Form
a. Write an equation in point-slope form
that describes the price y of a taxi ride
based on the number of miles x.
b. The taxi fare is calculated by taking a
base rate and adding a fee for each mile
you travel. What is the base rate? What is
the fee for each mile?
c. Chandra took a taxi ride that cost $19.35.
How far did she travel?
7. Short Response At a ski resort, a
chairlift begins at an elevation of 8662 ft.
The lift rises 29 ft for every 100 ft of
horizontal distance that it travels. Jacob says
he cannot write an equation that models
the path of the chairlift without knowing the
total horizontal distance it travels. Do you
agree or disagree? Explain.
Copyright © by Holt McDougal. All rights reserved.
The table shows the weight of a person on
Earth and the weight of the same person on
Mars. Use the table for 2−5.
6. A 3-mile taxi ride costs $7.80. A 7-mile taxi
ride costs $16.20.
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8-6
Date
MA.8.A.1.1 …Interpret tables,
graphs, and models to represent,
analyze, and solve problems related
to linear equations…
Direct Variation
Model Direct Variation
Human proportions vary from one person to the next. However, they are
consistent enough that artists often use standard approximations in their work.
One example is the approximation that an adult human is about 8 heads tall.
In this activity, you will explore the artist Leonardo da Vinci’s observation that the
kneeling height of a person is _34 of the person’s standing height.
Activity
1 Use a meter stick. Measure the standing and kneeling heights of six other
students. Record your results in Columns 2 and 3 of the table below.
2 In Column 4 of the table, record the ratio of kneeling height to standing
height for each student. Round ratios to the nearest thousandth.
Standing Height
(cm)
Kneeling Height
(cm)
Kneeling Height
______________
Standing Height
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Name
3 Compare your ratios with one another and with those of other students.
Are the results consistent? Explain.
8-6 Direct Variation 313
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4 Make a scatter plot of your data, showing
the relationship between standing heights (horizontal
axis) and kneeling heights (vertical axis). You should
plot a single point for each student whose heights
you measured.
5 Let k represent kneeling height and let s
represent standing height. Write an equation
for kneeling height as a function of standing
height, based on Leonardo da Vinci’s
observation.
Apply It!
200
Kneeling Height (cm)
Explore It!
150
100
50
50
100
150
200
Standing Height (cm)
6 Graph the function you wrote in Step 5 on your
scatter plot.
Try This
Find the height. Assume that kneeling height is _34 standing height.
1. standing height = 164 cm
2. kneeling height = 132 cm
kneeling height =
3. standing height = 192 cm
kneeling height =
5. standing height = 174.4 cm
kneeling height =
standing height =
4. kneeling height = 153 cm
standing height =
6. kneeling height = 116.1 cm
standing height =
7. Do your results support or disprove Leonardo da Vinci’s observation?
Explain.
8. Suppose someone’s kneeling height is 141 cm. Explain how you can estimate
his or her standing height.
314 8-6 Direct Variation
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Draw Conclusions
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analyze, and solve problems related
to linear equations…
Learn It!
Direct Variation (Student Textbook pp. 369–373)
Lesson Objective
Recognize direct variation by graphing tables of data and checking for
constant ratios
Vocabulary
direct variation
constant of variation
Example 1
Determine whether the data set shows direct variation.
Adam’s Growth Chart
Age (mo)
3
6
9
12
Length (in.)
22
24
25
27
Method 1 Make a graph that shows the relationship
between Adam’s age and his length.
28
Method 2 Compare ratios to see if a
26
direct
27
22 ___
__
3
12
Length (in.)
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A.
occurs.
≠
24
22
20
0
The ratios are
.
0
3
6
9
12
Age (mo)
15
Both methods show the relationship is
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8-6 Direct Variation 315
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Time (min)
10
20
30
40
Distance (mi)
25
50
75
100
Method 1 Make a graph.
50
25 __
__
20
10
Method 2 Compare ratios.
Distance Traveled by Train
Distance (mi)
Determine whether the data set shows direct variation.
B.
Distance Traveled by Train
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150
125
100
75
50
25
10 20 30 40 50
Time (min)
The ratios are
.
a direct variation.
Both methods show the relationship
Check It Out!
1a.
D
Determine
whether the data sets show direct variation.
1b.
Kyle’s Basketball Shots
Medication Guidelines
Distance (ft)
20
30
40
Weight (lb)
60
70
80
Baskets
5
3
0
Dose (mg)
30
35
40
Fi d each
h equation of direct variation, given that y varies directly with x.
Find
A. y is 54 when x is 6
y = kx
with x.
y varies
=k·
Substitute for x and y.
=k
Solve for k.
y=
Substitute
for k in the original equation.
B. x is 12 when y is 15
y = kx
with x.
y varies
=k·
Substitute for x and y.
=k
Solve for k.
y=
Substitute
316 8-6 Direct Variation
for k in the original equation.
