402
CHAPTER 8
Graphs, Functions, and Systems of Equations and Inequalities
59. A student was asked to find the distance between the
points 5, 8 and 2, 14, and wrote the following:
d 5 82 2 142 .
Explain why this is incorrect.
60. A circle can be drawn on a piece of posterboard by
fastening one end of a string, pulling the string taut
with a pencil, and tracing a curve as shown in the figure. Explain why this method works.
of the receiving stations and the distances to the epicenter are contained in the following three equations:
x 22 y 12 25, x 22 y 22 16, and x 12 y 22 9. Graph the circles
and determine the location of the earthquake epicenter.
64. Without actually graphing, state whether the graphs of
x 2 y 2 4 and x 2 y 2 25 will intersect. Explain
your answer.
65. Can a circle have its center at 2, 4 and be tangent to
both axes? (Tangent to means touching in one point.)
Explain.
66. Suppose that the endpoints of a line segment have
coordinates x1, y1 and x2, y2.
(a) Show that the distance between x1, y1 and
x1 x2 y1 y2
,
is the same as the distance
2
2
x1 x2 y1 y2
,
between x2, y2 and
.
2
2
61. Crawfish Racing This figure shows how the crawfish race is held at the Crawfish Festival in Breaux
Bridge, Louisiana. Explain why a circular “racetrack” is appropriate for such a race.
(b) Show that the sum of the distances between
x1 x2 y1 y2
x1, y1 and
, and x2, y2 and
,
2
2
x1 x2 y1 y2
,
is equal to the distance be2
2
tween x1, y1 and x2, y2.
(c) From the results of parts (a) and (b), what conclusion can be made?
62. Epicenter of an Earthquake Show algebraically
that if three receiving stations at 1, 4, 6, 0, and
5, 2 record distances to an earthquake epicenter
of 4 units, 5 units, and 10 units respectively, the epicenter would lie at 3, 4.
63. Epicenter of an Earthquake Three receiving stations
record the presence of an earthquake. The locations
8.2
67. If the coordinates of one endpoint of a line segment
are 3, 8 and the coordinates of the midpoint of
the segment are 6, 5, what are the coordinates of the
other endpoint?
68. Which one of the following has a circle as its graph?
A. x 2 y 2 9
B. x 2 9 y 2
2
2
C. y x 9
D. x 2 y 2 9
69. For the three choices that are not circles in Exercise 63,
explain why their equations are not those of circles.
Lines and Their Slopes
Linear Equations in Two Variables
In the previous chapter we studied
linear equations in a single variable. The solution of such an equation is a real number. A linear equation in two variables will have solutions written as ordered pairs.
Unlike linear equations in a single variable, equations with two variables will, in
general, have an infinite number of solutions.
An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.
8.2
403
Lines and Their Slopes
To find ordered pairs that satisfy the equation, select any number for one of
the variables, substitute it into the equation for that variable, and then solve for the
other variable. For example, suppose x 0 in the equation 2x 3y 6. Then
2x 3y 6
20 3y 6
0 3y 6
3y 6
y 2,
Graphing calculators can generate
tables of ordered pairs. Here is an
example for 2x 3y 6. We
must solve for y to get Y1 6 2X3 before generating
the table.
Let x 0.
giving the ordered pair 0, 2. Other ordered pairs satisfying 2x 3y 6 include
6, 2, 3, 0, 3, 4, and 9, 4.
The equation 2x 3y 6 is graphed by first plotting all the ordered pairs mentioned above. These are shown in Figure 10(a). The resulting points appear to lie on
a straight line. If all the ordered pairs that satisfy the equation 2x 3y 6 were
graphed, they would form a straight line. In fact, the graph of any first-degree equation in two variables is a straight line. The graph of 2x 3y 6 is the line shown in
Figure 10(b).
y
(–3, 4)
y
6
(0, 2)
(0, 2)
y1 = (6 – 2x)/3
2
(3, 0)
10
–4
–10
10
–2 0
–2
2x + 3y = 6
(–3, 4)
4
2
4
6
(3, 0)
x
8
0
10
(6, –2)
(6, –2)
–4
x
(9, – 4)
(9, – 4)
–6
–10
(b)
(a)
This is a calculator graph of the
line shown in Figure 10(b).
FIGURE 10
Linear Equation in Two Variables
An equation that can be written in the form
Ax By C
(A and B not both 0)
is a linear equation in two variables. This form is called standard
form.
All first-degree equations with two variables have straight-line graphs. Since a
straight line is determined if any two different points on the line are known, finding
two different points is enough to graph the line.
Two points that are useful for graphing lines are the x- and y-intercepts. The
x-intercept is the point (if any) where the line crosses the x-axis, and the y-intercept
is the point (if any) where the line crosses the y-axis. (Note: In many texts, the
An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.
404
CHAPTER 8
Graphs, Functions, and Systems of Equations and Inequalities
intercepts are defined as numbers, and not points. However, in this book we will
refer to intercepts as points.) Intercepts can be found as follows.
4x – y = –3 or y1 = 4x + 3
10
–10
10
To find the x-intercept of the graph of a linear equation, let y 0.
To find the y-intercept, let x 0.
–10
The display at the bottom of the
screen supports the fact that
34 , 0 is the x-intercept of
the line in Figure 11. We could
locate the y-intercept similarly.
EXAMPLE
equation.
(
4x 0 3
4x 3
3
x
4
(0, 3)
)
1
x
0
Find the x- and y-intercepts of 4x y 3, and graph the
To find the x-intercept, let y 0.
y
– 3_ , 0
4
Intercepts
4x – y = –3
To find the y-intercept, let x 0.
Let y 0.
x-intercept is 40 y 3
y 3
3
,0 .
4
y3
Let x 0.
y-intercept is 0, 3.
The intercepts are the two points 34, 0 and 0, 3. Use these two points to draw
the graph, as shown in Figure 11.
A line may not have an x-intercept, or it may not have a y-intercept.
EXAMPLE
FIGURE 11
10
–10
2
Graph each line.
(a) y 2
Writing y 2 as 0x 1y 2 shows that any value of x, including x 0, gives
y 2, making the y-intercept 0, 2. Since y is always 2, there is no value of x
corresponding to y 0, and so the graph has no x-intercept. The graph, shown
in Figure 12(a), is a horizontal line.
y1 = 2
10
y
y
–10
Compare this graph with the one in
Figure 12(a).
y=2
x
0
x = –1
(–1, 0)
x
0
Vertical
line
Horizontal
line
10
–10
x = –1
(0, 2)
10
(a)
(b)
–10
FIGURE 12
This vertical line is not an
example of a function (see Section
8.4), so we must use a draw
command to obtain it. Compare
with Figure 12(b).
(b) x 1
The form 1x 0y 1 shows that every value of y leads to x 1, and so no
value of y makes x 0. The graph, therefore, has no y-intercept. The only way
a straight line can have no y-intercept is to be vertical, as shown in Figure 12(b).
An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.
8.2
y
(x2, y2)
y2 – y1
0
(x1, y1)
x2 – x1
x
(x2, y1)
FIGURE 13
Lines and Their Slopes
405
Slope Two different points determine a line. A line also can be determined by a
point on the line and some measure of the “steepness” of the line. The measure of
the steepness of a line is called the slope of the line. One way to get a measure of the
steepness of a line is to compare the vertical change in the line (the rise) to the horizontal change (the run) while moving along the line from one fixed point to another.
Suppose that x 1 , y1 and x 2 , y2 are two different points on a line. Then, going
along the line from x 1 , y1 to x 2 , y2 , the y-value changes from y1 to y2 , an amount
equal to y2 y1 . As y changes from y1 to y2 , the value of x changes from x 1 to x 2 by
the amount x 2 x 1 . See Figure 13. The ratio of the change in y to the change in x is
called the slope of the line. The letter m is used to denote the slope.
Slope
If x 1 x 2 , the slope of the line through the distinct points x 1 , y1 and
x 2 , y2 is
m
EXAMPLE 3
Find the slope of the line that passes through the points 2, 1
and 5, 3.
If 2,1 x 1 , y 1 and 5, 3 x 2 , y 2 , then
y







