What You've Learned
California Content Standards
5.0 Solve multi-step problems, including word problems, involving linear
equations in one variable and provide justification for each step.
16.0 Understand the concepts of a relation and a function and give pertinent
information about given relations and functions.
18.0 Determine whether a relation defined by a graph, a set of ordered pairs,
or a symbolic expression is a function and justify the conclusion .
I • for Help to the Lesson in green.
Adding and Subtracting Real Numbers
(Lessons 1-4 and 1-5}
Simplify each expression.
1. -5 + 7
2. 2 - ( -3)
3.
-i + ~
4. 11 + ( -4)
5. 11 - 81
6.
l-2- 41
Solving Equations
(Lesson 2-3}
Solve each equation. Check your solution.
7. 3x + 4x = 8 - x
8. 12- 3d= d
Transforming Equations
9. 6x - 8 = 7 + x
(Review page 90}
Solve each equation for y.
10. 2y -
X
11. 3x = y + 2
= 4
Identifying Coordinates
12. -2y - 2x = 4
(Lesson 4-1}
Name the coordinates of each point on the graph at the right.
13. A
14. B
c
16. D
17. E
18. F
15.
Graphing Functions
y
•E
r4X
D
-4
-2
B• -2
•c
(Lesson 4-3}
Make a table of values and graph each function.
19. y = -~x
228
Chapter 5
20. y = 2x + 1
A•
2. F
21. y
=X-
5
.A. A glass pyramid covers the
entrance to the Louvre
Museum in Paris, France.
Architects use slopes of
parallel lines when designing
buildings.
What You'll Learn Next
California Content Standards
6.0 Graph a linear equation and compute the x- andy-intercepts.
7.0 Verify that a point lies on a line, given an equation of the line. Derive
linear equations by using the point-slope formula.
8.0 Understand the concepts of parallel lines and perpendicular lines and how
those slopes are related. Find the equation of a line perpendicular to a given
line that passes through a given point.
·)~
•
•
•
•
•
linear equation (p. 239)
linear function (p. 239)
linear parent function (p. 239)
negative reciprocal (p. 260)
parallel lines (p. 259)
•
•
•
•
•
parent function (p. 239)
perpendicular lines (p. 260)
point-slope form (p. 252)
rate of change (p. 230)
slope (p. 232)
ish Audio Online
•
•
•
•
slope-intercept form (p. 240)
standard form (p. 246)
x-intercept (p. 246)
y-intercept (p. 239)
Rate of Change and Slope
California Content Standards
6.0 Graph a linear equation and compute the x- and y-intercepts.lntroduce
7.0 Derive linear equations by using the point-slope formula. Introduce
8.0 Understand the concepts of parallel lines and perpendicular lines and
how those slopes are related. Introduce
What You'll Learn
• To find rates of change from
tables and graphs
Evaluate each function rule for x
= -5.
3. y = -~x + 3
+5
2. y = 2x
l.y=7 - x
• To find slope
for Help Lessons 4-3 and 1-5
@ Check Skills You'll Need
Write in simplest form .
. . . And Why
To find the rate of change of
an airplane's altitude, as in
Example 2
7 - 3
4· 3 - 1
6. 8 :- ( ~4)
5. 63 - 5
- 0
6
8. - :- ( ~4)
-1 - 2
7. ---o-=-5
II>~ New Vocabulary • rate of change
0- 1
9. 10
• slope
r--·~-·-··-----·-<>····.··--,.----·-------------------
Finding Rates of Change
CA Standards
Investigation
Exploring Rate of Change
The diagram at the right shows
the side view of a ski lift.
100
,...-...
1. What is the vertical change from
A to B? From B to C? From C
toD?
.....
Q)
~
'--"
80
1:::::
.s
.t::
rJ)
60
0
p..
2. What is the horizontal change
from A to B? From B to C?
~
.~
40
.....
l-<
Q)
>
FromD to D?
3. Find the ratio of the vertical
change to the horizontal change
for each section of the ski lift.
20
0
20
40
60
80 100
Horizontal Position (feet)
4. Which section is the steepest?
How does the ratio for that section compare to the ratios of the other sections?
In the graph above, AB and BC have different rates of change.
A rate is a comparison of two
quantities measured in
different units.
Rate of change allows you to see the relationship between two quantities that are
changing. If one quantity depends on the other, then the following is true.
change in the dependent variable
rate of ch ange = .
230
Chapter 5
Linear Equations and Their Graphs
. .. . .
. .
. ..
t
'
I
i
I
l
J
Cost of Renting
a Computer
Rental
Number
of Days
Charge
Finding Rate of Change Using a Table
For the data at the left, is the rate of change for each pair of consecutive days the
same? What does the rate of change represent?
1
2
$60
$75
3
$90
4
$105
5
$120
--
change in cost
rate of change = change in number of days
_)
Cost depends on the number of days.
75 - 60 - 15
2 - 1 - 1
90 - 75 - 15
3 - 2 - 1
105 - 90 - 15
4 - 3 - 1
120 - 105 - 15
5 - 4
- 1
The rate of change for each consecutive pair of days is 1{. The rate of change is the
same for all the data. It costs $15 for each day a computer is rented after the first day.
@CA Standards Check
G) a. Find the rate of change using Days 5 and 2.
b. Critical Thinking Does finding the rate of change for just one pair of days mean
that the rate of change is the same for all the data? Explain.
The graphs of all the ordered pairs (number of
days, cost) in Example llie on a line as shown
at the right. So, the data are linear.
You can use a graph to find a rate of change.
Recall that the independent variable is plotted
on the horizontal axis and the dependent
variable is plotted on the vertical axis.
___.,_ 120
~ 100
::g 80
~ 60
~ 40
u 20
0
(/)
a
2 3 4 5
Days
vertical change
change in the dependent variable
rate of change = horizontal change = change in the independent variable
Finding Rate of Change Using a Graph
The graph shows the altitude of an airplane
as it comes in for a landing. Find the rate of
change. Explain what this rate of change means.
rate of _ vertical change
change- horizontal change
1000 - 0
=
60 _ 180
_ 1000
- -120
Divide the vertical change
by the horizontal change.