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Example 2
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F
Find the equation of direct variation, given that y varies
d
directly with x.
2a. y is 7 when x is 3
2b. y is 18 when x is 12
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Example 3
M P
Mrs.
Perez h
has $4000 in a CD and $4000 in a money market account.
The amount of interest she has earned since the beginning of the year is
organized in the following table. Determine whether there is a direct variation
between either data set and time. If so, find the equation of direct variation.
Time (mo)
Interest from CD ($)
Interest from Money Market ($)
0
0
0
1
17
19
2
34
37
3
51
55
4
68
73
A. interest from CD and time
34 = 17
_____
interest
from CD ___
17 =
_______________
1
time
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All the ratios are equivalent to
, and (0,
The equation of direct variation is
_____ =
_____ =
4
) is included.
.
B. interest from money market and time
interest
from MM
_____________
time
The ratios
Check It Out!
19 ≠ ______
______
2
equal, so there
a direct variation.
D
Determine whether there is a direct
vvariation between either data set and
the number of cards. If so, find the
equation of direct variation.
3a. cards and retail
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Cards
Retail ($) Shipping ($)
200
20
5
300
30
6
400
40
7
3b. cards and shipping
8-6 Direct Variation 317
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Date
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LA.8.2.2.3 The student will organize
information to show understanding or
relationships…
Direct Variation
Think and Discuss
1. Describe the slope and the y-intercept of a direct variation equation.
2. Compare and contrast proportional and non-proportional linear relationships.
3. Get Organized Complete the graphic organizer. Fill in the boxes by writing
an example of an equation that is a direct variation. Then provide a table and
graph for the direct variation.
Equation
Direct
Variation
Copyright © by Holt McDougal. All rights reserved.
Table
Graph
y
5
x
-5
0
-5
318 8-6 Direct Variation
5
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8-6
Date
MA.8.A.1.1 …Interpret tables,
graphs, and models to represent,
analyze, and solve problems related
to linear equations…
Practice It!
Direct Variation
Make a graph to determine whether the data set shows direct variation.
1.
x
y
4
3
8
6
0
0
-4
-3
y
6
4
2
x
-6
2. Write the equation of direct variation for Exercise 1.
-4
O
-2
2
4
6
-2
-4
-6
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Find each equation of direct variation, given that y varies directly with x.
3. y is 27 when x is 3
4. y is 8 when x is -40
5. y is -54 when x is -12
6. y is 21 when x is 49
7. y is -31.5 when x is 14
8. y is 180 when x is -216
9. Bridgett’s gross pay per year is $40,300. Her net pay is $27,001 as a result of
her payroll deductions. Last year Bridgett’s gross salary was $37,700. Her net
pay was $25,259. Write an equation of direct variation indicating Bridgett’s
net pay y as a function of gross pay x.
10. The weight of an object on Venus is directly proportional to its weight on Earth.
A 110-pound person on Earth would weigh 99 pounds on Venus. Write an
equation of direct variation indicating a person’s weight on Venus as a function
of weight on Earth.
8-6 Direct Variation 319
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Date
MA.8.A.1.1 …Interpret tables,
graphs, and models to represent,
analyze, and solve problems related
to linear equations…
Apply It!
Direct Variation
The table shows how many gallons of water
are wasted per year by a leaking faucet. Use
the table for 1−3.
Number of Drops
per Minute
20
30
60
Gallons of Water
Wasted per Year
694
1041
2082
5. Julianne makes the following statement:
“In 2 seconds, the object falls 64 feet. So
in 4 seconds, the object must fall twice
that distance, or 128 feet.” Do you agree or
disagree with this statement? Explain.
1. Is there a direct variation between the data
sets? Why or why not?
2. Suppose you have a faucet that leaks at the
rate of 16 drops per minute. How many
gallons of water does this waste per year?
The table shows the distance in feet traveled
by a falling object in various amounts of time.
Use the table for 4−5.
Time (s)
Distance (ft)
0.5
1
1.5
2
4
16
36
64
4. Is there a direct variation between the
data sets? If so, write the equation of direct
variation. If not, explain why there is no
direct variation.
320 8-6 Direct Variation
Time (min)
15
18
27
33
Capacity
425
510
765
935
Explain how to estimate the maximum
number of people that can ride Kumba
each day. Be sure to state any assumptions
that you make.
Copyright © by Holt McDougal. All rights reserved.
3. The leaky faucet in Mike’s kitchen wastes
115_23 gallons each month (not each year). At
what rate does the faucet drip?
6. Extended Response Kumba is a
roller coaster at Tampa’s Busch Gardens
amusement park. The table shows the
maximum number of people that can ride
Kumba in various amounts of time.
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8-7
Date
MA.8.A.1.4 Identify the solution to
a system of linear equations using
graphs.
