4.3
Forms of Linear Equations
OBJECTIVES
4.3
1. Write the equation of a line given its slope and
y intercept
2. Find the slope and y intercept of a line from its
equation
3. Graph the equation of a line
4. Write the equation of a line given a point and a
slope
5. Write the equation of a line given two points
6. Express the equation of a line as a linear function
7. Graph a linear function
The special form
Ax By C
NOTE The coordinates are
(0, b) because the x coordinate
on the y axis is zero and the y
coordinate of the intercept is b.
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, we
will look at several forms for a linear equation. For the first of these forms, we use the concept 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) be
any other point on that line. Using (0, b) as (x1, y1) and (x, y) as (x2, y2) in the slope formula,
we have
y
L
Slope m
P(x, y)
y b
(0, b)
x 0
x
m
yb
x0
(1)
yb
x
(2)
© 2001 McGraw-Hill Companies
or
m
Multiplying both sides of equation (2) by x gives
mx y b
or
y mx b
(3)
Equation (3) will be satisfied by any point on line L, including (0, b). It is called the slopeintercept form for a line, and we can state the following general result.
227
228
CHAPTER 4
GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS
NOTE The slopeintercept form for the
equation of a line is
the most convenient
form for entering an equation
into the calculator.
Definitions: Slope-Intercept Form for the Equation of a Line
The equation of a line with y intercept (0, b) and with slope m can be written as
y mx b.
Example 1
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.
CHECK YOURSELF 1
2
Write the equation of the line with slope and y intercept (0, 3).
3
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 that
we 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 the
equation in the slope-intercept form and identify the slope of the line as the x coefficient.
Example 2
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
2
y x1
3
Subtract 2x from both sides.
Divide by 3.
2
We now see that the slope of the line is and the y intercept is (0, 1).
3
CHECK YOURSELF 2
Find the slope and y intercept of the line with equation.
3x 4y 8
© 2001 McGraw-Hill Companies
NOTE The x coefficient is 2; the
y intercept is (0, 3).
FORMS OF LINEAR EQUATIONS
SECTION 4.3
229
We can also use the slope-intercept form to determine whether the graphs of given equations will be parallel, intersecting, or perpendicular lines.
Example 3
Verifying That Two Lines Are Perpendicular
NOTE Two lines are
Show that the graphs of 3x 4y 4 and 4x 3y 12 are perpendicular lines.
First, we solve each equation for y.
perpendicular if their slopes are
negative reciprocals, so
3x 4y 4
1
m1 m2
4y 3x 4
3
y x1
4
(4)
4x 3y 12
3y 4x 12
4
y x4
3
(5)
3 4
We now look at the product of the two slopes: 1. Any two lines whose slopes
4 3
have a product of 1 are perpendicular lines. These two lines are perpendicular.
CHECK YOURSELF 3
Show that the graphs of the equations
3x 2y 4
and
2x 3y 9
are perpendicular lines.
The slope-intercept form can also be used in graphing a line, as Example 4 illustrates.
Example 4
y
© 2001 McGraw-Hill Companies
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
3 units to the right
(0, 1)
Down 2 units
x
(3, 1)
2
2
to
3
3
move to the right 3 units and
down 2 units.
NOTE We treat as
2
y x1
3
2
To graph the line, plot the y intercept at (0, 1). Now, because the slope m is equal to ,
3
from (0, 1) we move to the right 3 units and then down 2 units, to locate a second point on
the graph of the line, here (3, 1). We can now draw a line through the two points to complete the graph.
230
CHAPTER 4
GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS
CHECK YOURSELF 4
Graph the line with equation
3x 4y 8
Hint: You worked with this equation in Check Yourself 2.
The following algorithm summarizes the use of graphing with the slope-intercept form.
Step by Step: Graphing by Using the Slope-Intercept Form
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.
NOTE The desired form for the
equation is
y mx b
y
Slope is m
Q(x, y)
Often in mathematics it is useful to be able to write the equation of a line, given its slope
and 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) be
any other point on the line. Once again we can use the definition of slope and write
y y1
x x1
P(x1, y1)
x
m
y y1
x x1
(6)
Multiplying both sides of equation (6) by x x1, we have
m(x x1) y y1
or
(7)
Equation (7) is called the point-slope form for the equation of a line, and all points lying
on the line [including (x1, y1)] will satisfy this equation. We can state the following general
result.
NOTE Recall that a vertical line
has undefined slope. The
equation of a line with
undefined slope passing
through the point (x1, y1) is
given by x x1. For example,
the equation of a line with
undefined slope passing
through (3, 5) is x 3.