= -831
Simplify.
The rate of change is
@CA Standards Check
+--- change in altitude
+--- change in time
Airplane Landing
~2000
(I)
::;
~ 1000
,g
(60, 1000)
~
(180, O)
.
0
Use two pomts.
60
120
180
Time (seconds)
-81. The airplane descends 81 feet each second.
® Find the rate of change of the data in the graph.
The graph shows the distance traveled by an
automobile over time.
A Moving Automobile
]~
l
250
200 l L J
150
~
I
E-< ~ 100
.. ~ :
~
50
•
!=:"--"
0
.s
123456
(\j
1-c
___.,_
(/)
I
s
(/)
5
Lesson 5-1
Time (hours)
Rate of Change and Slope
231
,.--Flliding Slope
The slope of a line is its rate of change.
vertical change
slope = hnri •umto:~J
rise
run
l"h-:lnOP
Finding Slope Using a Graph
The grade of a road is the
ratio of rise to run expressed
as a percent. For example, a
road with 100% grade is at a
45° angle with level ground.
Find the slope of the line.
rise
slope = run
- ___2__::___1
- 4 - ( -1)
_z
-s
{i! CA Standards Check
The slope of the line is ~ .
•
Q) Find the slope of each line.
a.
b.
5
y
( -3, 2)
-2
01
2
X
-2
2
X
You can use any two points on a line to find its
slope. You use subscripts to distinguish between two
points. In the diagram, (x 1, y 1) are the coordinates of P,
and (x 2,y2) are the coordinates of Q. To find the slope of
PQ, you can use the following formula.
~
-
Formula
.
Slope
y2 - y1
slope = nse
=
where x 2 - x 1
run
x2 - xl '
=I=
0
When calculating slope, the x-coordinate you use first in the denominator must
belong to the same ordered pair as they-coordinate you use first in the numerator.
To set up the subtraction in
the slope formula, think of
moving from the coordinates
of 8 to the coordinates of A.
Finding Slope Using Points
7-1
Find the slope of the line through A( -2, 1) and B(6 , 7).
A{-2~
~
6,7)
slope
6- {-2)
y2- yl
= x
-
2
_ XJ
7-1
Substitute (6, 7) for (x2, Y2)
and (-2, 1) for (x1, Y1).
- 6-(-2)
_Q
Simplify.
- 8
•
:
{i! CA Standards Check
232
~
e The slope of AB is i or i·
4 } Find
the slope of the line through C(2, 5) and D(4, 7).
Chapter 5 Linear Equations and Their Graphs
You can also analyze the graphs of horizontal and vertical lines. The next
example shows why the slope of a horizontal line is 0, and the slope of a
vertical line is undefined.
Horizontal and Vertical Lines
Find the slope of each line.
y - y
a.
slope = - 2- -1
y
(1, 2)
01
-2
t
= 2-2
4- 1
Substitute (4, 2) for (x2, Y2)
and (1, 2) for (x1, Y1).
-3
=0
Simplify.
_Q
X
2
-2
The slope of the horizontal line is 0.
Video Tutor Help
Visit: PHSchool.com
Web Code: bae-0775
b.
T
J
,_ I.
I
slope =
(4,2 J
(4~ -l)t .
=
X
01
-2
-2
(iff CA Standards Check
y2 - yl
x2 - xl
Substitute (4, 2) for (x2, Y2)
and (4, -1) for (x1, Y1).
2 - ( -1)
4- 4
-- ~0
Simplify.
Division by zero is undefined. So, the slope
of the vertical line is undefined.
~ Find the slope of each line.
a.
-4
-2
01
y
b.
y
l
2
X
-6
-4
0
X
- 2
The following summarizes what you have learned about slope.
Slopes of Lines
Summary
A line with
positive slope
slants upward
from left to right.
X
~
~
1
~
01
X
y
_,
X
~
A line with
negative slope
slants downward
from left to right.
t
'~
~
A line with a
slope of 0 is
horizontal.
+
0
Lesson 5-1
X
A line with an
undefined slope
is vertical.
Rate of Change and Slope
233
24.0 Derive linear equations by using the point-slope formula. Develop
( -~~~------------~.---'--~-~---- ~- - ~~~~---~
-----------
-----~-----
You can use similar triangles to show that the slope of a line is the same no matter
which two points you use to calculate it. In the graph below, MBC ~ 6.DEF. Recall
that, in similar triangles, the ratios of the lengths of corresponding sides are equal.
So AB
BC _
- DE
EF , wh ere AB
BC an d DE
EF represent the slope o f t h e 1·me.
Use points A(4, 3) and C(O, 1).
_ 3 - 1_ 2 or 1
Slope -_ AB
BC- 4- o- 4
2
Use points D(2, 2) and F( -4, -1).
_ DE_ 2 - ( - 1) _ 3
1
slope- EF - 2 - ( -4) - 6 or 2
So the slope of the line is the same regardless of
which two points you use.
For more exercises, see Extra Skills and Word Problem Practice.
Practice by Example
Examples 1, 2
(page 231)
The rate of change is constant in each tabl~nd graph. Find the rate of change.
Explain what the rate of change means for ea'cb situation.
1.
for
Time
(hours)
Temperature
(Of)
1
2.
People
Cost
(dollars)
-2
2
7.90
4
7
3
11.85
7
16
4
15.80
10
25
5
19.75
13
34
6
23.70
Help
3.
4.
A Tank of Gas
Emissions: Generating
Electricity for TV Use
15
~
r:n4L
§
~~~3
oo'"O
~ ~10
.-e -~
,......
2
ro
g~5
. ,
~'--'
~
0
0
'
u·a &1
~ 0
. ...... :::::::1
1 2 3 4 5 6
Hours of Use
100 200 300
Miles Traveled
r:n
5.
l-<
Descent of a Skydiver
~
(1)
Vp...
