118 CHAPTER 1
Linear Equations and Inequalities
1.6 EXERCISES
1–36. are the Quick
In Problems 65–76, plot each pair of points and graph the line
containing them. Determine the slope of the line. See Objective 4.
s that follow each EXAMPLE
65. (0, 0); (1, 5)
Building Skills
In Problems 37–44, graph each linear equation by plotting points.
See Objective 1.
37. x - 2y = 6
38. 2x - y = -8
39. 3x + 2y = 12
40. -5x + y = 10
41.
2
x + y = 6
3
42. 2x -
43. 5x - 3y = 6
66. (0, 0); 1-2, 52
67. 1-2, 32; 11, -62
68. 13, -12; 1-2, 112
69. 1-2, 32; 13, 72
3
y = 10
2
70. 11, -42; 1-1, 32
71. 1-3, 22; 14, 22
44. -7x + 3y = 9
72. 1-3, 12; 12, 12
In Problems 45–54, graph each linear equation by finding its
intercepts. See Objective 2.
73. 110, 22; 110, -32
45. 3x + y = 6
46. -2x + y = 4
74. 14, 12; 14, -32
47. 5x - 3y = 15
48. -4x + 3y = 24
1 5
9 11
75. a , b; a , b
2 3
4 6
49.
1
1
x - y = 1
3
2
50.
51. 2x + y = 0
53.
1
1
x + y = 2
4
5
7 5
13 13
76. a , b; a , b
3 2
9 4
52. 4x + 3y = 0
2
1
x - y = 0
3
2
3
3
54. - x + y = 0
2
4
In Problems 77–86, graph the line containing the given point
and having slope m. Do not find the equation of the line.
See Objective 6.
In Problems 55–60, graph each linear equation. See Objective 3.
77. m = 3; 11, 22
55. x = -5
56. x = 5
79. m = -2; 1-3, 12
57. y = 1
58. y = 6
81. m =
59. 2y + 8 = -6
60. 3y + 20 = -10
3
83. m = - ; 12, 52
2
In Problems 61–64, (a) find the slope of the line and (b) interpret
the slope. See Objective 4.
61.
y
4
4
(3, 4)
3
⫺4 (0, 0)
x
⫺4
63.
⫺4
(⫺1, 4)
⫺4
4 x
(2, ⫺4)
⫺5
4 x
(3, ⫺2)
86. m is undefined; 1-5, 22
5
; 1-2, 32
2
2
88. m = - ; 11, -32
3
In Problems 89–92, find an equation of the line. Express your
answer in slope-intercept form. See Objective 7.
⫺4
(⫺2, ⫺3)
90.
y
y
4
(4, 2)
y
(4, 4)
(⫺4, 1)
(⫺2, 0)
x
4 x
⫺4
4
; 1-2, -52
3
1
84. m = - ; 13, 32
2
89.
64.
y
5
82. m =
85. m = 0; 11, 22
87. m =
(0, 0)
⫺3
80. m = -4; 1-1, 52
In Problems 87 and 88, the slope and a point on a line are given.
Use the information to find three additional points on the line.
Answers may vary.
62.
y
1
; 1-3, 42
3
78. m = 2; 1-1, 42
(5, ⫺3)
x
Section 1.6 Linear Equations in Two Variables 119
91.
92.
y
(1, 3)
y
(2, 4)
In Problems 115–126, find the slope and y-intercept of each line.
Graph the line. See Objective 8.
115. y = 2x - 1
(⫺4, 3)
x
x
(2, ⫺2)
116. y = 3x + 2
117. y = - 4x
118. y = - 7x
In Problems 93–102, find an equation of the line with the given
slope and containing the given point. Express your answer in
slope-intercept form, if possible. See Objective 7.
