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8.3 • Slope of a Line
333
Deﬁnition 8.1
If points P1 and P2 with coordinates (x1, y1 ) and (x2, y2 ) , respectively, are any two
different points on a line, then the slope of the line (denoted by m) is
mϭ
y2 Ϫ y1
, x1 ϶ x2
x2 Ϫ x1
Using Definition 8.1, we can easily determine the slope of a line if we know the coordinates
of two points on the line.
Classroom Example
Find the slope of the line determined
by each of the following pairs of
points:
(a) (3, 3) and (5, 6)
(b) (5, 3) and (3, Ϫ7)
(c) (Ϫ6, Ϫ1) and (Ϫ2, Ϫ1)
EXAMPLE 1
Find the slope of the line determined by each of the following pairs of points:
(a) (2, 1) and (4, 6)
(b) (3, 2) and (Ϫ4, 5)
(c) (Ϫ4, Ϫ3) and (Ϫ1, Ϫ3)
Solution
(a) Let (2, 1) be P1 and (4, 6) be P2 as in Figure 8.26; then we have
mϭ
y2 Ϫ y1
6Ϫ1
5
ϭ
ϭ
x2 Ϫ x1
4Ϫ2
2
y
P2 (4, 6)
P1 (2, 1)
x
Figure 8.26
(b) Let (3, 2) be P1 and (Ϫ4, 5) be P2 as in Figure 8.27.
mϭ
y2 Ϫ y1
5Ϫ2
3
3
ϭ
ϭ
ϭϪ
x2 Ϫ x1
Ϫ4 Ϫ 3
Ϫ7
7
P2 (−4, 5)
y
P1 ( 3, 2)
x
Figure 8.27
334
Chapter 8 • Coordinate Geometry and Linear Systems
(c) Let (Ϫ4, Ϫ3) be P1 and (Ϫ1, Ϫ3) be P2 as in Figure 8.28.
mϭ
y2 Ϫ y1
Ϫ3 Ϫ (Ϫ3)
0
ϭ
ϭ ϭ0
x2 Ϫ x1
Ϫ1 Ϫ (Ϫ4)
3
y
x
P2 (−1, −3)
P1 (− 4, −3)
Figure 8.28
The designation of P1 and P2 in such problems is arbitrary and does not affect the value
of the slope. For example, in part (a) of Example 1 we will let (4, 6) be P1 and (2, 1) be P2.
Then we obtain the same result for the slope as the following:
mϭ
y2 Ϫ y1
1Ϫ6
Ϫ5
5
ϭ
ϭ
ϭ
x2 Ϫ x1
2Ϫ4
Ϫ2
2
The parts of Example 1 illustrate the three basic possibilities for slope; that is, the slope of
a line can be positive, negative, or zero. A line that has a positive slope rises as we move from
left to right, as in part (a). A line that has a negative slope falls as we move from left to right, as
in part (b). A horizontal line, as in part (c), has a slope of 0. Finally, we need to realize that the
concept of slope is undefined for vertical lines. This is because, for any vertical line, the
y2 Ϫ y1
change in x as we move from one point to another is zero. Thus the ratio
will have a
x2 Ϫ x1
denominator of zero and be undefined. So in Definition 8.1, the restriction x1 ϶ x2 is made.
Classroom Example
Find the slope of the line determined
by the equation 5x ϩ 8y ϭ 4.
EXAMPLE 2
Find the slope of the line determined by the equation 3x ϩ 4y ϭ 12.
Solution
Since we can use any two points on the line to determine the slope of the line, let’s find the
intercepts.