~0~
.....
~t:
...c:~~
-~~
v ro
:r::
234
Chapter 5
l-<
ro
0...
'--'
2000 ~
IO O L
0
20
Price of Oregano
.....
r:n
.
~2~
t ·_
:
l-<
u0 ~0 1
~
r
;
+
+
0123456
40
60
Time (seconds)
Linear Equations and Their Graphs
6.
Weight (ounces)
Example 3
(page 232)
Find the slope of each line.
7.
T T
T .._ v
8.
r
9.
y
-6
-4
01
X
- 2
Example 4
(page 232)
Example 5
(page 233)
Find the slope of the line that passes through each pair of points.
10. (3 , 2) , (5 , 6)
11. (5 , 6) , (3 , 2)
12. (4,8) , (8, 11)
13. ( - 4, 4) , (2, - 5)
14. ( -2, 1), (1 , - 2)
15. ( - 3, 1), (3, - 5)
16. ( - 8, 0) , (1 , 5)
17. (0, 0) , (3 , 5)
18. ( -4, - 5) , ( - 9, 1)
19. (5 , 0) , (0, 2)
20. ( - 7, 1), (7, 8)
21. (0, - 1), (1, - 6)
State whether the slope is zero or undefined.
22.
23.
y
2
-6
-4
-2
0
24. (3' 4) ' ( - 3' 4)
Apply Your Skills
2
4
01
- 4 ~ -2
2
X
-2
X
25. (4, 3), (4, - 3)
26. ( - 5,
!),(- 5, 3)
Find the rate of change for each situation.
27. A baby is 18 in. long at birth and 27 in. long at ten months.
28. The cost of group museum tickets is $48 for four people and $78 for ten people.
29. You drive 30 mi in one hour and 120 mi in four hours.
Find the slope of the line passing through each pair of points.
30. ( -7, 1), (7, 8)
33.
(!,8) , (1, - 2)
31. ( 4, 1~), ( - 2, ~)
34. ( - 5,
32. (0, 3.5), ( - 4, 2.5)
!),(- 5, 3)
35. (0.5, 6.25), (3, -1.25)
Through the given point, draw the line with the given slope.
36. K(3 , 5)
slope - 2
'? 4 t
Piant
;
Growth
2,
:.::-3
i·v 1
2 B
:::c:
A
c
:
0123456
Time (weeks)
37. M(5 , 2)
slope -!
38.. Q( -2, 3)
slope~
39. R(2, - 3)
slope-~
40. a. Which line in the graph at the left is the steepest?
b. During the 6-week period, which plant had the greatest rate of change? The
least rate of change? How do you know?
41. a. Find the slope of the line through A( 4, - 3) and B(1 , - 5) using A for (x2 , y 2 )
and B for (x 1, y 1).
b. Find the slope of the line in part (a) using B for (x 2, y 2) and A for (x 1, y 1).
c. Critical Thinking Explain why it does not matter which point you use for
(x2 , y 2 ) and which point you use for (x 1, y 1) when you calculate a slope.
Lesson 5-1
Rate of Change and Slope
235
42. An extension ladder has a label that says, "Do not place base of ladder less than
5 ft from the vertical surface." What is the greatest slope possible if the ladder can
safely extend to reach a height of 12 ft? Of 18 ft?
43. Writing If two points on a line have positive coordinates, is the slope
necessarily positive? Explain.
Homework Video Tutor
Find the slope of the sides of each figure.
44.
4
45.
J
tY
,
R
~
p
Visit: PHSchool.com
Web Code: bae-050 1
K
X
4
M
X
s
46. a.
b.
c.
d.
L
Graph the direct variation y = -~x.
What is the constant of variation?
What is the slope?
What is the relationship between the constant of variation and the slope?
47. a. Name two points on a line with a slope of
i.
b. Name two points on a line with a slope of-!.
Each pair of points lies on a line with the given slope. Find x or y.
48. (2, 4), (x , 8); slope = -2
49. (2, 4), (x, 8); slope = -~
50. (4, 3), (x, 7); slope = 2
51. (x, 3), (2, 8); slope = -~
52. (-4,y),(2, 4y);slope = 6
53. (3, 5), (x, 2); undefined slope
Math Reasoning In Exercises 54-60, tell whether each statement is true or false.
If false, give a counterexample.
For a guide to solving
Exercise 54, see p. 238.
54. A rate of change must be either positive or zero.
55. All horizontal lines have the same slope.
56. A line with slope 1 always passes through the origin.
57. Two lines may have the same slope.
58. The slope of a line that passes through Quadrant III must be negative.
59. A line with slope 0 never passes through point (0, 0).
60. Two points with the same x-coordinate are always on the same vertical line.
61. The graph shows how much it costs to rent
carousel equipment.
a. Estimate the slope of the line. What does
that number mean?
b. Customers pay $2 for a ride.
What is the average number of customers
needed to cover the rental costs?
The rental cost of a carousel
varies depending on whether
the horses are 2 or 3 abreast.
236
Chapter 5
62. Error Analysis A friend says the slope of a line
passing through (1, 7) and (3, 9) is equal to
the ratio~ ~.What is your friend's error?
=
Linear Equations and Their Graphs
Rent for a Carousel
_..-. . 2000
<J)
!-<
ro
~ 1500
~
"§
1000
u
]
s::
<l)
~
500
0 E--------~
0 1 2 3 4 5 6 7 8
Hours
Challenge
Find the slope of the line passing through each pair of points.
63. (a, - b) , ( - a, - b)
64. ( -rn, n) , (3m, -n)
65. (2a, b) , ( c, 2d)
Do the points in each set lie on the same line? Explain your answer.