Definitions: Point-Slope Form for the Equation of a Line
The equation of a line with slope m that passes through point (x1, y1) is given by
y y1 m(x x1)
© 2001 McGraw-Hill Companies
y y1 m(x x1)
FORMS OF LINEAR EQUATIONS
SECTION 4.3
231
Example 5
Finding the Equation of a Line
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
CHECK YOURSELF 5
3
Write the equation of the line that passes through point (2, 4) with a slope of .
2
Write your result in slope-intercept form.
Because we know that two points determine a line, it is natural that we should be able to
write the equation of a line passing through two given points. Using the point-slope form
together with the slope formula will allow us to write such an equation.
Example 6
Finding the Equation of a Line
Write the equation of the line passing through (2, 4) and (4, 7).
First, we find m, the slope of the line. Here
m
NOTE We could just as well
74
3
42
2
Now we apply the point-slope form with m have chosen to let
(x1, y1) (4, 7)
© 2001 McGraw-Hill Companies
The resulting equation will be
the same in either case. Take
time to verify this for yourself.
3
and (x1, y1) (2, 4):
2
y4
3
(x 2)
2
3
Distribute the .
2
y4
3
x3
2
Write the result in slope-intercept form.
y
3
x1
2
CHECK YOURSELF 6
Write the equation of the line passing through (2, 5) and (1, 3). Write your result
in slope-intercept form.
CHAPTER 4
GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS
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.
Example 7
Finding the Equation of a Line
(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 the
form
yb
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 is
the x coordinate for the given point. The equation is
x4
CHECK YOURSELF 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 have
particular significance in other fields of mathematics, such as statistics. Example 8 shows
such an alternate approach.
Example 8
Finding the Equation of a Line
Write the equation of the line through points (2, 3) and (4, 5).
First, we find m, as before.
m
53
2
1
4 (2)
6
3
We now make use of the slope-intercept equation, but in a slightly different form.
© 2001 McGraw-Hill Companies
232
FORMS OF LINEAR EQUATIONS
SECTION 4.3
233
Because y mx b, we can write
b y mx
NOTE We substitute these
values because the line must
pass through (2, 3).
1
Now, letting x 2, y 3, and m , we can calculate b.
3
1
b3
(2)
3
3
2
11
3
3
1
11
and b , we can apply the slope-intercept form to write the equation of
3
3
the desired line. We have
With m y
1
11
x
3
3
CHECK YOURSELF 8
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 appropriate geometric conditions, such as a point on the line and the slope of that line. In some
applications, the slope may be given not directly but through specified parallel or perpendicular lines.
Example 9
Finding the Equation of a Line
Find the equation of the line passing through (4, 3) and parallel to the line determined
by 3x 4y 12.
First, we find the slope of the given parallel line, as before.
3x 4y 12
4y 3x 12
NOTE The slope of the given
© 2001 McGraw-Hill Companies
3
line is .
4
NOTE The line must pass
through (4, 3), so let
(x1, y1) (4, 3).
3
y x3
4
3
Now, because the slope of the desired line must also be , we can use the point-slope form
4
to write the required equation.
3
y (3) [x (4)]
4
This simplifies to
3
y x6
4
and we have our equation in slope-intercept form.
234
CHAPTER 4
GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS
CHECK YOURSELF 9
Find the equation of the line passing through (5, 4) and perpendicular to the line
with equation 2x 5y 10. Hint: Recall that the slopes of perpendicular lines are
negative reciprocals of each other.
test, which was used to
determine whether a graph
was the graph of a function.
Because a vertical line cannot pass through a line with defined slope more than once,
any nonvertical line can be represented as a function.
Example 10
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-intercept
form).
3y 2x 6
y
2
x2
3
This can be rewritten as
f (x) 2
x2
3
CHECK YOURSELF 10
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 finding 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 (the
third 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
© 2001 McGraw-Hill Companies
NOTE Recall the vertical line
FORMS OF LINEAR EQUATIONS
SECTION 4.3
235
We will use the three points (0, 5), (1, 2), and (2, 1) to graph the line.
y
(2, 1)
x
(1, 2)
(0, 5)
CHECK YOURSELF 11
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 substitute values for x. In Example 11, we substituted the values 0, 1, and 2. Sometimes it is
useful 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
© 2001 McGraw-Hill Companies
Distributing the 2, then simplifying, we have
f(2 h) 4 2h 3
2h 7
CHECK YOURSELF 12
Let f(x) 4x 2. Evaluate f as indicated.
