4.3 forms of linear equations - mcgraw hill education · forms of linear equations section 4.3 229...

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Forms of Linear Equations4.3

227

4.3 OBJECTIVES

1. Write the equation of a line given its slope and y intercept

2. Find the slope and y intercept of a line from itsequation

3. Graph the equation of a line4. Write the equation of a line given a point and a

slope5. Write the equation of a line given two points6. Express the equation of a line as a linear function7. Graph a linear function

The special form

Ax � By � C

in which A and B cannot both be zero, is called the standard form for a linear equation.In Section 4.2, we determined the slope of a line from two ordered pairs. In this section, wewill look at several forms for a linear equation. For the first of these forms, we use the con-cept of slope to write the equation of a line.

First, suppose we know the y intercept (0, b) of a line L, and its slope m. Let P(x, y) beany other point on that line. Using (0, b) as (x1, y1) and (x, y) as (x2, y2) in the slope formula,we have

(1)

or

(2)

Multiplying both sides of equation (2) by x gives

or

(3)

Equation (3) will be satisfied by any point on line L, including (0, b). It is called the slope-intercept form for a line, and we can state the following general result.

y � mx � b

mx � y � b

m �y � b

x

m �y � b

x � 0

y � b

x � 0

P(x, y)

(0, b)

y

x

L

Slope m

NOTE The coordinates are(0, b) because the x coordinateon the y axis is zero and the ycoordinate of the intercept is b.

228 CHAPTER 4 GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS

Finding the Equation of a Line

Write the equation of the line with slope 2 and y intercept (0, 3).Here m � 2 and b � 3. Applying the slope-intercept form, we have

y � 2x � 3

as the equation of the specified line.It is easy to see that whenever a linear equation is written in slope-intercept form, the

slope of the line is simply the x coefficient and the y intercept is determined by the constant.

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NOTE The slope-intercept form for theequation of a line isthe most convenient

form for entering an equationinto the calculator.

NOTE The x coefficient is 2; they intercept is (0, 3).

Finding the Slope and y Intercept of a Line

Find the slope and y intercept of the line with equation

2x � 3y � 3

To write the equation in slope-intercept form, we solve for y.

2x � 3y � 3

3y � �2x � 3 Subtract 2x from both sides.

y � � x � 1 Divide by 3.

We now see that the slope of the line is and the y intercept is (0, 1).�2

3

2

3

Example 2

The equation of a line with y intercept (0, b) and with slope m can be written asy � mx � b.

Definitions: Slope-Intercept Form for the Equation of a Line

Example 1

C H E C K Y O U R S E L F 1

Write the equation of the line with slope � and y intercept (0, �3).23

Note that the slope-intercept form now gives us a second (and generally more efficient)means of finding the slope of a line whose equation is written in standard form. Recall thatwe determined two specific points on the line and then applied the slope formula. Now,rather than using specific points, we can simply solve the given equation for y to rewrite theequation in the slope-intercept form and identify the slope of the line as the x coefficient.


Find the slope and y intercept of the line with equation.

3x � 4y � 8

FORMS OF LINEAR EQUATIONS SECTION 4.3 229

We can also use the slope-intercept form to determine whether the graphs of given equa-tions will be parallel, intersecting, or perpendicular lines.

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NOTE Two lines areperpendicular if their slopes arenegative reciprocals, so

m1 � �1

m2

Verifying That Two Lines Are Perpendicular

Show that the graphs of 3x � 4y � 4 and �4x � 3y � 12 are perpendicular lines.First, we solve each equation for y.

3x � 4y � 4

4y � �3x � 4

y � � x � 1 (4)

�4x � 3y � 12

3y � 4x � 12

y � x � 4 (5)

We now look at the product of the two slopes: . Any two lines whose slopes

have a product of �1 are perpendicular lines. These two lines are perpendicular.

�3

4 �

4

3� �1

4

3

3

4

The slope-intercept form can also be used in graphing a line, as Example 4 illustrates.

Example 3


Show that the graphs of the equations

�3x � 2y � 4 and 2x � 3y � 9

are perpendicular lines.

Example 4

Graphing the Equation of a Line

Graph the line 2x � 3y � 3.In Example 2, we found that the slope-intercept form for this equation is

To graph the line, plot the y intercept at (0, 1). Now, because the slope m is equal to

from (0, 1) we move to the right 3 units and then down 2 units, to locate a second point onthe graph of the line, here (3, �1). We can now draw a line through the two points to com-plete the graph.