93. m = 2; 10, 02
94. m = - 1; 10, 02
95. m = - 3; 1- 1, 12
96. m = 4; 12, -12
4
97. m = ; 13, 22
3
1
98. m = ; 12, 12
2
5
99. m = - ; 1-2, 42
4
4
100. m = - ; 11, -32
3
101. m undefined; (6, 1)
102. m = 0; (3, -2)
In Problems 103–114, find an equation of the line containing the
given points. Express your answer in slope-intercept form, if
possible. See Objective 9.
103. (0, 0); (5, 7)
104. (0, 0); 14, -32
105. (3, 2); (4, 7)
106. (1, 3); (3, 7)
107. 1-2, 12; 15, -22
119. 2x + y = 3
120. - 3x + y = 1
121. 4x + 2y = 8
122. 3x + 6y = 12
123. x - 4y - 2 = 0
124. 2x - 5y - 10 = 0
125. x = 3
126. y = - 4
Applying the Concepts
127. Find an equation for the x-axis.
128. Find an equation for the y-axis.
129. Maximum Heart Rate The data below represent
the maximum number of heartbeats that a healthy
individual should have during a 15-second interval
of time while exercising for different ages.
(a) Plot the ordered pairs (x, y) on a graph and
connect the points with straight lines.
(b) Compute and interpret the average rate of change
in the maximum number of heartbeats between
20 and 30 years of age.
(c) Compute and interpret the average rate of change
in the maximum number of heartbeats between
50 and 60 years of age.
(d) Based upon your results to parts (a), (b), and
(c), do you think that the maximum number of
heartbeats is linearly related to age? Why?
108. 1 -3, 12; (1, 6)
109. 1-1, -32; (⫺1, 5)
Age, x
Maximum
Number of
Heartbeats, y
110. 1-3, -42; 11,-42
20
50
111. (1, 3); 1-3, -72
30
47.5
40
45
50
42.5
60
40
70
37.5
112. 1-5, 12; 11, -12
113. (2, 4); 1-4, 42
114. (3, 1); 13, -42
SOURCE:
American Heart Association
120 CHAPTER 1
Linear Equations and Inequalities
130. Raisins The following data represent the weight (in
grams) of a box of raisins and the number of raisins
in the box.
(a) Plot the ordered pairs (x, y) on a graph and
connect the points with straight lines.
(b) Compute and interpret the average rate of
change in the number of raisins between
42.3 and 42.5 grams.
(c) Compute and interpret the average rate of
change in the number of raisins between
42.7 and 42.8 grams.
(d) Based upon your results to parts (a), (b), and (c),
do you think that the number of raisins is linearly
related to weight? Why?
Weight
(in grams), x
Number of
Raisins, y
42.3
82
42.5
86
42.6
89
42.7
91
42.8
93
132. U.S. Population The following data represent the
population of the United States between 1930 and
2000.
(a) Plot the ordered pairs (x, y) on a graph and
connect the points with straight lines.
(b) Compute and interpret the average rate of
change in population between 1930 and 1940.
(c) Compute and interpret the average rate of
change in population between 1990 and 2000.
(d) Based upon your results to parts (a), (b), and (c),
do you think that population is linearly related
to the year? Why?
Jennifer Maxwell, student at
Joliet Junior College
SOURCE:
131. Average Income An individual’s income varies with
age. The following data show the average income of
individuals of different ages in the United States for
2005.
(a) Plot the ordered pairs (x, y) on a graph and
connect the points with straight lines.
(b) Compute and interpret the average rate of
change in average income between 20 and
30 years of age.
(c) Compute and interpret the average rate of
change in average income between 50 and
60 years of age.
(d) Based upon your results to parts (a), (b), and
(c), do you think that average income is linearly
related to age? Why?
Age, x
Average Income, y
20
$10,469
30
$31,161
40
$40,964
50
$43,627
60
$40,654
70
$21,784
SOURCE:
Statistical Abstract, 2008
Year, x
Population, y
1930
123,202,624
1940
132,164,569
1950
151,325,798
1960
179,323,175
1970
203,302,031
1980
226,542,203
1990
248,709,873
2000
281,421,906
SOURCE:
U.S. Census Bureau
133. Measuring Temperature The relationship between
Celsius (°C) and Fahrenheit (°F) degrees for measuring temperature is linear. Find an equation relating °C and °F if 0°C corresponds to 32°F and 100°C
corresponds to 212°F. Use the equation to find the
Celsius measure of 60°F.