–5 – 2 = –7
(–5, 3)
0
4 = – 4_
m = __
–7
y2 y1
rise change in y
.
run change in x
x2 x1

 3 – (–1) =x4

m
(2, –1)
7
y 2 y 1 3 1
4
4
.
x2 x1
5 2
7
7
See Figure 14. On the other hand, if 2, 1 x 2 , y 2 and 5, 3 x 1 , y 1 , the
slope would be
m
FIGURE 14
4
4
1 3
,
2 5
7
7
the same answer. This example suggests that the slope is the same no matter which
point is considered first. Also, using similar triangles from geometry, it can be shown
that the slope is the same no matter which two different points on the line are chosen.
If we apply the slope formula to a vertical or a horizontal line, we find that
either the numerator or denominator in the fraction is 0.
EXAMPLE
4
Find the slope, if possible, of each of the following lines.
(a) x 3
By inspection, 3, 5 and 3, 4 are two points that satisfy the equation
x 3. Use these two points to find the slope.
m
4 5
9
3 3
0
Undefined slope
Since division by zero is undefined, the slope is undefined. This is why the definition of slope includes the restriction that x 1 x 2 .
An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.
406
CHAPTER 8
Graphs, Functions, and Systems of Equations and Inequalities
(b) y 5
Find the slope by selecting two different points on the line, such as 3, 5 and
1, 5, and by using the definition of slope.
m
Highway slopes are measured in
percent. For example, a slope of
8% means that the road gains 8
feet in altitude for each 100 feet
that the road travels horizontally.
Interstate highways cannot exceed
a slope of 6%. While this may not
seem like much of a slope, there
are probably stretches of interstate
highways that would be hard work
for a distance runner.
55
0
0
3 1
4
Zero slope
In Example 2, x 1 has a graph that is a vertical line, and y 2 has a graph
that is a horizontal line. Generalizing from those results and the results of Example
4, we can make the following statements about vertical and horizontal lines.
Vertical and Horizontal Lines
A vertical line has an equation of the form x a, where a is a real number, and its slope is undefined. A horizontal line has an equation of the
form y b, where b is a real number, and its slope is 0.
If we know the slope of a line and a point contained on the line, then we can
graph the line using the method shown in the next example.
EXAMPLE 5
Graph the line that has slope 23 and goes through the point
1, 4.
First locate the point 1, 4 on a graph as shown in Figure 15. Then, from the
definition of slope,
change in y
2
m
.
change in x
3
Move up 2 units in the y-direction and then 3 units to the right in the x-direction to
locate another point on the graph (labeled P). The line through 1, 4 and P is the
required graph.
y
Right 3




