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3: Slope of a Line

3: Slope of a Line

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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





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





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.





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.





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:





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





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.





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



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