If x ϭ 0, then 3(0) ϩ 4y ϭ 12
4y ϭ 12
yϭ3
Thus (0, 3) is on the line
If y ϭ 0, then 3x ϩ 4(0) ϭ 12
3x ϭ 12
xϭ4
Thus (4, 0) is on the line
Using (0, 3) as P1 and (4, 0) as P2, we have
mϭ
y2 Ϫ y1
0Ϫ3
Ϫ3
3
ϭ
ϭ
ϭϪ
x2 Ϫ x1
4Ϫ0
4
4
8.3 • Slope of a Line
335
We need to emphasize one final idea pertaining to the concept of slope. The slope of a line
3
is a ratio of vertical change to horizontal change. A slope of means that for every 3 units of
4
vertical change, there is a corresponding 4 units of horizontal change. So starting at some
point on the line, we could move to other points on the line as follows:
3
6
ϭ
4
8
3
15
ϭ
4
20
3
3
2
ϭ
4
2
3
Ϫ3
ϭ
4
Ϫ4
by moving 6 units up and 8 units to the right
by moving 15 units up and 20 units to the right
1
by moving 1 units up and 2 units to the right
2
by moving 3 units down and 4 units to the left
5
Likewise, a slope of Ϫ indicates that starting at some point on the line, we could move to
6
other points on the line as follows:
5
Ϫ ϭ
6
5
Ϫ ϭ
6
5
Ϫ ϭ
6
5
Ϫ ϭ
6
Classroom Example
Graph the line that passes through the
1
point (Ϫ1, 3) and has a slope of .
4
Ϫ5
6
5
Ϫ6
Ϫ10
12
15
Ϫ18
by moving 5 units down and 6 units to the right
by moving 5 units up and 6 units to the left
by moving 10 units down and 12 units to the right
by moving 15 units up and 18 units to the left
EXAMPLE 3
1
Graph the line that passes through the point (0, Ϫ2) and has a slope of .
3
Solution
vertical change
1
ϭ , we can locate
horizontal change
3
another point on the line by starting from the point (0, Ϫ2) and moving 1 unit up and 3 units
to the right to obtain the point (3, Ϫ1). Because two points determine a line, we can draw the
line (Figure 8.29).
To begin, plot the point (0, Ϫ2). Because the slope ϭ
y
x
(0, −2)
Figure 8.29
(3, −1)
336
Chapter 8 • Coordinate Geometry and Linear Systems
1
Ϫ1
ϭ
, we can locate another point by moving 1 unit down and
3
Ϫ3
3 units to the left from the point (0, Ϫ2).
Remark: Because m ϭ
Classroom Example
Graph the line that passes through
the point (3, Ϫ2) and has a slope
of Ϫ3.
EXAMPLE 4
Graph the line that passes through the point (1, 3) and has a slope of Ϫ2.
Solution
To graph the line, plot the point (1, 3). We know that m ϭ Ϫ2 ϭ
Ϫ2
. Furthermore, because
1
vertical change
Ϫ2
ϭ
, we can locate another point on the line by starting from
horizontal change
1
the point (1, 3) and moving 2 units down and 1 unit to the right to obtain the point (2, 1).
Because two points determine a line, we can draw the line (Figure 8.30).
the slope ϭ
y
(1, 3)
(2, 1)
x
Figure 8.30
Ϫ2
2
ϭ
we can locate another point by moving 2 units up and 1
1
Ϫ1
unit to the left from the point (1, 3).
Remark: Because m ϭ Ϫ2 ϭ
Applications of Slope
The concept of slope has many real-world applications even though the word “slope” is often
not used. For example, the highway in Figure 8.31 is said to have a “grade” of 17%. This means
that for every horizontal distance of 100 feet, the highway rises or drops 17 feet. In other
17
words, the absolute value of the slope of the highway is
.
100
17 feet
100 feet
Figure 8.31
8.3 • Slope of a Line
Classroom Example
A certain highway has a 4% grade.
How many feet does it rise in a horizontal distance of 1 mile?
337
EXAMPLE 5
A certain highway has a 3% grade. How many feet does it rise in a horizontal distance of
1 mile?
Solution
3
. Therefore, if we let y represent the unknown vertical dis100
tance and use the fact that 1 mile ϭ 5280 feet, we can set up and solve the following proportion:
A 3% grade means a slope of
y
3
ϭ
100
5280
100y ϭ 3(5280) ϭ 15,840
y ϭ 158.4
The highway rises 158.4 feet in a horizontal distance of 1 mile.