�2

3,

y � �2

3x � 1

y

x

3 units to the right

Down 2 units

(3, �1)

(0, 1)

NOTE We treat to

move to the right 3 units anddown 2 units.

�23

as �2�3


The following algorithm summarizes the use of graphing with the slope-intercept form.

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Graph the line with equation

3x � 4y � 8

Hint: You worked with this equation in Check Yourself 2.

Step 1 Write the original equation of the line in slope-intercept form.Step 2 Determine the slope m and the y intercept (0, b).Step 3 Plot the y intercept at (0, b).Step 4 Use m (the change in y over the change in x) to determine a second

point on the desired line.Step 5 Draw a line through the two points determined above to complete the

graph.

Step by Step: Graphing by Using the Slope-Intercept Form

Often in mathematics it is useful to be able to write the equation of a line, given its slopeand any point on the line. We will now derive a third special form for a line for this purpose.

Suppose a line has slope m and passes through the known point P(x1, y1). Let Q(x, y) beany other point on the line. Once again we can use the definition of slope and write

(6)

Multiplying both sides of equation (6) by x � x1, we have

m(x � x1) � y � y1

or

y � y1 � m(x � x1) (7)

Equation (7) is called the point-slope form for the equation of a line, and all points lyingon the line [including (x1, y1)] will satisfy this equation. We can state the following generalresult.

m �y � y1

x � x1

Q(x, y)

P(x1, y1)

y � y1

x � x1

y

x

Slope is m

NOTE The desired form for theequation is

y � mx � b

NOTE Recall that a vertical linehas undefined slope. Theequation of a line withundefined slope passingthrough the point (x1, y1) isgiven by x � x1. For example,the equation of a line withundefined slope passingthrough (3, 5) is x � 3.

The equation of a line with slope m that passes through point (x1, y1) is given by

y � y1 � m(x � x1)

Definitions: Point-Slope Form for the Equation of a Line

FORMS OF LINEAR EQUATIONS SECTION 4.3 231©

200

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Example 5


Write the equation for the line that passes through point (3, �1) with a slope of 3.Letting (x1, y1) � (3, �1) and m � 3 in point-slope form, we have

y � (�1) � 3(x � 3)

or

y � 1 � 3x � 9

We can write the final result in slope-intercept form as

y � 3x � 10


Write the equation of the line that passes through point (�2, 4) with a slope ofWrite your result in slope-intercept form.

32

.

Because we know that two points determine a line, it is natural that we should be able towrite the equation of a line passing through two given points. Using the point-slope formtogether with the slope formula will allow us to write such an equation.


Write the equation of the line passing through (2, 4) and (4, 7).First, we find m, the slope of the line. Here

Now we apply the point-slope form with and (x1, y1) � (2, 4):

Distribute the .

Write the result in slope-intercept form.

y �3

2x � 1

y � 4 �3

2 x � 3

32

y � 4 �3

2 (x � 2)

m �3

2

m �7 � 4

4 � 2�

3

2

Example 6


Write the equation of the line passing through (�2, 5) and (1, 3). Write your result in slope-intercept form.

NOTE We could just as wellhave chosen to let

(x1, y1) � (4, 7)

The resulting equation will bethe same in either case. Taketime to verify this for yourself.



(a) Find the equation of a line passing through (7, �2) with a slope of zero.

We could find the equation by letting m � 0. Substituting into the slope-intercept form,we can solve for b.

y � mx � b

�2 � 0(7) � b

�2 � b

So,

y � 0x � 2 y � �2

It is far easier to remember that any line with a zero slope is a horizontal line and has theform

y � b

The value for b will always be the y coordinate for the given point.

(b) Find the equation of a line with undefined slope passing through (4, �5).

A line with undefined slope is vertical. It will always be of the form x � a, in which a isthe x coordinate for the given point. The equation is

x � 4

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Example 7


(a) Find the equation of a line with zero slope that passes through point (�3, 5).

(b) Find the equation of a line passing through (�3, �6) with undefined slope.

Alternate methods for finding the equation of a line through two points do exist and haveparticular significance in other fields of mathematics, such as statistics. Example 8 showssuch an alternate approach.

Example 8


Write the equation of the line through points (�2, 3) and (4, 5).First, we find m, as before.

We now make use of the slope-intercept equation, but in a slightly different form.

m �5 � 3

4 � (�2)�

2

6�

1

3

A line with slope zero is a horizontal line. A line with an undefined slope is vertical.Example 7 illustrates the equations of such lines.