Using Graphs and Tables to Solve Linear Systems
Explore Graphs of Linear Systems
In this activity, you will explore the graphs of pairs of
linear equations.
y
Activity 1
4
1 Use a straightedge to draw a pair of lines on the coordinate plane
so that the lines intersect at a point with integer coordinates and
each line intersects at least one other point with integer coordinates.
2
x
-4
2 Write the equations for the lines that you graphed.
4
2
-2
-2
Equation 1
-4
Equation 2
3 Write the coordinates of the point of intersection.
Copyright © by Holt McDougal. All rights reserved.
4 Is the ordered pair that you wrote in Step 3 a solution of Equation 1?
5 Is the ordered pair that you wrote in Step 3 a solution of Equation 2 ?
Try This
y
1. Write the equations of the lines graphed at the right.
Equation 1
4
Equation 2
2. The point of intersection is (1, -1). Show that this ordered pair
is a solution of both of these equations.
2
x
-4
2
-2
-2
4
(1, -1)
-4
8-7 Using Graphs and Tables to Solve Linear Systems 321
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y
Activity 2
1 Parallel lines have the same slope. Graph two parallel lines.
Make sure that you can write the equations for both lines.
6
4
2 Write the equations for the lines that you graphed.
2
Equation 1
Equation 2
x
-4
-2
O
2
4
2
4
-2
3 Do you think it is possible to find an ordered pair that is a
solution of both of these equations? Justify your answer.
-4
y
Try This
3. The graphs of lines y = -3x + 2 and y = -3x - 1
are parallel. Graph the lines, and determine whether
there is a point that is a solution of both equations. Justify your
answer.
4
2
x
-4
-2
O
-2
-4
Draw Conclusions
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4. Make a conjecture : Given two linear equations whose graphs intersect,
describe how you might be able to determine what ordered pair is a solution
of both equations.
5. Use your conjecture : Graph the linear equations y = x + 3
and y = -6x -4, and find an ordered pair that is a solution of
both linear equations. Show that it is a solution of both equations.
y
4
2
x
-4
2
-2
-2
-4
322 8-7 Using Graphs and Tables to Solve Linear Systems
4
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Date
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MA.8.A.1.4 Identify the solution to
a system of linear equations using
graphs.
Using Graphs and Tables to Solve Linear Systems (Student Textbook pp. 376–380)
Lesson Objective
Graph and solve systems of linear equations
Vocabulary
system of equations
solution of a system of equations
Example 1
Let t =
in hours.
Let d =
in nautical miles.
boat distance: d =
t+
cutter distance: d =
t
d
Distance (nautical mi)
Copyright © by Holt McDougal. All rights reserved.
A fishing boat leaves the harbor traveling east at 16 knots (nautical miles
per hour). After it travels 40 nautical miles, a Coast Guard cutter follows
the boat, traveling at 26 knots. After how many hours will the Coast Guard
cutter catch up with the fishing boat?
Graph each equation.
The point of intersection appears to be
.
150
135
120
105
90
75
60
45
30
15
t
1 2 3 4 5 6 7 8 9 10
Check
Time (h)
t
d = 16t + 40
d
(t, d )
t
d = 26t
d
(t, d )
3
16(3) + 40
88
(3, 88)
3
26(3)
78
(3, 78)
4
16(4) + 40
104
(4, 104) ✔
4
26(4)
104
(4, 104) ✔
5
16(5) + 40
120
(5, 120)
5
26(5)
130
(5, 130)
The Coast Guard cutter will catch up after
from the harbor.
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, 104 nautical miles
8-7 Using Graphs and Tables to Solve Linear Systems 323
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Widgets
1. A machine starts producing 50 widgets per hour. After it produces 50 widgets,
another machine starts producing 75 widgets per hour. After how many hours
will the machines have produced the same number of widgets?
250
200
150
100
50
1 2 3 4 5
Time (hr)
Example 2
Solve
linear system by graphing. Check your answer.
S
l each
h li
A. y = 2x - 7
3x + y = 3
Step 1: Solve
equations for y.
y = 2x - 7
Step 2: Graph.
y
3x + y = 3
4
x
-4
O
-2
2
4
-2
-4
-6
The lines appear to intersect at (
).
Check
y = 2x - 7
2(
=
3x + y = 3
)-7
✔
3(
)+(
)3
=3✔
324 8-7 Using Graphs and Tables to Solve Linear Systems
Lesson Tutorial Videos
Copyright © by Holt McDougal. All rights reserved.
2
y = -3x + 3
Explore It!
Learn It!
Summarize It!
Practice It!
Apply It!
Solve each linear system by graphing. Check your answer.
B. 2x + y = 9
y - 9 = -2x
Step 1: Solve
equations for y.
2x + y = 9
Step 2: Graph.
y - 9 = -2x
10
y=
8
y=
y
6
Check
4
-2x + 9 = -2x + 9
2
x
✔ always true
=
-1 O
-1
2
4
6
8
The lines are
has
Check It Out!
solutions.