A roofer, when making an estimate to replace a roof, is concerned about not only the total
area to be covered but also the “pitch” of the roof. (Contractors do not define pitch the same
way that mathematicians define slope, but both terms refer to “steepness.”) The two roofs in
Figure 8.32 might require the same number of shingles, but the roof on the left will take
longer to complete because the pitch is so great that scaffolding will be required.
Figure 8.32
The concept of slope is also used in the construction of flights of stairs. The terms “rise”
and “run” are commonly used, and the steepness (slope) of the stairs can be expressed as the
10
ratio of rise to run. In Figure 8.33, the stairs on the left with the ratio of
are steeper than
11
7
the stairs on the right, which have a ratio of .
11
Technically, the concept of slope is involved in most situations where the idea of an
incline is used. Hospital beds are constructed so that both the head-end and the foot-end can
be raised or lowered; that is, the slope of either end of the bed can be changed. Likewise,
treadmills are designed so that the incline (slope) of the platform can be raised or lowered as
desired. Perhaps you can think of several other applications of the concept of slope.
rise of
10 inches
rise of
7 inches
run of
11 inches
Figure 8.33
run of
11 inches
338
Chapter 8 • Coordinate Geometry and Linear Systems
Concept Quiz 8.3
For Problems 1–10, answer true or false.
1. The concept of slope of a line pertains to the steepness of the line.
2. The slope of a line is the ratio of the horizontal change to the vertical change moving
from one point to another point on the line.
3. A line that has a negative slope falls as we move from left to right.
4. The slope of a vertical line is 0.
5. The slope of a horizontal line is 0.
6. A line cannot have a slope of 0.
Ϫ5
5
7. A slope of
is the same as a slope of Ϫ .
2
Ϫ2
8. A slope of 5 means that for every unit of horizontal change there is a corresponding 5
units of vertical change.
1
9. The slope of the line determined by the equation Ϫx Ϫ 2y ϭ 4 is Ϫ .
2
5
10. The slope of the line determined by the points (Ϫ1, 4) and (2, Ϫ1) is .
3
Problem Set 8.3
For Problems 1–20, find the slope of the line determined by
each pair of points. (Objective 1)
1. (7, 5), (3, 2)
2. (9, 10), (6, 2)
3. (Ϫ1, 3), (Ϫ6, Ϫ4)
4. (Ϫ2, 5), (Ϫ7, Ϫ1)
5. (2, 8), (7, 2)
6. (3, 9), (8, 4)
7. (Ϫ2, 5), (1, Ϫ5)
8. (Ϫ3, 4), (2, Ϫ6)
9. (4, Ϫ1), (Ϫ4, Ϫ7)
10. (5, Ϫ3), (Ϫ5, Ϫ9)
11. (3, Ϫ4), (2, Ϫ4)
12. (Ϫ3, Ϫ6), (5, Ϫ6)
13. (Ϫ6, Ϫ1), (Ϫ2, Ϫ7)
14. (Ϫ8, Ϫ3), (Ϫ2, Ϫ11)
15. (Ϫ2, 4), (Ϫ2, Ϫ6)
16. (Ϫ4, Ϫ5), (Ϫ4, 9)
17. (Ϫ1, 10), (Ϫ9, 2)
18. (Ϫ2, 12), (Ϫ10, 2)
19. (a, b), (c, d)
20. (a, 0), (0, b)
21. Find y if the line through the points (7, 8) and (2, y)
4
has a slope of .
5
22. Find y if the line through the points (12, 14) and (3, y)
4
has a slope of .
3
23. Find x if the line through the points (Ϫ2, Ϫ4) and
3
(x, 2) has a slope of Ϫ .
2
24. Find x if the line through the points (6, Ϫ4) and (x, 6)
5
has a slope of Ϫ .
4
For Problems 25–32, you are given one point on a line and
the slope of the line. Find the coordinates of three other
points on the line.