P
Up 2 

(–1, 4)
1
–4
–2
0
x
2
4
FIGURE 15
The line graphed in Figure 14 has a negative slope, 47, and the line goes
down from left to right. In contrast, the line graphed in Figure 15 has a positive
An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.
8.2
Negative
slope
y
407
slope, 23, and it goes up from left to right. These are particular cases of a general
statement that can be made about slopes. (Figure 16 shows lines of positive, zero,
negative, and undefined slopes.)
Zero
slope
x
0
Positive
slope
Lines and Their Slopes
Positive and Negative Slopes
Undefined
slope
A line with a positive slope goes up (rises) from left to right, while a line
with a negative slope goes down (falls) from left to right.
FIGURE 16
Parallel and Perpendicular Lines
The slopes of a pair of parallel or perpendicular lines are related in a special way. The slope of a line measures the steepness of the line. Since parallel lines have equal steepness, their slopes also must be
equal. Also, lines with the same slope are parallel.
Slopes of Parallel Lines
Two nonvertical lines with the same slope are parallel; two nonvertical
parallel lines have the same slope. Furthermore, any two vertical lines
are parallel.
EXAMPLE 6
Are the lines L 1 , through 2, 1 and 4, 5, and L 2 , through
3, 0 and 0, 2, parallel?
The slope of L 1 is
51
4
2
m1 .
4 2
6
3
The slope of L 2 is
m2 2 0 2
2
.
03
3
3
Since the slopes are equal, the lines are parallel.
Perpendicular lines are lines that meet at right angles. It can be shown that the
slopes of perpendicular lines have a product of 1, provided that neither line is vertical. For example, if the slope of a line is 34, then any line perpendicular to it has
slope 43, because 34 43 1.
Slopes of Perpendicular Lines
If neither is vertical, two perpendicular lines have slopes that are negative
reciprocals; that is, their product is 1. Also, two lines with slopes that are
negative reciprocals are perpendicular. Every vertical line is perpendicular
to every horizontal line.
An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.
CHAPTER 8
Graphs, Functions, and Systems of Equations and Inequalities
EXAMPLE 7
Are the lines L 1 , through 0, 3 and 2, 0, and L 2 , through
3, 0 and 0, 2, perpendicular?
The slope of L 1 is
0 3
3
m1 .
20
2
The slope of L 2 is
m2 2
2 0
.
0 3
3
Since the product of the slopes of the two lines is 32 23 1, the lines
are perpendicular.
Average Rate of Change We have seen how the slope of a line is the ratio
of the change in y (vertical change) to the change in x (horizontal change). This idea
can be extended to real-life situations as follows: the slope gives the average rate of
change of y per unit of change in x, where the value of y is dependent upon the value
of x. The next example illustrates this idea of average rate of change. We assume a
linear relationship between x and y.
EXAMPLE 8
The graph in Figure 17 approximates the percent of U.S. households owning multiple personal computers in the years 1997–2001. Find the average
rate of change in percent per year for the years 1998 to 2001.
HOMES WITH MULTIPLE PCS
30
(2001, 24.4)
25
Percent
408
20
15
10
(1998, 13.6)
5
0
1997
1998
1999
Year
2000
2001
Source: The Yankee Group.
FIGURE 17
To use the slope formula, we need two pairs of data. From the graph, if we let
x 1998, then y 13.6 and if x 2001, then y 24.4, so we have the ordered
pairs 1998, 13.6 and 2001, 24.4. By the slope formula,
average rate of change 24.4 13.6
10.8
y2 y1
3.6 .
x 2 x 1 2001 1998
3
This means that the number of U.S. households owning multiple computers increased by an average of 3.6% each year in the period from 1998 to 2001.
An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.