Solve each linear system by graphing. Check your answer.
2a. y = -4x + 1
2
5x + y = - 1
Copyright © by Holt McDougal. All rights reserved.
, so the system
2b. 6y + 12 = -2x
x + 3y = - 6
20
y
y
1
16
-4
-2
O
x
1
12
-3
8
4
x
-4
-3
-2
-1
1
-2
Lesson Tutorial Videos
8-7 Using Graphs and Tables to Solve Linear Systems 325
Explore It!
8-7
Learn It!
Name
Summarize It!
Practice It!
Class
Summarize It!
Apply It!
Date
LA.8.2.2.3 The student will organize
information to show understanding or
relationships…
Using Graphs and Tables to Solve Linear Systems
Think and Discuss
1. Explain why finding the exact solution of a linear system of equations by
graphing may present a challenge.
2. Describe the solution of a system of linear equations, where the lines have
the same slope but different y-intercepts.
3. Get Organized Complete the graphic organizer. Fill in the boxes by
writing the steps for solving a system of linear equations by graphing. (You
may use any number of boxes.)
1.
2.
326 8-7 Using Graphs and Tables to Solve Linear Systems
3.
Copyright © by Holt McDougal. All rights reserved.
Solving a System of
Linear Equations by Graphing
Explore It!
Learn It!
Practice It!
Summarize It!
Name
Apply It!
Class
8-7
Date
Practice It!
MA.8.A.1.4 Identify the solution to
a system of linear equations using
graphs.
Using Graphs and Tables to Solve Linear Systems
Tell whether the ordered pair is a solution of the given system.
x + 3y = 6
1. (3, 1) 4x - 5y = 7
x + 3y = 6
2. (6, -2)
4x - 5y = 7
3x - 2y = 14
5x - y = 32
3x - 2y = 14
5x - y = 32
Solve each system by graphing. Check your answer.
y=x+6
4. y = - 3x + 6 Solution:
y=x+4
3. y = - 2x + 1 Solution:
y
y
4
8
2
6
x
-4
-2
O
2
4
4
Copyright © by Holt McDougal. All rights reserved.
-2
2
x
-4
-6
-4
-2
O
2
4
6
-2
5. Maryann and Carlos are each saving for new scooters.
So far Maryann has saved $9, and can earn $6 per hour
babysitting. Carlos has saved $3, and can earn $9 per
hour working at his family’s restaurant. After how
many hours of work will Maryann and Carlos have
saved the same amount? What will be that amount?
8-7 Using Graphs and Tables to Solve Linear Systems 327
Explore It!
8-7
Learn It!
Summarize It!
Name
Class
Apply It!
Apply It!
Practice It!
Date
MA.8.A.1.4 Identify the solution to
a system of linear equations using
graphs.
Using Graphs and Tables to Solve Linear Systems
Ari and Kate open savings accounts at the
same time. They each make regular weekly
deposits. The table shows the amount in each
account after various numbers of weeks. Use
the table for 1−6.
Number of Weeks
Amount in Ari’s Account ($)
2
3
4
4. How many weeks does it take until Ari
and Kate have the same amount in their
accounts? How much money is in each
account?
5
120 130 140 150
Amount in Kate’s Account ($) 50 70 90 110
5. After 14 weeks, who has saved more
money? How much more money has he or
she saved compared to the other person?
1. Write an equation that gives the amount y
in Ari’s account after x weeks.
6. How many weeks does it take until the total
amount in both accounts is $410?
2. Write an equation that gives the amount y
in Kate’s account after x weeks.
Distance from
Pensacola (mi)
400
350
300
250
200
150
100
50
1 2 3 4 5 6 7 8
Time (h)
328 8-7 Using Graphs and Tables to Solve Linear Systems
Copyright © by Holt McDougal. All rights reserved.
3. Graph the system of linear equations using
your equations from Problems 1 and 2.
Label the graphs so you know which graph
corresponds to which person’s account.
7. Short Response Carlos is driving on
a highway. The graph shows his distance
from Pensacola at various times. Jessica
enters the highway at the same time as
Carlos. Her distance y from Pensacola after
x hours is given by y = 25 + 50x. Will Jessica
pass Carlos on the highway? If so, when? If
not, why not?
Explore It!
Learn It!
Name
Summarize It!
Practice It!
Class
Apply It!
8-8
Date
MA.8.A.1.3 Use tables, graphs, and
models to represent, analyze, and
solve real-world problems related to
systems of linear equations.
Explore It!
Using Algebra to Solve Linear Systems
Explore Linear Systems With Tables
You can use a table of values to investigate a linear system.
Activity
In this activity, you will explore the linear system y = 2x + 5 and y = 5x - 7.
1 For each value of x in the table, evaluate y for each equation. Record the
y-values in the table.
x
y = 2x + 5
y = 5x - 7
0
y
10
8
1
6
4
y = 2x + 5
2
2
3
-8 -6 -4 -2
-2
y = 5x - 7
x
2 4 6 8
-4
4
Copyright © by Holt McDougal. All rights reserved.