25. (3, 2), m ϭ
2
3
26. (4, 1), m ϭ
5
6
27. (Ϫ2, Ϫ4), m ϭ
1
2
28. (Ϫ6, Ϫ2), m ϭ
2
5
29. (Ϫ3, 4), m ϭ Ϫ
3
4
30. (Ϫ2, 6), m ϭ Ϫ
3
7
31. (4, Ϫ5), m ϭ Ϫ2
32. (6, Ϫ2), m ϭ 4
For Problems 33–40, sketch the line determined by each pair
of points and decide whether the slope of the line is positive,
negative, or zero.
33. (2, 8), (7, 1)
34. (1, Ϫ2), (7, Ϫ8)
35. (Ϫ1, 3), (Ϫ6, Ϫ2)
36. (7, 3), (4, Ϫ6)
Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
8.4 • Writing Equations of Lines
37. (Ϫ2, 4), (6, 4)
38. (Ϫ3, Ϫ4), (5, Ϫ4)
62. 7x Ϫ 6y ϭ Ϫ42
39. (Ϫ3, 5), (2, Ϫ7)
40. (Ϫ1, Ϫ1), (1, Ϫ9)
63. y ϭ Ϫ3x Ϫ 1
For Problems 41–48, graph the line that passes through the
given point and has the given slope. (Objective 3)
41. (3, 1), m ϭ
2
3
42. (Ϫ1, 0), m ϭ
43. (Ϫ2, 3), m ϭ Ϫ1
46. (Ϫ3, 4), m ϭ Ϫ
47. (2, Ϫ2), m ϭ
3
2
48. (3, Ϫ4), m ϭ
65. y ϭ 4x
2
1
67. y ϭ x Ϫ
3
2
44. (1, Ϫ4), m ϭ Ϫ3
1
4
64. y ϭ Ϫ2x ϩ 5
66. y ϭ 6x
3
4
45. (0, 5), m ϭ Ϫ
339
3
2
3
1
68. y ϭ Ϫ x ϩ
4
5
5
2
For Problems 69–73, solve word problems that involve
slope. (Objective 4)
For Problems 49–68, find the coordinates of two points on
the given line, and then use those coordinates to find the
slope of the line. (Objective 2)
49. 3x ϩ 2y ϭ 6
69. Suppose that a highway rises a distance of 135 feet in a
horizontal distance of 2640 feet. Express the grade of
the highway to the nearest tenth of a percent.
70. The grade of a highway up a hill is 27%. How much
change in horizontal distance is there if the vertical
height of the hill is 550 feet? Express the answer to the
nearest foot.
50. 4x ϩ 3y ϭ 12
51. 5x Ϫ 4y ϭ 20
52. 7x Ϫ 3y ϭ 21
3
for some stairs,
5
and the measure of the rise is 19 centimeters, find the
measure of the run to the nearest centimeter.
71. If the ratio of rise to run is to be
53. x ϩ 5y ϭ 6
54. 2x ϩ y ϭ 4
55. 2x Ϫ y ϭ Ϫ7
2
for some stairs, and
3
the measure of the run is 28 centimeters, find the measure
of the rise to the nearest centimeter.
72. If the ratio of rise to run is to be
56. x Ϫ 4y ϭ Ϫ6
57. y ϭ 3
58. x ϭ 6
1
73. A county ordinance requires a 2 % “fall” for a sewage
4
pipe from the house to the main pipe at the street. How
much vertical drop must there be for a horizontal distance of 45 feet? Express the answer to the nearest tenth
of a foot.
59. Ϫ2x ϩ 5y ϭ 9
60. Ϫ3x Ϫ 7y ϭ 10
61. 6x Ϫ 5y ϭ Ϫ30
Thoughts Into Words
74. How would you explain the concept of slope to someone
who was absent from class the day it was discussed?
2
75. If one line has a slope of , and another line has a slope
3
of 2, which line is steeper? Explain your answer.