66. A(1 , 3) , B( 4, 2) , C( - 2, 4)
67. G(3, 5) , H( -1 , 3) , 1(7, 7)
68. D( -2, 3), E(O, - 1), F(2, 1)
69. P(4, 2), Q( -3, 2), R(2, 5)
70. G(1 , -2), H( - 1, - 5), 1(5, 4)
71. S(-3, 4), T(0, 2), X( - 3, 0)
·_:.:. M.Y~~.i."filf:! B~ffi)ic:fi! ·.~· ~ .~ .~~i.~fi!)1!;2 . .·.
·. ,.
c;i;·;" ··.; ;iy(\.;~i:ih·:;."':
'r;
~·
For California Standards Tutorials, visit PHSchool.com. Web Code: baq-9045
Alg1 7.0
72. The line shown on the graph contains the points
(-3, -2) and (2, 2). What is the slope of the line?
®
l
® ~
Alg116.0
CD
i
®
4
73. Which ordered pair could be removed from the relation below to change it into
a function?
{(2, 3) , (3, 2), (2, 5) , (5 , 4), (1 , 6) , ( 6, - 1), (25, 31)}
®
Alg118.0
(2, 5)
®
74. For the graph of y =
IxI +
CD (6, -1)
(25, 31)
®
(3 , 2)
1, which of the following statements is true?
®
®
The graph represents a function whose range is all real numbers.
The graph represents a function that assigns to each member of the
domain exactly one member of the range.
CD The graph does not represent a function.
® The graph represents neither a function nor a relation.
~1~~~~\;:J.Ievifi!·vv
for
Help
Lesson 4-4
:<,;~.~~"';i;&.v:tw'.~~;~;;':I.t:~~~;< {.:'t~i·: . ::;~·;J;
·.o·· .,:;·· ·.· . • ,.,,."'··· ·•
Write a function rule for each situation.
75. the total cost of renting a movie for n days if it costs $3.50/day
76. the total profit if supplies and wages cost $232, and each item q sells for $4.95
Lesson 3-6
Solve each equation. If there is no solution, write no solution.
11. Ic I
Lesson 2-2
+4= -3
78.
- 71 rn I
= - 28
79.
51 j 1- 6 =
19
Solve.
80. x + 3 + 2x = -6
81. 3(2t + 5) = -9
82. 9 = y + 2(4y - 5)
83. 4n - 7(n - 9) = 42
84. 2(7 - q) - 4 = 0
85. ~(p
nline Lesson Quiz Visit: PH School. com, Web Code: baa-0501
+ 10) = 0
237
Understanding Math Problems Read through the problem below. Then follow
along with what Vera thinks as she solves the problem. Check your understanding
with the exercises at the bottom of the page.
24.3 Use counterexamples to show that
an assertion is false and
recognize that a single
counterexample is sufficient
to refute an assertion .
Master
Tell whether the statement is true or false . If false, give a counterexample.
A rate of change must be either positive or zero.
What Vera Writes
What Vera fhittks
Yes, the rate of chat1ge for n, !J at1d (S, 4J
is positive.
First, I will determine whether or not a rate of
change can be positive.
4-Z Z 1
~ -1 = Z' or
Then I will determine whether or not a rate of
change can be zero.
~
Can a rate of change be something other than
positive or zero? Yes, a rate of change can be
negative or undefined. I only need to give one
counterexample to prove that the statement is
false.
[ I will write my answer in a sentence.
Yes, the rate of chat1ge for n, !J at1d
(4, !J is zero.
Z-Z
0 o
4-1 =}'or
fhe rate of chat1ge for (!, ~) at1d (4, -U is
t1either positive t1or zero.
-1- ~- -4
'I
4 - Z - T' or -~.-
)
•
fhe statetMettt is false. fhe rate of chat1ge
for (!, ~) at1d (4, -U is t1egative.
EXERCISES
Tell whether each statement is true or false. If false, give a counterexample.
1. A line always passes through three quadrants.
2. A vertical line always crosses they-axis.
3. You can use any two points on a line to determine its slope.
238
Guided Problem Solving
Understand ing Math Pro blems
Slope-Intercept Form
California Content Standards
6.0 Graph a linear equation. Develop
7.0 Verify that a point lies on a line, given an equation of the line. Develop
What You'll Learn
• To write linear equations in
slope-intercept form
• To graph linear equations
.. . And Why
To use a graph to relate total
cost to number of items
purchased, as in Example 5
for Help
@ Check Skills You'll Need
Lessons 1-2 and 1-6
Evaluate each expression.
1. 6a
+ 3 for a = 2
2. - 2x - 5 for x = 3
3. ~ x
+ 2 for x = 16
4. 0.2x
+ 2 for x = 15
6. - 4p
+ 9 for p = 2
5. 8 - 5n for n = 3
·>~ New Vocabulary • linear function • parent function • linear parent function
• linear equation
• y-intercept
• slope-intercept form
r--v\lriting Linear Equations
The word linear contains the
word "line."
In Lesson 4-5, you studied direct variations such as y = 3x. The graph of a
direct variation is a straight line. All direct variations are linear functions. A
linear function is a function that graphs a line. Direct variations are only part
of the family of linear functions. For example,
+ 1 is a linear function but not a direct
variation because it does not go through (0, 0).
y = -~x
A parent function is the simplest equation of a
function. The equation y = x or f(x) = x is the
linear parent function.
A linear equation is an equation that models a
linear function. In a linear equation, the variable
cannot be raised to a power other than 1. So y = 2x
is the equation of a linear function, but y = x 2 or y = 2x are not.
The equation of a line gives important information about
its graph. Consider the table and graph of y = -2x + 1.
y
X
-2x+ 1
0
-2(0) + 1
1
1
-2(1) + 1
-1
2
-2(2) + 1
-3
1
Two points on the line are (0, 1) and (2, -3). The slope is ~ ~ ~J) = -~or -2.
The y-intercept is they-coordinate of the point where a line crosses they-axis.
Since y = -2x + 1 crosses they-axis at (0, 1), they-intercept is 1.
If you know the slope of a line and its y-intercept, you can write the equation of
the line. The letter m refers to the slope.