16
14
12
-6
-8
5
2 What pattern do you see in the column of y-values for y = 2x + 5?
3 What pattern do you see in the column of y-values for y = 5x - 7?
4 What value of x gives the same value of y for both equations?
5 What is the solution of the system? Write the solution as an
ordered pair (x, y).
8-8 Using Algebra to Solve Linear Systems 329
Explore It!
Learn It!
Summarize It!
Practice It!
Apply It!
Try This
Solve each system by completing the table.
1. y = 3x - 13
y = 2x - 8
x
2. y = -2x + 4
y = 3x - 11
y = 3x - 13
y = 2x - 8
x
1
1
2
2
3
3
4
4
5
5
y = -2x + 4
3. The table below does not show a solution to the linear system:
y = 3x - 11
y = 3x + 2
y = 4x - 5
Use patterns to continue the y-values in each column.
What is the solution?
y = 3x + 2
y = 4x - 5
0
2
-5
1
5
-1
2
8
3
3
11
7
Draw Conclusions
4. List some advantages and disadvantages of using tables to solve
a linear system.
330 8-8 Using Algebra to Solve Linear Systems
Copyright © by Holt McDougal. All rights reserved.
x
Learn It!
Explore It!
Summarize It!
Name
Practice It!
Apply It!
Class
8-8
Date
MA.8.A.1.3 Use tables, graphs, and
models to represent, analyze, and
solve real-world problems related to
systems of linear equations.
Learn It!
Using Algebra to Solve Linear Systems (Student Textbook pp. 381–385)
Lesson Objective
Solve systems of equations
Example 1
Solve
S
l each
h system of equations.
B. y = 2x + 9
y = -8 + 2x
A. y = 4x - 6
y=x+3
y = 4x - 6
y=x+3
y = 2x + 9
y = -8 + 2x
=
=
Solve the equation to find x.
4x - 6
=
x+3
3x - 6
=
3x
=
x
=
-
3
≠ -8
+
+
The system of equations has
solutions.
You can also see in a graph that this
To find y, substitute
of the original equations.
for x in one
solutions.
system has
y
8
y=x+3
=
.
2x + 9 = -8 + 2x
-
-
Copyright © by Holt McDougal. All rights reserved.
Solve the equation to find
4
+3
x
=
The solution is
-8
.
-4
O
4
8
-4
-8
Lesson Tutorial Videos
8-8 Using Algebra to Solve Linear Systems 331
Learn It!
Explore It!
Check It Out!
Summarize It!
Practice It!
Apply It!
Solve each system of equations.
1a. y = x - 5
y = 2x - 8
1b. y = 2x
y=x+6
Example 2
Solve each system of equations.
A. x + 4y = -10
x - 3y = 11
Solve both equations for x.
x + 4y = -10
x - 3y = 11
-
+
x
=
-
x
- 10 - 4y =
+
+
Add
to both sides.
= 11 +
-
Subtract
from both sides.
= 7y
7y
-21 = ______
______
Divide both sides by 7.
=y
To find x, substitute
x + 4y
= -10
x + 4(
) = -10
x
for y in one of the
equations.
= -10
x=2
The solution is (
)
332 8-8 Using Algebra to Solve Linear Systems
Lesson Tutorial Videos
Copyright © by Holt McDougal. All rights reserved.
-
=
11 + 3y
+
-10
+
Explore It!
Learn It!
Summarize It!
Practice It!
Apply It!
Solve the system of equations.
B. -2x + 10y = -8
x - 5y = 4
Solve both equations for y.
-2x + 10y = -8
+
x - 5y = 4
+
-
10y = -8 + 2x
- 5y = 4 - x
10y
+ 2x
______
_______
= -8
-5y
4-x
______
= ______
y = ______
+ ______
x
10
10
y = ______
+ _____
x
-5
-5
y = ______ x - ______
y = ______ x - ______
4
1 x - __
4 = __
1 x - __
__
5
5
5
5
-
-
Subtract
from both sides.
Copyright © by Holt McDougal. All rights reserved.
- __45 = - __45 always true
solutions.
The system of equations has
Check It Out!
Solve each system of equations.
2a. 2x + y = 0
2x + 3y = 8
Lesson Tutorial Videos
2b. y = x - 1
-3x + 3y = 4
8-8 Using Algebra to Solve Linear Systems 333
Explore It!
8-8
Learn It!
Summarize It!
Name
Practice It!
Class
Apply It!
Date
Summarize It!
LA.8.2.2.3 The student will organize
information to show understanding or
relationships…
Using Algebra to Solve Linear Systems
Think and Discuss
1. Discuss the steps for using algebra to solve a system of equations.
2. Describe different ways to check that (-1, 0) is a solution of the system of
equations below.
x + 2y = -1
-3x + 4y = 3
Solving a System of
Linear Equations
Solve by
Graphing
Solve Using
Algebra
Advantage
Advantage
Disadvantage
Disadvantage
334 8-8 Using Algebra to Solve Linear Systems
Copyright © by Holt McDougal. All rights reserved.