134. Building Codes As a result of the Americans with
Disabilities Act (ADA, 1990), the building code
states that access ramps must have a slope not
1
steeper than . Interpret what this result means.
12
135. Which of the following equations might have the
graph shown? (More than one answer is possible.)
(a)
(b)
(c)
(d)
(e)
y = 3x - 1
y = -2x + 3
y = 2x + 3
3x - 2y = 4
-3x + 2y = -4
y
x
136. Which of the following equations might have the
graph shown? (More than one answer is possible.)
(a) y = 2x - 5
(b) y = -x + 2
2
(c) y = - x - 3
3
(d) 4x + 3y = -5
(e) -2x + y = -4
y
x
Section 1.7
observe? In general, describe the graph of y = ax
with a 6 0.
Explaining the Concepts
137. Name the five forms of equations of lines given in
this section.
138. What type of line has one x-intercept, but no
y-intercept?
139. What type of line has one y-intercept, but no
x-intercept?
The Graphing Calculator
145. To see the role that the slope m plays in the graph of
a linear equation y = mx + b, graph the following
lines on the same screen.
140. What type of line has one x-intercept and one
y-intercept?
1
x + 2
2
Y1 = 0x + 2
Y2 =
Y3 = 2x + 2
Y4 = 6x + 2
State some general conclusions about the graph of
y = mx + b for m Ú 0. Now graph
141. Are there any lines that have no intercepts? Explain
your answer.
1
Y1 = - x + 2
2
142. Exploration Graph y = 2x, y = 2x + 3,
y = 2x + 7, and y = 2x - 4 on the same Cartesian
plane. What pattern do you observe? In general,
describe the graph of y = 2x + b.
1
143. Exploration Graph y = x, y = x, and y = 2x on the
2
same Cartesian plane. What pattern do you observe?
In general, describe the graph of y = ax with a 7 0.
Parallel and Perpendicular Lines 121
Y2 = -2x + 2
Y3 = -6x + 2
State some general conclusions about the graph of
y = mx + b for m 6 0.
146. To see the role that the y-intercept b plays in the
graph of a linear equation y = mx + b, graph the
following lines on the same screen.
1
144. Exploration Graph y = - x, y = -x, and y = -2x
2
on the same Cartesian plane. What pattern do you
Y1 = 2x
Y2 = 2x + 2
Y3 = 2x + 5
Y4 = 2x - 4
State some general conclusions about the graph of
y = 2x + b.
1.7 Parallel and Perpendicular Lines
OBJECTIVES
Preparing for Parallel and Perpendicular Lines
Before getting started, take this readiness quiz. If you get a problem wrong, go back to the section
cited and review the material.
1
Define Parallel Lines
2
Find Equations of Parallel Lines
3
Define Perpendicular Lines
P1. Determine the reciprocal of 3.
[Section R.3, pp. 23–24]
4
Find Equations of Perpendicular
Lines
3
P2. Determine the reciprocal of - .
5
[Section R.3, pp. 23–24]
Work Smart
The words “if and only if” given in
the definition mean that there are
two statements being made.
If two nonvertical lines are
parallel, then their slopes are equal
and they have different y-intercepts.
If two nonvertical lines have
equal slopes and different yintercepts, then they are parallel.
1
Preparing for...Answers P1.
3
5
P2. 3
1
Define Parallel Lines
When two lines (in the Cartesian plane) do not intersect (that is, they have no points in
common), they are said to be parallel.
PARALLEL LINES
Two nonvertical lines are parallel if and only if their slopes are equal and they have
different y-intercepts. Vertical lines are parallel if they have different x-intercepts.
Figure 52(a) on the next page shows nonvertical parallel lines. Figure 52(b) on the
next page shows vertical parallel lines.