Lesson S-2
Slope-Intercept Form
239
Slope-Intercept Form of a linear Equation
Definition
The slope-intercept form of a linear equation is y = mx + b.
t
slope
t
y-intercept
Identifying Slope and y-lntercept
What are the slope andy-intercept of y = 3x - 5?
y = mx
t
+ b
Use the slope-intercept form.
\
: y = 3x + (- 5)
(i/ CA Standards Check
Think of y = 3x - 5 as y = 3x
+ (- 5).
e The slope is 3; they-intercept is -5.
<D a. Find the slope andy-intercept of y = tx - ~·
b. Critical Thinking For the equation in Example 1, what happens to the graph of
the line and to the equation if they-intercept is moved down 3 units?
Writing an Equation
: Write an equation of the line with slope~ andy-intercept 6.
1
e; y
(i/ CA Standards Check
Use the slope-intercept form.
y= mx+ b
3
= gx + 6
Substitute
i for m and 6 for b.
® Write an equation of a line with slope m =
~ andy-intercept b =
-1.
Writing an Equation From a Graph
Which equation models the linear function shown
in the graph?
nline ~
active math
'";- t nd,
~
Hl l
;::;-::; ~~,---
I
_,;.~
I~
®
y = 2x-
CO Y = -~x + 2
@ y=2x-~
~x
Find the slope. Two points on the line are (0, 2)
and(4,-1).
4 - 0
4
They-intercept is 2. Write an equation in slope-intercept form.
-
- - ·=-·
.------..-"j;'j--.-,:--.:o--v-.-.-.:i .-.. .. ,
~·*
y =
slope = -l - 2 = _ _3_
. . .,. ·="""
L-...........-.
-
-~x + 2
®
y= mx+ b
For: Slope-Intercept Activity
Use: Interactive Textbook, 5-2
(i/ CA Standards Check
y =
-~ x + 2
o The equation is y
Substitute
-i for m and 2 for b.
= -~x + 2. So the answer is A.
Q) a. Write the equation of the line using the points (0, 1)
and (2, 2).
b. Critical Thinking Does the equation of the line
change if you use ( -2, 0) instead of (2, 2)? Explain.
240
Chapter 5
Linear Equations and Their Graphs
X
Each point on the graph of an equation is an ordered pair that makes the equation
true. To verify that a point lies on a line, substitute its coordinates for x and y in the
equation. If doing so gives a true statement, then the point lies on the line.
Verifying Points on a Line
a. Determine whether (3, - 5) lies on the graph of y = - 3x + 4.
+4
- 3(3) + 4
- 9+4
y = - 3x
-5
:1::.
-5
:1::.
Substitute 3 for x and -5 for y.
Multiply.
- 5 = - 5 .I
Add.
(3 , - 5) is a solution.
b. Determine whether (8, 4) lies on the graph of 3y = 2x- 1.
3y = 2x - 1
3(4)
:1::.
2(8) - 1 Substitute 8 for x and 4 for y.
12
:1::.
16 - 1
Multiply.
12 =I= 15 X
Subtract.
(8, 4) is not a solution.
{iff CA Standards Check
@) a.
Determine whether (1 , 6) lies on the graph of y = 4x - 2.
b. Determine whether (5, -7) lies on the graph of 3y + 5x = 4.
The graph of a linear equation is a line that indicates all the solutions of the
equation. You can use the slope and y-intercept to graph a line.
Graphing Equations
You buy vegetables at a farmer 's market for $2 per pound. The equation y = 2x
represents the situation where x is the number of pounds andy is the total cost of
your purchase. Graph the equation.
Step 2
The slope is 2, or
Use the slope to plot
a second point.
Step 1
They-intercept is 0.
So plot a point at
(0, 0).
y
f
~
In the United States, tomatoes
cost an average of $1.22 per
pound.
21
-2
01
-2
{iff CA Standards Check
1-
~
j
~I
2
~
f.
t
I
~
+
I
I
trright
2
-
OT
~ -
Draw a line through
the two points.
1
up2
-2
X
t
~
.
Step 3
J
~
2
X
-2
X
-2
@ You buy various snacks for $1.50 per bag. You save $2 using your store rewards
card. The equation y = ~x - 2 represents the situation, where x is the number
of bags andy is the total cost of your purchase. Graph the equation.
lesson 5-2
Slope-Intercept Form
241
For more exercises, see Extra Skills and Word Problem Practice.
Practice by Example
Example 1
(page 240)
for
Help
Example 2
(page 240)
Example 3
(page 240)
Find the slope and y-intercept of each equation.
3. y =X-~
-1- x + 2
1. y = - 2x + 1
2. y =
4. y = 5x + 8
5. y = ~ x + 1
6. y = -4x
7. y = -x- 7
8. y = -0.7x- 9
9. y = -~x- 5
Write an equation of a line with the given slope andy-intercept.
10. m = ~, b = 3
11. m = 3, b = ~
12. m = ~, b = 3
13. m = 0, b = 1
14. m = -1, b = -6
15. m = -~, b = 5
16. m = 0.3, b = 4
17. m = 0.4, b = 0.6
18. m = -7, b =
1 b 19.m --5,
--52
1 b - 5
20.m --4,
-4
21.m -3,
- 8 b -3
- 2
i
Write the slope-intercept form of the equation for each line.
22.
23.
24.
-4
X
-2
25.
26.
27.
-2
Exa~ple 4
(page 241)
01
2
X
(page 241)
~
Determine whether the ordered pair lies on the graph of the given equation.
28. (-3,4);y = -2x + 1
29. (2,9);y = 6x- 3
31. (-5,6); -8x + 4y = 10 32. (-1,0); -4y = 2x + 3
Example 5
j
~
30. (4,3);2x- 5y = -7
33. (0, -2);6y = x- 3
Use the slope andy-intercept to graph each equation.