3. Get Organized Complete the graphic organizer. Fill in the boxes by writing
an advantage and a disadvantage of each method of solving a system of linear
equations.
Explore It!
Learn It!
Name
Summarize It!
Practice It!
Class
Apply It!
8-8
Date
MA.8.A.1.3 Use tables, graphs, and
models to represent, analyze, and
solve real-world problems related to
systems of linear equations.
Practice It!
Using Algebra to Solve Linear Systems
Copyright © by Holt McDougal. All rights reserved.
Solve each system of equations.
1. y = x - 2
y = 4x + 1
2. y = x - 4
y = -x + 2
3. y = 3x + 1
y = 5x - 3
4. y = 3x - 1
y = 2x + 2
5. 2x - y = 6
x + y = -3
6. 2x + y = 8
y=x-7
7. 2x + 3y = 0
x + 2y = -1
8. 3x - 2y = 7
x + 3y = -5
9. -2x + y = 0
5x + 3y = -11
10. _12x + _13y = 5
_1 x + y = 10
4
11. The length of a rectangle is 3 more than its width. The perimeter of the rectangle
is 58 cm. Write and solve a system of equations to find the rectangle’s dimensions.
12. At a craft fair, all candles cost the same amount, and all necklaces cost the same
amount. Bianca bought 3 candles and 2 necklaces for a total of $19.50. Irina
bought 5 candles and 1 necklace for a total of $18.50. Find the cost of each candle
and the cost of each necklace.
8-8 Using Algebra to Solve Linear Systems 335
Explore It!
8-8
Learn It!
Summarize It!
Name
Practice It!
Class
Apply It!
Date
MA.8.A.1.3 Use tables, graphs, and
models to represent, analyze, and
solve real-world problems related to
systems of linear equations.
Apply It!
Using Algebra to Solve Linear Systems
After college, Julia is offered two different
jobs. The table summarizes the pay offered
with each job. Use the table for 1−5.
Job
Yearly Salary
Yearly Increase
A
$20,000
$2500
B
$25,000
$2000
6. Two of Florida’s smallest towns are Indian
Creek and Bascom. As of 2007, the total
population of the two towns is just 170.
There are 52 more people in Bascom than
in Indian Creek. What are the populations
of the two towns?
Write an equation for each job that gives the
pay y after x years.
1. Job A
2. Job B
3. Is (8, 40,000) a solution of the system of
equations you wrote in Problems 1 and 2?
Why or why not?
5. No matter which job Julia takes, she plans to
stay at the job for only a few years. Assuming
pay is the only consideration, which job
should Julia take? Why?
8. Wei bought pens and notebooks. The pens
cost $0.75 each and the notebooks cost
$2.25 each. She spent a total of $14.25 and
bought 9 items altogether. How much did
she spend on pens? on notebooks?
9. Gridded Response
Tyrell has some nickels
and dimes. He has
20 coins all together,
and the total value of
the coins is $1.60. What
is the ratio of dimes to
nickels? (Hint: Express
the ratio as an improper
fraction.)
336 8-8 Using Algebra to Solve Linear Systems
Copyright © by Holt McDougal. All rights reserved.
4. Solve the system of equations you wrote in
Problems 1 and 2. Explain what the solution
means.
7. Brad sold tickets to a school play. He sold
adult tickets for $6.50 each and children’s
tickets for $4 each. He sold a total of 26
tickets and collected a total of $129. How
much money did he make from selling the
children’s tickets?
8-5
Name
Class
Date
THROUGH
8-8
Got It?
Ready to Go On?
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Quiz for Lessons 8-5 through 8-8
8-5
Point-Slope Form (Student Textbook pp. 365–368)
Write the point-slope form of the equation with the given slope that
passes through the indicated point.
1. slope -3, passing through (7, 2)
8-6
2. slope 2, passing through (-5, 3)
Direct Variation (Student Textbook pp. 369–373)
Find each equation of direct variation, given that y varies directly with x.
3. y is 10 when x is 2
4. y is 16 when x is 4
5. y is 2.5 when x is 2.5
6. y is 2 when x is 8
8-7
Using Graphs and Tables to Solve Linear Systems (Student Textbook pp. 376–380)
Solve each linear system by graphing. Check your answer.
Copyright © by Holt McDougal. All rights reserved.
9. y - 1 = 2x
-y = -2x - 1
8. y = 3x + 2
y = 3x - 2
10. 2y = 8
3y = 2x + 6
11. A balloon begins rising from the ground at the rate of 4 meters
per second at the same time a parachutist’s chute opens at a
height of 200 meters. The parachutist descends at 6 meters per
second. Graph to find the time it will take for them to be at the
same height and find that height.