Answers to the Concept Quiz
1. True
2. False
3. True
4. False
9. True
10. False
5. True
76. Why do we say that the slope of a vertical line is
undefined?
3
77. Suppose that a line has a slope of and contains the point
4
(5, 2). Are the points (Ϫ3, Ϫ4) and (14, 9) also on the
line? Explain your answer.
6. False
7. False
8. True
340
Chapter 8 • Coordinate Geometry and Linear Systems
8.4
Writing Equations of Lines
OBJECTIVES
1
Find the equation of a line given
a. a slope and a point
b. two points on the line
c. a point on the line and the equation of a line parallel or perpendicular
to it
2
Become familiar with the point-slope form and the slope-intercept form
of the equation of a straight line
3
Know the relationships for slopes of parallel and perpendicular lines
There are two basic types of problems in analytic or coordinate geometry:
1. Given an algebraic equation, find its geometric graph.
2. Given a set of conditions pertaining to a geometric figure, determine its algebraic
equation.
We discussed problems of type 1 in the first two sections of this chapter. Now we want to consider a few problems of type 2 that deal with straight lines. In other words, given certain facts
about a line, we need to be able to write its algebraic equation.
Classroom Example
Find the equation of the line that has
2
a slope of and contains the point
3
(Ϫ3, 2).
EXAMPLE 1
Find the equation of the line that has a slope of
3
and contains the point (1, 2).
4
Solution
3
First, we draw the line as indicated in Figure 8.34. Since the slope is , we can find a second
4
point by moving 3 units up and 4 units to the right of the given point (1, 2). (The point (5, 5)
merely helps to draw the line; it will not be used in analyzing the problem.) Now we choose
a point (x, y) that represents any point on the line other than the given point (1, 2). The slope
3
determined by (1, 2) and (x, y) is .
4
Thus
yϪ2
3
ϭ
xϪ1
4
3(x Ϫ 1) ϭ 4(y Ϫ 2)
3x Ϫ 3 ϭ 4y Ϫ 8
3x Ϫ 4y ϭ Ϫ5
y
(5, 5) is 3 units
up and 4 units to
the right of (1, 2)
(5, 5)
(x, y)
(1, 2)
given point
x
Figure 8.34
8.4 • Writing Equations of Lines
Classroom Example
Find the equation of the line that
contains (Ϫ1, 4) and (3, Ϫ2).
EXAMPLE 2
341
Find the equation of the line that contains (3, 4) and (Ϫ2, 5).
Solution
y
First, we draw the line determined by the two given
points in Figure 8.35. Since we know two points, we
can find the slope.
mϭ
y2 Ϫ y1
5Ϫ4
1
1
ϭ
ϭ
ϭϪ
x2 Ϫ x1
Ϫ2 Ϫ 3
Ϫ5
5
P2 (−2, 5)
P(x, y)
P1 (3, 4)
x
Figure 8.35
Now we can use the same approach as in Example 1. We form an equation using a variable
1
point (x, y), one of the two given points (we choose P1(3, 4)), and the slope of Ϫ .
5
yϪ4
1
1
1
ϭ
Ϫ ϭ
5
Ϫ5
xϪ3
Ϫ5
x Ϫ 3 ϭ Ϫ5y ϩ 20
x ϩ 5y ϭ 23
Classroom Example
Find the equation of the line that has a
2
slope of Ϫ and a y intercept of Ϫ3.
5
EXAMPLE 3
Find the equation of the line that has a slope of
1
and a y intercept of 2.
4
Solution
1
A y intercept of 2 means that the point (0, 2) is on the line. Since the slope is , we can find
4
another point by moving 1 unit up and 4 units to the right of (0, 2). The line is drawn in
Figure 8.36. We choose variable point (x, y) and proceed as in the preceding examples.
yϪ2
1
ϭ
xϪ0
4
x ϭ 4y Ϫ 8
x Ϫ 4y ϭ Ϫ8
y
(x, y)
(0, 2)
x
Figure 8.36