Because y � mx � b, we can write

b � y � mx

Now, letting x � �2, y � 3, and we can calculate b.

With we can apply the slope-intercept form to write the equation of

the desired line. We have

y �1

3x �

11

3

m �1

3 and b �

11

3,

� 3 �2

3�

11

3

b � 3 � �1

3�(�2)

m �1

3,

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Repeat the Check Yourself 6 exercise, using the technique illustrated in Example 8.

We now know that we can write the equation of a line once we have been given appro-priate geometric conditions, such as a point on the line and the slope of that line. In someapplications, the slope may be given not directly but through specified parallel or perpen-dicular lines.

Example 9


Find the equation of the line passing through (�4, �3) and parallel to the line determinedby 3x � 4y � 12.

First, we find the slope of the given parallel line, as before.

3x � 4y � 12

4y � �3x � 12

y � � x � 3

Now, because the slope of the desired line must also be , we can use the point-slope formto write the required equation.

This simplifies to

y � � x � 6

and we have our equation in slope-intercept form.

3

4

y � (�3) � �3

4[x � (�4)]

�3

4

3

4

NOTE The line must passthrough (�4, �3), so let (x1, y1) � (�4, �3).

NOTE The slope of the given

line is � .34

NOTE We substitute thesevalues because the line mustpass through (�2, 3).


Because a vertical line cannot pass through a line with defined slope more than once,any nonvertical line can be represented as a function.

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Example 10

C H E C K Y O U R S E L F 1 0

Rewrite each equation as a function of x.

(a) y � �2x � 5 (b) 3x � 5y � 15

The process of finding the graph of a linear function is identical to the process of find-ing the graph of a linear equation.

Example 11

Graphing a Linear Function

Graph the function

f(x) � 3x � 5

We could use the slope and y intercept to graph the line, or we can find three points (thethird is just a check point) and draw the line through them. We will do the latter.

f(0) � �5 f(1) � �2 f(2) � 1

Writing Equations as Functions

Rewrite each linear equation as a function of x.

(a) y � 3x � 4

This can be rewritten as

f(x) � 3x � 4

(b) 2x � 3y � 6

We must first solve the equation for y (recall that this will give us the slope-interceptform).

�3y � �2x � 6

This can be rewritten as

f (x) �2

3x � 2

y �2

3x � 2


Find the equation of the line passing through (5, 4) and perpendicular to the linewith equation 2x � 5y � 10. Hint: Recall that the slopes of perpendicular lines arenegative reciprocals of each other.

NOTE Recall the vertical linetest, which was used todetermine whether a graphwas the graph of a function.


We will use the three points (0, �5), (1, �2), and (2, 1) to graph the line.

y

x(2, 1)

(1, �2)

(0, �5)

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Let f(x) � 4x � 2. Evaluate f as indicated.

(a) f(b) (b) f(4 � h)


Graph the function

f(x) � 5x � 3

One benefit of having a function written in f(x) form is that it makes it fairly easy to sub-stitute values for x. In Example 11, we substituted the values 0, 1, and 2. Sometimes it isuseful to substitute nonnumeric values for x.

Example 12

Substituting Nonnumeric Values for x

Let f(x) � 2x � 3. Evaluate f as indicated.

(a) f(a)

Substituting a for x in our equation, we see that

f(a) � 2a � 3

(b) f (2 � h)

Substituting 2 � h for x in our equation, we get

f(2 � h) � 2(2 � h) � 3

Distributing the 2, then simplifying, we have

f(2 � h) � 4 � 2h � 3

� 2h � 7


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C H E C K Y O U R S E L F A N S W E R S

1. 2.

3. 4.

5. 6. 7. (a) y � 5; (b) x � �3

8. 9.

10. (a) f(x) � �2x � 5; (b) 11.

12. (a) 4b � 2; (b) 4h � 14

y

x

8

�2

�4

6

4

2

2 4 6 8�2�4�6�8

�6

�8

f (x) � �3

5x � 3

y � �5

2x �

33

2y � �

2

3x �

11

3

y � �2

3x �

11

3y �

3

2x � 7

y

x

8

�4

�6

�8

6

4

2

4 6 8�2�4�6�8�2

2

m1 �3

2 and m2 � �

2

3; (m1)(m2) � �1

m �3

4 and the y intercept is (0,�2) y � �

2

3 x � 3

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Exercises

In exercises 1 to 8, match the graph with one of these equations: (a) y � 2x;

(b) y � x � 1; (c) y � �x � 3; (d) y � 2x � 1; (e) y � �3x � 2; (f)

(g) (h) y � �4x.