Height (m)
7. y = x - 1
y = 2x - 3
220
200
180
160
140
120
100
80
60
40
20
5 10 15 20 25 30
8-8
Time (s)
Using Algebra to Solve Linear Systems (Student Textbook pp. 381–385)
Solve each system of equations.
12. y = -3x + 2
y = 4x - 5
13. y = 5x - 3
y = 2x + 6
14. 2x + y = 12
3x - y = 13
15. 4x - 3y = 33
x = -4y - 25
16. The sum of two numbers is 18. Their difference is 8.
a. If the numbers are x and y, write a system of equations to
describe their sum and their difference.
b. Solve the system to find the numbers.
Chapter 8 Linear Functions 337
8-5
THROUGH
8-8
Name
Class
Date
Connect It!
MA.8.A.1.1; MA.8.A.1.2;
MA.8.A.1.3; MA.8.A.1.4
Connect the concepts of Lessons 8-5 through 8-8
Saving for Surfing
Tyler and Maria want to buy surfboards. They
both begin putting some money into their savings
accounts each week.
1. The table gives data on Tyler’s savings. Does the
data show direct variation? Why or why not?
3. Maria’s total savings y are given by the equation
y = 90 + 15x, where x is the number of weeks.
What are the slope and y-intercept of the
equation’s graph? What do these represent?
Total Savings ($)
2. Show how to use point-slope form to find an
equation for Tyler’s savings.
Number of
Weeks (x)
2
5
6
8
Total
Savings (y)
50
125
150
200
300
275
250
225
200
175
150
125
100
75
50
25
1 2 3 4 5 6 7 8 9 10 11 12
Number of Weeks
4. Use a graph to find out when Tyler and Maria will both have
the same amount in their account. What will that amount be?
5. Surfboards cost $300. Who will be the first to have enough saved?
All Systems Go!
Arrange the numbers 2, 3, 4, and 5 in the four boxes. Your goal
is to make a system of equations that has the solution (2, 3).
x+
y = 18
Think About the Puzzler
x+
y = 19
1. What strategy did you use to help you arrange the numbers?
338 Chapter 8 Linear Functions
Copyright © by Holt McDougal. All rights reserved.
0
FLORIDA
Name
Class
Study It!
Vocabulary
CHAPTER
Date
8
Multi-Language
Glossary
Go to thinkcentral.com
(Student Textbook page references)
constant of variation . . . . (369)
point-slope form . . . . . . . . (365)
slope-intercept form . . . . (357)
direct variation . . . . . . . . . (369)
rate of change . . . . . . . . . . (347)
function notation . . . . . . . (361)
rise . . . . . . . . . . . . . . . . . . . . (351)
solution of a system
of equations . . . . . . . . . . . . (376)
linear equation . . . . . . . . . (346)
run . . . . . . . . . . . . . . . . . . . . (351)
system of equations . . . . . (376)
linear function . . . . . . . . . (361)
slope . . . . . . . . . . . . . . . . . . . (351)
x-intercept . . . . . . . . . . . . . (356)
y-intercept . . . . . . . . . . . . . (356)
Complete the sentences below with vocabulary words from the list above.
1. y = mx + b is the
of a line, and y - y1 = m(x - x1) is the
.
2. Two variables related by a constant ratio are in
Lesson 8-1
.
Graphing Linear Equations
G
MA.8.A.1.6, MA.8.A.1.1,
MA.8.A.1.5
((Student Textbook pp. 346–350)
Copyright © by Holt McDougal. All rights reserved.
Graph y = x - 2. Tell whether it is linear.
x
y
-1
-3
0
2
-1
2
0
y
x
-2
-2
1
Determine whether the rates of change
are constant or variable.
+2 +4 +1
2
4
-2
x
-2
0
4
y
1
5
13 15
-4
+4
y = x - 2 is linear; its graph is a line.
+8
5
8 = __
4 = __
2=2
__
4
2
1
+2
The rates of change are constant.
Graph each equation. Tell whether it is linear.
4. y = - 2x2
3. y = 4x - 2
x
y
y
2
-1 -6
0
-2
1
2
2
6
x
-2
-2
2
0
y
y
-1 -2
x
O
5. y = 2 -3x
0
1
-2
2
-8
Lesson Tutorial Videos @ thinkcentral.com
x
2
-2
-2
-4
x
y
-1
5
0
2
1
-1
2
-4
y
2
x
-2
O
2
-2
Chapter 8 Linear Functions 339
Lesson 8-2
Slope of a Line (Student Textbook pp. 351–355)
S
MA.8.A.1.2, MA.8.A.1.1
Find the slope of the line that passes through (-1, 2) and (1, 3).
Let (x1, y1) be (-1, 2) and (x2, y2) be (1, 3).
y2 - y1
3-2
1
1
______
_______
__
__
x2 - x1 = 1 - (-1) = 2 The slope of the line that passes through (-1, 2) and (1, 3) is 2 .