(a) f(b)
(b) f(4 h)
CHAPTER 4
GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS
CHECK YOURSELF ANSWERS
2
3
1. y x 3
2. m and the y intercept is (0,2)
3
4
3
2
3. m1 and m2 ; (m1)(m2) 1
4.
2
3
y
8
6
4
2
8 6 4 2
2
2
4 6 8
x
4
6
8
3
2
11
5. y x 7
6. y x 7. (a) y 5; (b) x 3
2
3
3
2
11
5
33
8. y x 9. y x 3
3
2
2
3
y
10. (a) f(x) 2x 5; (b) f (x) x 3
11.
5
8
6
4
2
8 6 4 2
2
2
4 6 8
x
4
6
8
12. (a) 4b 2; (b) 4h 14
© 2001 McGraw-Hill Companies
236
Name
4.3
Exercises
Section
Date
In exercises 1 to 8, match the graph with one of these equations: (a) y 2x;
2
(b) y x 1; (c) y x 3; (d) y 2x 1; (e) y 3x 2; (f) y x 1;
3
4
(g) y x 1; and (h) y 4x.
3
1.
2.
1.
2.
3.
y
y
ANSWERS
y
3.
4.
x
x
x
5.
6.
7.
8.
4.
5.
y
6.
y
y
x
7.
x
8.
y
© 2001 McGraw-Hill Companies
x
y
x
x
237
ANSWERS
In exercises 9 to 20, write each equation in slope-intercept form. Give its slope and y
intercept.
9.
10.
9. x y 5
10. 2x y 3
11.
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
12.
13.
14.
15.
16.
17.
In exercises 21 to 38, write the equation of the line passing through each of the given points
with the indicated slope. Give your results in slope-intercept form, when possible.
18.
19.
20.
21. (0, 2), m 3
22. (0, 4), m 2
3
2
24. (0, 3), m 2
23. (0, 2), m 21.
22.
26. (0, 5), m 25. (0, 4), m 0
3
5
23.
27. (0, 5), m 24.
5
4
3
4
28. (0, 4), m 25.
29. (1, 2), m 3
30. (1, 2), m 3
31. (2, 3), m 3
32. (1, 4), m 4
27.
28.
33. (5, 3), m 29.
30.
2
5
35. (2, 3), m is undefined
31.
32.
37. (5, 0), m 33.
238
4
5
34. (4, 3), m 0
36. (2, 5), m 1
4
38. (3, 0), m is undefined
© 2001 McGraw-Hill Companies
26.
ANSWERS
In exercises 39 to 48, write the equation of the line passing through each of the given pairs
of points. Write your result in slope-intercept form, when possible.
34.
35.
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)
39.
47. (0, 4) and (2, 1)
48. (4, 1) and (3, 1)
40.
36.
37.
38.
In exercises 49 to 58, write the equation of the line L satisfying the given geometric
conditions.
41.
42.
49. L has slope 4 and y intercept (0, 2).
43.
2
50. L has slope and y intercept (0, 4).
3
44.
51. L has x intercept (4, 0) and y intercept (0, 2).
45.
46.
3
52. L has x intercept (2, 0) and slope .
4
47.
48.
53. L has y intercept (0, 4) and a 0 slope.
49.
54. L has x intercept (2, 0) and an undefined slope.
50.
55. L passes through point (3, 2) with a slope of 5.
51.
3
2
56. L passes through point (2, 4) with a slope of .
52.
53.
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 y 2
x 1.
3
54.
55.
© 2001 McGraw-Hill Companies
56.
In exercises 59 to 70, write the equation of each line in function form.
57.
59. L has y intercept (0, 4) and is perpendicular to the line with equation y 2x 1.
58.
60. L has y intercept (0, 2) and is parallel to the line with equation y 1.
59.
60.
61. L has y intercept (0, 3) and is parallel to the line with equation y 2.
61.
239
62.
63.
62. L has y intercept (0, 2) and is perpendicular to the line with equation 2x 3y 6.
64.
63. L passes through point (3, 2) and is parallel to the line with equation y 2x 3.
65.
64. L passes through point (4, 3) and is parallel to the line with equation y 2x 1.
66.
65. L passes through point (3, 2) and is parallel to the line with equation y 67.
66. L passes through point (2, 1) and is perpendicular to the line with equation
4
x 4.
3
y 3x 1.
67. L passes through point (5, 2) and is perpendicular to the line with equation
68.
y 3x 2.
69.