1. 2. 3.

4. 5. 6.

7. 8.y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y � �4

3x � 1; and

y �2

3x � 1;

4.3

Name

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

237

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In exercises 9 to 20, write each equation in slope-intercept form. Give its slope and yintercept.

9. x � y � 5 10. 2x � y � 3

11. 2x � y � �2 12. x � 3y � 6

13. x � 3y � 9 14. 4x � y � 8

15. 2x � 3y � 6 16. 3x � 4y � 12

17. 2x � y � 0 18. 3x � y � 0

19. y � 3 � 0 20. y � 2 � 0

In exercises 21 to 38, write the equation of the line passing through each of the given pointswith the indicated slope. Give your results in slope-intercept form, when possible.

21. (0, 2), m � 3 22. (0, �4), m � �2

23. 24. (0, �3), m � �2

25. (0, 4), m � 0 26.

27. (0, �5), m � 28. (0, �4), m � �

29. (1, 2), m � 3 30. (�1, 2), m � 3

31. (�2, �3), m � �3 32. (1, �4), m � �4

33. (5, �3), m � 34. (4, 3), m � 0

35. (2, �3), m is undefined 36. (2, �5), m �

37. (5, 0), m � � 38. (�3, 0), m is undefined4

5

1

4

2

5

3

4 5

4

(0, 5), m � �3

5

(0, 2), m �3

2

ANSWERS

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

238

In exercises 39 to 48, write the equation of the line passing through each of the given pairsof points. Write your result in slope-intercept form, when possible.

39. (2, 3) and (5, 6) 40. (3, �2) and (6, 4)

41. (�2, �3) and (2, 0) 42. (�1, 3) and (4, �2)

43. (�3, 2) and (4, 2) 44. (�5, 3) and (4, 1)

45. (2, 0) and (0, �3) 46. (2, �3) and (2, 4)

47. (0, 4) and (�2, �1) 48. (�4, 1) and (3, 1)

In exercises 49 to 58, write the equation of the line L satisfying the given geometricconditions.

49. L has slope 4 and y intercept (0, �2).

50. L has slope and y intercept (0, 4).

51. L has x intercept (4, 0) and y intercept (0, 2).

52. L has x intercept (�2, 0) and slope

53. L has y intercept (0, 4) and a 0 slope.

54. L has x intercept (�2, 0) and an undefined slope.

55. L passes through point (3, 2) with a slope of 5.

56. L passes through point (�2, �4) with a slope of

57. L has y intercept (0, 3) and is parallel to the line with equation y � 3x � 5.

58. L has y intercept (0, �3) and is parallel to the line with equation .

In exercises 59 to 70, write the equation of each line in function form.

59. L has y intercept (0, 4) and is perpendicular to the line with equation y � �2x � 1.

60. L has y intercept (0, 2) and is parallel to the line with equation y � �1.

61. L has y intercept (0, 3) and is parallel to the line with equation y � 2.

y �2

3x � 1

�3

2.

3

4.

�2

3

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sANSWERS

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

55.

56.

57.

58.

59.

60.

61.

239

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62. L has y intercept (0, 2) and is perpendicular to the line with equation 2x � 3y � 6.

63. L passes through point (�3, 2) and is parallel to the line with equation y � 2x � 3.

64. L passes through point (�4, 3) and is parallel to the line with equation y � �2x � 1.

65. L passes through point (3, 2) and is parallel to the line with equation

66. L passes through point (�2, �1) and is perpendicular to the line with equationy � 3x � 1.

67. L passes through point (5, �2) and is perpendicular to the line with equationy � �3x � 2.

68. L passes through point (3, 4) and is perpendicular to the line with equation

.

69. L passes through (�2, 1) and is parallel to the line with equation x � 2y � 4.

70. L passes through (�3, 5) and is parallel to the x axis.

71. Geometry. Find the equation of the perpendicular bisector of the segment joining(�3, �5) and (5, 9). Hint: First determine the midpoint of the segment. The midpoint

would be The perpendicular bisector passes through

that point and is perpendicular to the line segment connecting the points.

72. Geometry. Find the equation of the perpendicular bisector of the segment joining(�2, 3) and (8, 5).

In exercises 73 to 76, without graphing, compare and contrast the slopes and y intercepts ofthe equations.