Find the slope of the line that passes through each pair of points.
6. (4, 2) and (8, 5)
Lesson 8-3
7. (4, 3) and (5, -1)
8. (-5, 7) and (-1, -2)
Using Slopes and Intercepts (Student Textbook pp. 356–360)
U
MA.8.A.1.2,
MA.8.A.1.1
Write 3x + 4y = 12 in slope-intercept form. Identify the slope and y-intercept.
3x + 4y = 12
y = - __34x + 3
m = - __34 b = 3 The slope is - __34 and the y-intercept is 3.
Write each equation in slope-intercept form. Identify the slope and y-intercept.
9. 5y = 6x - 10
10. 2x + 3y = 12
Lesson 8-4
LLinear Functions (Student Textbook pp. 361–364)
MA.8.A.1.2, MA.8.A.1.1
Write the rule for each linear function.
y
-2
-10
-1
-3
0
4
1
11
y
The y-intercept b is f (0) = 4.
The y-intercept b is 2.
2
-10 -(-3)
-7 = 7
m = _________ = ___
-1
-2-(-1)
-1 = __
1
m = ___
-2
2
x
2
-2
The rule is f (x)= 7x + 4 .
The rule is f (x) =__12x + 2.
-2
Write the rule for each linear function.
11.
x
y
-2
-3
-1
-2
0
-1
1
0
Lesson 8-5
12.
4
y
2
x
-2
O
2
4
-2
Point-Slope Form (Student Textbook pp. 365–368)
P
MA.8.A.1.2, MA.8.A.1.1
Write the point-slope form of the line with slope -4 that passes through (3, -2).
y - y1 = m(x -x1)
y - (-2) = -4(x -3)
y + 2 = -4(x-3)
Substitute 3 for x1,
Point-Slope Form
340 Chapter 8 Linear Functions
2 for y1, and
4 for m.
Lesson Tutorial Videos @ thinkcentral.com
Copyright © by Holt McDougal. All rights reserved.
x
Write the point-slope form of each line with the given conditions.
13. slope -4, passes through (-2, 3)
Lesson 8-6
14. passes through (0, -3) and (-6, 2)
Direct Variation (Student Textbook pp. 369–373)
D
MA.8.A.1.1
Find the equation of direct variation if y varies directly with x, and
y is 32 when x is 4.
y = kx
32 = k · 4
8=k
y varies directly with x.
Substitute 4 for x and 32 for y.
Solve for k.
Substitute 8 for k in the direct variation
equation.
y = 8x
Find each equation of direct variation, given that y varies directly as x.
15. y is 42 when x is 7
Lesson 8-7
16. y is 8 when x is 56.
Using Graphs and Tables to Solve Linear Systems
U
MA.8.A.1.4
MA.8.A.1.3
((Student Textbook pp. 376–380)
Solve the linear system by graphing. Check your answer.
y
4y - 12 = x
4y - 3x = 4
Solve for y and graph.
y = __14 x + 3
Check
4
(4, 4)
2
x
y = __34 x + 1
2
4
x
y
x
y
3
3_34
3
3_14
4
4✔
4
4✔
5
4_14
5
4_34
Copyright © by Holt McDougal. All rights reserved.
Solve each linear system by graphing. Check your answer.
17. 2y + 2x = 6
y = -x
18. x = -y + 4
y - 4 = -x
y
4
y
4
2
2
x
-4
O
-2
2
x
4
-2
-2
Lesson 8-8
Using Algebra to Solve Linear Systems
U
O
2
4
6
MA.8.A.1.3, MA.8.A.1.4
(Student
(
Textbook pp. 381–385)
Solve the system of equations.
1. Solve the equations for one variable.
2. Set the expressions for y (or x) equal to each other, and solve for x (or y).
3. Substitute for x (or y), and solve for y (or x).
Solve each system of equations.
19. 2x - y = -2
x+y=8
20. 4x + y = 10
x - 2y = 7
Lesson Tutorial Videos @ thinkcentral.com
21. y = x - 2
-x + y = 2
Chapter 8 Linear Functions 341
Name
Class
Write About It!
Date
LA.8.3.1.2 The student will prewrite
by making a plan for writing that
addresses purpose, audience, main
idea, logical sequence, and time
frame for completion
Think and Discuss
Answer these questions to summarize the important concepts from Chapter 8
in your own words.
1. Tell whether y = -5x + 4 is a linear equation. Explain.
2. Explain how to find the slope of a line that passes through (4, 9) and (–1, 8).
3. Explain how to write the point-slope form of a line with slope 6 passing through (5, -2).
Before the Test
I need answers to these questions:
342 Chapter 8 Linear Functions
Copyright © by Holt McDougal. All rights reserved.
4. Explain the three solution possibilities and their meaning when graphing a system
of two linear equations.