34. y =
1- X + 4
35. y = ~X
-
1
36. y = - 5x + 2
37. ·y = 2x + 5
38. y = ,x + 4
39. y = - x + 2
40. y
41. y = -~x
42. y =~X- 3
44. y = -~x + 4
45. y = -0.5x + 2
=
4x- 3
43. y = -~x + 2
46. You buy magnets online for $2 each. Shipping costs $5. The equation y = 2x + 5
represents the situation, where x is the number of magnets andy is the total cost
of your purchase. Graph the equation.
242
Chapter 5
Linear Equations and Their Graphs
'
J
Apply Your Skills
Find the slope and y-intercept of each equation.
=!
47. y - 2 = -3x
48. y + !x = 0
49.y- 9x
50. y = 3x- 9
51. 2y- 6 = 3x
52. -2y = 6(5 - 3x)
53. y- d = ex
54. y = (2 - a)x + a
55. 2y + 4n = - 6x
Use the slope andy-intercept to graph each equation.
<
i
56.y = 7- 3x
57. 2y + 4x = 0
58. 3y + 6 = - 2x
59.y + 2 = Sx- 4
60. 4x + 3y = 2x - 1
61. -2(3x - 4) + y = 0
62. Error Analysis Fred drew the graph at the right for the
equation y = -2x + 1. What error did he make?
y
63. a. A candle begins burning at timet = O.lts original
height is
in. After 30 min the height of the candle is
8 in. Draw a graph showing the change in the height of
the candle.
b. Write an equation that relates the height of the candle
to the time it has been burning.
c. How many minutes after the candle is lit will it burn out?
1T
/'
Error Analysis Find and correct the student's mistake in the work below.
Verify whether the point lies on the graph of the given line.
64.
65.
:Y- 1.5x = 0.72
(4,"
- 1.5
6.72
t .
I.
(4)
5.22(
20.88
=
0.72
?
=0.72
?
=
0.72
Determine whether the ordered pair lies on the graph of the given equation.
66. (-4,2);y = -~x + 1
'
67. (-6,5);y = -!x + 2
68. (0, -1);y =x-i
69. At the left is the graph of y = ix - 2. If the slope is doubled and they-intercept
stays the same, which of the graphs below represents the new linear function?
®
CD
y
o'
-1
X
GD~JL~--~~~x
y
o'
®
X
_______tr
01
2
-2
Visit: PHSchool.com
.I
Web Code: bae-0502
Lesson 5-2
Slope-Intercept Form
243
70. When the Bryants leave town for a vacation, they put their cat in a kennel.
The kennel charges $15 for a first-day flea bath and $5 per day. The equation
t = 15 + 5d relates the total charge t to the number of days d.
a. Rewrite the equation in slope-intercept form.
b. Graph the equation.
c. Reasoning Explain why the line you graph should lie only in Quadrant I.
71. Writing Explain the steps you would use to graph y = ~ x
+ 5.
72. Critical Thinking Which graphed line has the greater slope? Explain.
A.
80
The basic annual expense for
cat owners is about $120.
r
.ty
60
I
I
I
t
I
I
I
I
.
+
+
..
62
I
t
+
~
I
X
0
1
fy
64
1 ,
t
68
66
I
.
t
I
40
zol
l
I
I
B.
2
3
4
5
--
60
0
X
1
2
3
4
5
Given two points on a line, write the equation of the line in slope-intercept form.
73. (3, 5), (5, 9)
74. (5,-13), (2,-1)
75. (-4, 10), (6,5)
76. (8, 7), (-12, 2)
77. (-7,4),(11,-14)
78. (-1, -9), (2, 0)
79. Math Reasoning Is the following statement sometimes, always, or never true?
Explain.
The point (0, b) lies on the graph of the equation y = mx + b.
80. a. What is the slope of each line?
b. What is they-intercept of each line?
c. The lines in the graph are parallel. What
appears to be true about the slopes of
parallel lines?
81. Write a linear equation. Identify the
slope andy-intercept. Then graph your
equation.
Challenge
Find the value of a such that the graph of the equation has the given slope.
82. y = 2ax + 4; m = -1
83. y = -~ax - 5; m = ~
84. y =
i ax + 3; m = {6
85. a. Graph these equations on the same grid.
y = 3
y = -3
X = 2
X = -2
b. What geometric figure did you draw? Justify your answer.
c. Draw a diagonal of the figure. What is the equation of this line? Explain.
86. A group of mountain climbers begin an expedition with 265 lb of food. They
plan to eat a total of 15 lb of food per day.
a. Write an equation in slope-intercept form relating the remaining food supply
r to the number of days d.
b. Graph your equation.
c. The group plans to eat the last of their food the day their expedition ends.
Use your graph to find how many days they expect the expedition to last.
244
Chapter 5
Linear Equations and Their Graphs
For California Standards Tutorials, visit PHSchool.com. Web Code: baq-9045
Alg1 7.0
87. Which point lies on the line represented by the equation 2x + 3y = -6?
®
Alg1 3.0
Alg1 6.0
®
(-1, 2)
CD (0, 2)
®
(- 3, 0)
®
- 1or4
88. What is the solution for the equation l2x - 51 = 9?
®
Alg1 6.0
(2, - 2)
®
-7, 2
-2or7
CD -3, 5
89. Which equation represents a line that has a slope of -3 and a y-intercept of 6?
®
y=3x - 6
CD y
®
y=6x-3
®
=
-3x + 6
y = - 6x
+3
90. Brian knows the coordinates of point P on a line. What additional information
does he need to find the slope of the line?
®
®
CD
®
The length of the line
The direction of the line
The coordinates of another point on the line
The distance between point P and another point on the line
4'i>'j· p:<q\Mfl:ttti@MIIBttB~~@fi?i1tnDif~llr1'11'f!Rfll~~~~iti~tl8
for
Help
Lesson 2-2
Lesson 3-4
Solve each equation.
91. 5 (X + 1) = 8
92. - 33 = 6h + 7 + 2h
93. - 2(3 - k) = 15
94. 17 = 4t - 2t + 1
95. 4(n + 1) - 3 = 16
96. 40 = 8p + 5 - p
98. 7x + 3 < 2x + 28
99. 4x + 4 > 2 + 2x
Solve each inequality.