73. (a) y � 2x � 1, (b) y � 2x � 5, (c) y � 2x

74. (a) y � 3x � 2, (b) y � x � 2, (c) y � �2x � 2

75. (a) y � 4x � 5, (b) y � �4x � 5, (c) y � 4x � 5

76. (a) y � 3x, (b) y � x, (c) y � �3x

In exercises 77 to 80, use your graphing utility to graph the following.

77. 3x � 5y � 30 78. 2x � 7y � 14

79. The line with slope and y intercept 80. The line with slope and at (0, 7). y intercept at (0, 3).

�1

5

2

3

1

3

3

4

��3 � 5

2,

�5 � 9

2 � � (1, 2).

y � �3

5x � 2

y �4

3x � 4.

62.

63.

64.

65.

66.

67.

68.

69.

70.

71.

72.

73.

74.

75.

76.

77.

78.

79.

80.

240

In exercises 81 to 88, use the graph to determine the slope and y intercept of the line. Thenwrite the equation of the line in slope-intercept form.

81. 82. 83.

84. 85. 86.

87. 88.

In exercises 89 to 94, graph the functions.

89. f(x) � 3x � 7 90. f(x) � �2x � 5 91. f(x) � �2x � 7y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

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sANSWERS

81.

82.

83.

84.

85.

86.

87.

88.

89.

90.

91.

241

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92. f(x) � �3x � 8 93. f(x) � �x � 1 94. f(x) � �2x � 5

In exercises 95 to 100, if f(x) � 4x � 3, find the following.

95. f(5) 96. f(0)

97. f(4) 98. f (�1)

99. f (�4) 100.

In exercises 101 to 106, if f(x) � 5x � 1, find the following.

101. f(a) 102. f (2r)

103. f(x � 1) 104. f(a � 2)

105. f(x � h) 106.f(x � h) � f (x)

h

f �1

2�

y

x

y

x

y

x

ANSWERS

92.

93.

94.

95.

96.

97.

98.

99.

100.

101.

102.

103.

104.

105.

106.

242

In exercises 107 to 110, if g(x) � �3x � 2, find the following.

107. g(m) 108. g(5n)

109. g(x � 2) 110. g(s � 1)

In exercises 111 to 114, let f(x) � 2x � 3.

111. Find f(1) 112. Find f(3)

113. Form the ordered pairs (1, f(1)) and (3, f(3)).

114. Write the equation of the line passing through the points determined by the orderedpairs in exercise 113.

Answers1. (e) 3. (a) 5. (b) 7. (h) 9. y � �x � 5, m � �1, y int: (0, 5)

11. y � 2x � 2, m � 2, y int: (0, 2) 13. y int: (0, 3)

15. y int: (0, �2) 17. y � 2x, m � 2, y int: (0, 0)

19. y � �3, m � 0, y int: (0, �3) 21. y � 3x � 2 23.

25. y � 4 27. 29. y � 3x � 1 31. y � �3x � 9

33. 35. x � 2 37. 39. y � x � 1

41. 43. y � 2 45. 47.

49. y � 4x � 2 51. 53. y � 4 55. y � 5x � 13

57. y � 3x � 3 59. 61. f(x) � 3 63. f(x) � 2x � 8

65. 67. 69.

71. 73. Same slope but different y int.

75. (a) and (b) have the same y intercept but (a) rises and (b) falls, both at the same rate.(a) and (c) have the same slope but different y int.

y � �4

7 x �

18

7

f (x) � �1

2xf (x) �

1

3x �

11

3f (x) �

4

3x � 2

f (x) �1

2x � 4

y � �1

2x � 2

y �5

2x � 4y �

3

2x � 3y �

3

4x �

3

2

y � �4

5x � 4y �

2

5x � 5

y �5

4x � 5

y �3

2x � 2

y �2

3x � 2, m �

2

3,

y � �1

3x � 3, m � �

1

3,

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ANSWERS

107.

108.

109.

110.

111.

112.

113.

114.

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77. 79.

81. Slope � 1; y int (0, 3); y � x � 3 83. Slope � 2; y int (0, 1); y � 2x � 185. Slope � �3; y int (0, 1); y � �3x � 187. Slope � �2; y int (0, �3); y � �2x � 3

89. 91.

93. 95. 17 97. 13 99. �19101. 5a � 1 103. 5x � 4105. 5x � 5h � 1 107. �3m � 2109. �3x � 4 111. 5113. (1, 5), (3, 9)

x

y

x

y

x

y

8

4

2

�2��4 2 4

y � 2_3x � 72

�2�2��4 2 4

y � 3_5 x � 6

244

4.3 forms of linear equations - mcgraw hill education · forms of linear equations section 4.3 229...

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