68. L passes through point (3, 4) and is perpendicular to the line with equation
3
y x 2.
5
70.
71.
69. L passes through (2, 1) and is parallel to the line with equation x 2y 4.
72.
70. L passes through (3, 5) and is parallel to the x axis.
73.
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
3 5 5 9
would be
,
(1, 2). The perpendicular bisector passes through
2
2
that point and is perpendicular to the line segment connecting the points.
74.
72. Geometry. Find the equation of the perpendicular bisector of the segment joining
(2, 3) and (8, 5).
75.
In exercises 73 to 76, without graphing, compare and contrast the slopes and y intercepts of
the equations.
76.
77.
73. (a) y 2x 1, (b) y 2x 5, (c) y 2x
78.
3
4
74. (a) y 3x 2, (b) y x 2, (c) y 2x 2
79.
1
3
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
79. The line with slope
at (0, 7).
240
78. 2x 7y 14
2
and y intercept
3
1
y intercept at (0, 3). 5
80. The line with slope and
© 2001 McGraw-Hill Companies
75. (a) y 4x 5, (b) y 4x 5, (c) y 4x 5
80.
ANSWERS
In exercises 81 to 88, use the graph to determine the slope and y intercept of the line. Then
write the equation of the line in slope-intercept form.
81.
82.
81.
82.
y
83.
y
y
83.
84.
x
x
x
85.
86.
87.
84.
85.
y
86.
y
y
88.
89.
x
x
x
90.
91.
87.
88.
y
y
x
x
In exercises 89 to 94, graph the functions.
89. f(x) 3x 7
90. f(x) 2x 5
© 2001 McGraw-Hill Companies
91. f(x) 2x 7
y
y
x
y
x
x
241
ANSWERS
92.
92. f(x) 3x 8
93. f(x) x 1
93.
94. f(x) 2x 5
y
y
y
94.
95.
x
x
x
96.
97.
98.
99.
In exercises 95 to 100, if f(x) 4x 3, find the following.
100.
95. f(5)
96. f(0)
97. f(4)
98. f (1)
101.
102.
103.
104.
99. f (4)
100. f
105.
106.
2
1
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.
© 2001 McGraw-Hill Companies
f(x h) f (x)
h
242
ANSWERS
107.
In exercises 107 to 110, if g(x) 3x 2, find the following.
108.
107. g(m)
108. g(5n)
109. g(x 2)
110. g(s 1)
109.
110.
111.
In exercises 111 to 114, let f(x) 2x 3.
112.
111. Find f(1)
112. Find f(3)
113.
113. Form the ordered pairs (1, f(1)) and (3, f(3)).
114.
114. Write the equation of the line passing through the points determined by the ordered
pairs in exercise 113.
Answers
1. (e)
3. (a)
5. (b)
7. (h)
1
3
11. y 2x 2, m 2, y int: (0, 2)
15. y 25. y 4
27. y 2
x5
5
3
3
41. y x 4
2
33. y 65.
© 2001 McGraw-Hill Companies
71.
75.
17. y 2x, m 2, y int: (0, 0)
21. y 3x 2
29. y 3x 1
4
5
35. x 2
37. y x 4
43. y 2
45. y 3
x3
2
23. y 3
x2
2
31. y 3x 9
39. y x 1
47. y 5
x4
2
1
53. y 4
55. y 5x 13
2
1
y 3x 3
59. f (x) x 4
61. f(x) 3
63. f(x) 2x 8
2
4
1
11
1
f (x) x 2
67. f (x) x 69. f (x) x
3
3
3
2
4
18
y x
73. Same slope but different y int.
7
7
(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.
49. y 4x 2
57.
5
x5
4
1
3
13. y x 3, m , y int: (0, 3)
2
2
x 2, m , y int: (0, 2)
3
3
19. y 3, m 0, y int: (0, 3)
9. y x 5, m 1, y int: (0, 5)
51. y x 2
243
77.
y 3_5 x 6
2
4
2
2
79.
y 2_3 x 7
8
4
4
2
2
4
2
2
4
81. Slope 1; y int (0, 3); y x 3
83. Slope 2; y int (0, 1); y 2x 1
85. Slope 3; y int (0, 1); y 3x 1
87. Slope 2; y int (0, 3); y 2x 3
89.
y
91.
x
93.
x
95.
101.
105.
109.
113.
y
y
17
97. 13
99. 19
5a 1
103. 5x 4
5x 5h 1
107. 3m 2
3x 4
111. 5
(1, 5), (3, 9)
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