97. 1 + Sx + 1 > x + 9
100. 4x + 3
~
101. - x + 5 < 3x - 1
2x - 7
,...0 -cliecl<poiiifQiii%1
102. 2x > 7x - 3 - 4x
[essons s=-1-tlirough- 5=2
Find the slope of the line passing through each pair of points.
1. (-1,3), (6,- 2)
2. (4, 5), (0, 2) . 3. (-2,-3), (- 1,-7)
4. (4,-4), (- 5, 5)
5. One year, people charged $534 billion on the two most-used types of credit
cards. Four years later, people charged $1.021 trillion on these same two types
of credit cards. What was the rate of change?
Graph each equation.
6. y = 4x - 1
7. y = -~x + 6
8. y =
-i x -
10
9. y = -0.75x
10. Writing How are the graphs of y = 3x + 5, y = ~ x + 5, andy = ~ x + 5 alike?
How are they different?
nline Lesson Quiz Visit; PHSchool.com, Web Code: baa-0502
245
Standard Form
California Content Standards
6.0 Graph a linear equation and compute the x- andy-intercepts. Develop
What You'll Learn
• To graph equations using
intercepts
• To write equations in
standard form
... And Why
(iff Check Skills You'll Need
Review page 90 and Lesson 2-2
Solve each equation for y.
1. 3x + y = 5
4. 20x + 4y = 8
2. y- 2x = 10
5. 9y + 3x = 1
3.x- y = 6
6. 5y - 2x = 4
Clear each equation of decimals.
To use an equation to model
a real-world situation that
involves exercise, as in
Example 5
7. 6.25x + 8.5 = 7.75
II>)~ New Vocabulary
•
8. 0.4 = 0.2x - 5
standard form of a linear equation
9. 0.9 - 0.222x
=
1
• x-intercept
Graphing Equations Using Intercepts
CA Standards
Investigation Intercepts
1. Graph the equation 3y - 2x = 12 by making a table of values.
2. What is they-intercept?
3. What is the value of x when the line crosses the x-axis?
4. In the equation 3y - 2x = 12, what is the value of y when x = 0?
What is the value of x when y = 0?
5. Using your answers to 2, 3, and 4, explain how you can make a graph of
3y - 2x = 12 without making a table.
I
I
The slope-intercept form is just one form of a linear equation. Another form is
standard form , which is Ouseful in making quick graphs.
J
I
f
,
tihtftitfbltj
~
Standard Form of a Linear Equation
are real numbers, and A and B are not both zero.
You can use the x- andy-intercepts to make a graph. The x-intercept is the
x-coordinate of the point where a line crosses the x-axis. To graph a linear
equation in standard form, you can find the x-intercept by substituting 0 for y and
solving for x. Similarly, to find they-intercept, substitute 0 for x and solve for y.
246
Chapter 5
Linear Equations and Their Graphs
Finding x- and y-lntercepts
Find the x- andy-intercepts of 3x + 4y = 8.
Step 1 To find the x-intercept,
substitute 0 for y and solve for x.
Step 2 To find they-intercept,
substitute 0 for x and solve for y.
3x + 4y = 8
3x + 4y = 8
3x + 4(0) = 8
3(0) + 4y = 8
3x = 8
4y = 8
_a
X - 3
y=2
The x-intercept is ~.
@CA Standards Check
They-intercept is 2.
}) Find the x- andy-intercepts of 4x
- 9y = -12.
If the x- andy-intercepts are integers, you can use them to make a quick graph.
Graphing Lines Using Intercepts
Graph 2x + 3y = 12 using intercepts.
Step 1 Find the intercepts.
Step 2 Plot (0, 4) and (6, 0).
D raw a line through the points.
2x + 3y = 12
2x + 3(0) = 12
2x = 12
X=
2(0)
+
Substitute 0 for y.
Solve for x.
6
3y = 12
Substitute 0 for x.
3y = 12
Solve for y.
y=4
@CA Standards Check
.J) Graph 5x + 2y = - 10 using the x- andy-intercepts.
In the standard form of an equation Ax + By = C, either A orB, but not both,
may be zero. If A or B is zero, the line is either horizontal or vertical.
Graphing Horizontal and Vertical Lines
a. Graphy = - 3.
b. Graph x = 2.
Ox + 1y = - 3
~ Write in standard form. 4For all values of x, y = -3.
1x + Oy = 2
For all values of y , x = 2.
2'y
-4
-2
0
2'y
2
4
X
-4
-2
Oj
2
4
X
-2
@CA Standards Check ~ Graph each equation.
a. y = 5
b. x = -4
Lesson S-3
Standard Form
247
, -wri-ting Equations in
sfanda.rii Form
You can change an equation from slope-intercept form to standard form. If
the equation contains fractions or decimals, multiply to write the equation
using integers.
Transforming to Standard Form
/
Write y = ~ x + 2 in standard form using integers.
y
= ~X + 2
4y
= 4(~ x +
4y = 3x + 8
- 3x + 4y
2)
Multiply each side by 4.
Use the Distributive Property.
=8
Subtract 3x from each side.
The standard form of y = ~x + 2 is -3x + 4y = 8.
{i! CA Standards Check
r
4~ Write y = -~ x -+ 1 in standard form using integers.
You can write equations for real-world situations using standard form.
Application
Write an equation in standard form to find the number of minutes someone who
weighs 150 lb would need to bicycle and swim laps in order to burn 300 Calories.
Use the data below.
Activity by a
150-lb Person
Doctors recommend
30 minutes of exercise
each day.
Bicycling
10
Bowling
4
Hiking
7
Running 5.2 mi/h
11
Swimming, laps
12
Walking 3.5 mi/h
5
Define Let x
Let y
Relate
Write
Calories Burned
per Minute
= the minutes spent bicycling.
= the minutes spent swimming laps.
10 ·minutes
12 ·minutes
·
pus
1
equa1s 300 ca1ones
bicycling
swimming laps
lOx
+
l2y
300
The equation in standard form is lOx + l2y = 300.
{i! CA Standards Check
248
® Write an equation in standard form to find the number of minutes someone who
weighs 150 lb would need to bowl and walk to burn 250 Calories.
Chapter 5 Linear Equations and Their Graphs
For more exercises, see Extra Skills and Word Problem Practice.
SG&SLJ
0
Practice by Example
Example 1
for
Help
(page 247)
Find the x- and y-intercepts of each equation.
1. X+ 2y = 18
4. - 6x
+ 3y = -9
7. -2x - 3y = -12
Example 2
(page 247)
2. 3x - y = 9
3. -5x + y = 30
5.4x + 12y = -18
6. 9x - 6y = -72
8. 7x - 2y = 4
9. -8x
+ lOy = 40
Match each equation with its graph.
10. 2x - 5y = 10
11. - 2x
A.
B.
y
+ 5y = 10
+ 5y = 10
12. 2x
c.
y
0
-2
Graph each equation using x- andy-intercepts.
13.x + y = 2
16. -3x
Example 3
(page 247)
+
14. X+ y = -5
y = 6
17. - 2x
15.
+ y = -6
X-
y = -7
18. 5x - 3y = 15
For each equation, tell whether its graph is a horizontal or a vertical line.
19. y = -1
20.
X
= 4
21. y = 2i
22.
X
= -3.75
25. y = -1.5
26.
X=
Graph each equation.
23.y = 3
Example-4
(page 248)
(page 248)
X=
-7
4.5
Write each equation in standard form using integers.
27.y = 3x + 1
28.y = 4x- 7
30. y =~X+ 5
31. y =
33. y
Example 5
24.
=~X+
i
34.
-ix- 4
y= - ~ x + lo
29. y =
ix- 3
32. y = -~x- 7
35. y = -3x
36. The sophomore class holds a car wash to raise money. A local merchant
donates all of the supplies. A wash costs $5 per car and $6.50 per van or truck.
a. Define a variable for the number of cars. Define a different variable for the
number of vans or trucks.
b. Write an equation in standard form to relate the number of cars and vans or
trucks the students must wash to raise $800.
37. Larry runs at an average rate of 8 mi/h. He walks at an average rate of 3 mi/h.
a. Define a variable for time spent walking. Define a different variable for time
spent running.
b. Write an equation in standard form to relate the times he could spend
running and walking if he travels a distance of 15 mi.
Lesson 5-3
Standard Form
249
Apply Your Skills
Graph each equation.
38. -3x + 2y = -6
39.x + y = 1
40. 2x - 3y = 18
41. y-
42. y = 2x + 5
43. y = -3x- 1
X=
44.2- y
-4
=X-
45. 9 + y = 8-
6
46. 6x = y
X
47. Suppose you are preparing a snack mix. You want the total protein from
peanuts and granola to equal 28 grams. Peanuts have 7 grams of protein per
ounce, and granola has 3 grams of protein per ounce.
a. Write an equation for the protein content of your mix.
b. Graph your equation. Use your graph to find how many ounces of granola
you should use if you use 1 ounce of peanuts.
48. You are sent to the store to buy sliced meat for a party. You are told to get
roast beef and turkey, and you are given $30. Roast beef is $4.29/lb and turkey
is $3.99/lb. Write an equation in standard form to relate the pounds of each
kind of meat you could buy at the store with $30.
Write each equation in slope-intercept form. Make a sketch of the graph. Include
the x- and y-intercepts.
A peanut contains about
0.24 gram of protein.
49. 8x- lOy= -100
50. -6x + 7y = 21
51. 12x + 15y = - 45
52. -5x + 9y = -15
53. 16x + lly = -88
54. 3x - 27y = 18
55. Writing Two of the forms of a linear equation are slope-intercept form and
standard form. Explain when each is more useful.
56. Critical Thinking The definition of standard form states that A and B can't
both be zero. Explain why.
57. Error Analysis A student says that the equation 3x + 2y = 6 is a standard
form of the equation y = ~ x + 3. What is the student's error?
y
Write an equation for each line on the graph.
58. a
60. c
59. b
61. d
a
10
-1
~1
1
X
62. a. Suppose your school is having a talent show to raise
b
money for new music supplies. You estimate that
c
d
200 students and 150 adults will attend. You estimate
$200 in expenses. Write an equation to find what ticket prices you should
set to raise $1000.
b. Graph your equation. Choose three possible prices you could set for
students' and adults' tickets. Which is the best choice? Explain.
Visit: PHSchool.com
Web Code: bae-0503
Challenge
63. Write an equation of a line that has the same slope as the line 3x - 5y = 7 and
the same y-intercept as the line 2y - 9x = 8.
64. Graph each of the four lines below on the same graph. What figure do the four
lines appear to form?
-2x
+ 3y = 10
3x
+ 2y = -2
-2x
+ 3y = -3
65. a. Graph 2x + 3y = 6 and 2x + 3y = 18.
b. What is the slope of each line?
c. How are the x- andy-intercepts of the two lines related?
250
Chapter 5
Linear Equations and Their Graphs
3x
+ 2y = 11
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66. What is they-intercept of the line represented by the equation 4x
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67. Christina recorded the height of her tomato plant for 3 weeks. She graphed a
line to model her data. If Christina's plant grew about 2 inches each week,
which graph could model her data?
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68. Andy can use the function c = 900n
3
+ 200 to find the cost c of ordering n new
computers for his business. What is the dependent variable?
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Determine whether the ordered pair is a solution of the equation.
69. (2, -3);y = -x- 1
70. (6, -1);y = 2x- 15
71. (-5, -7);y = -3x- 8
Write the slope-intercept form of the equation for each line.
72.
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y
4
74.
73.
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0
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4
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Lesson 2-4
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a _ 12
75. 515
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X
42 = ~
14
16
78• 4m
= 5m + 9
251