4: Writing Equations of Lines
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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
342
Chapter 8 • Coordinate Geometry and Linear Systems
It may be helpful for you to pause for a moment and look back over Examples 1, 2, and 3.
Notice that we used the same basic approach in all three examples; that is, we chose a variable
point (x, y) and used it, along with another known point, to determine the equation of the line.
You should also recognize that the approach we take in these examples can be generalized to
produce some special forms of equations for straight lines.
Point-Slope Form
Classroom Example
Find the equation of the line that has
a slope of 2 and contains the point
(Ϫ3, 2). Express the equation in
point-slope form.
EXAMPLE 4
Find the equation of the line that has a slope of m and contains the point (x1, y1 ) .
Solution
We choose (x, y) to represent any other point on the line
in Figure 8.37. The slope of the line given by
mϭ
y
y Ϫ y1
x Ϫ x1
(x, y)
(x1, y1)
enables us to obtain
y Ϫ y1 ϭ m(x Ϫ x1 )
x
Figure 8.37
We refer to the equation
y Ϫ y1 ϭ m(x Ϫ x1 )
as the point-slope form of the equation of a straight line. Instead of the approach we used in
Example 1, we could use the point-slope form to write the equation of a line with a given
slope that contains a given point—as the next example illustrates.
Classroom Example
Write the equation of the line that
5
has a slope of and contains the
6
point (3, Ϫ4).
EXAMPLE 5
Write the equation of the line that has a slope of
3
and contains the point (2, Ϫ4).
5
Solution
Substituting
3
for m and (2, Ϫ4) for (x1, y1) in the point-slope form, we obtain
5
y Ϫ y1 ϭ m(x Ϫ x1 )
3
y Ϫ (Ϫ4) ϭ (x Ϫ 2)
5
3
y ϩ 4 ϭ (x Ϫ 2)
5
5(y ϩ 4) ϭ 3(x Ϫ 2)
5y ϩ 20 ϭ 3x Ϫ 6
26 ϭ 3x Ϫ 5y
Multiply both sides by 5
8.4 • Writing Equations of Lines
343
Slope-Intercept Form
Now consider the equation of a line that has a slope of m and a y intercept of b (see Figure
8.38). A y intercept of b means that the line contains the point (0, b); therefore, we can use
the point-slope form.
y Ϫ y1 ϭ m(x Ϫ x1 )
y Ϫ b ϭ m(x Ϫ 0)
y Ϫ b ϭ mx
y ϭ mx ϩ b
y1 ϭ b and x1 ϭ 0
y
(0, b)
x
Figure 8.38
The equation
y ϭ mx ϩ b
is called the slope-intercept form of the equation of a straight line. We use it for three primary purposes, as the next three examples illustrate.
Classroom Example
Find the equation of the line that has a
2
slope of Ϫ and a y intercept of Ϫ3.
5
EXAMPLE 6
Find the equation of the line that has a slope of
1
and a y intercept of 2.
4
Solution
This is a restatement of Example 3, but this time we will use the slope-intercept form of a line
1
(y ϭ mx ϩ b) to write its equation. From the statement of the problem we know that m ϭ
4
and b ϭ 2. Thus substituting these values for m and b into y ϭ mx ϩ b, we obtain
y ϭ mx ϩ b
1
yϭ xϩ2
4
4y ϭ x ϩ 8
x Ϫ 4y ϭ Ϫ8
Same result as in Example 3
Remark: It is acceptable to leave answers in slope-intercept form. We did not do that in
Example 6 because we wanted to show that it was the same result as in Example 3.
344
Chapter 8 • Coordinate Geometry and Linear Systems
Classroom Example
Find the slope of the line with the
equation 4x Ϫ 10y ϭ 5.
EXAMPLE 7
Find the slope of the line with the equation 2x ϩ 3y ϭ 4.
Solution
We can solve the equation for y in terms of x, and then compare it to the slope-intercept form
to determine its slope.
2x ϩ 3y ϭ 4
3y ϭ Ϫ2x ϩ 4
2
4
yϭϪ xϩ
3
3
2
Compare this result to y ϭ mx ϩ b, and you see that the slope of the line is Ϫ . Furthermore,
3
4
the y intercept is .
3
Classroom Example
Graph the line detemined by the
3
equation y ϭ x ϩ 2.
4
EXAMPLE 8
2
Graph the line determined by the equation y ϭ x Ϫ 1.
3
Solution
y
Comparing the given equation to the general slope2
intercept form, we see that the slope of the line is ,
3
and the y intercept is Ϫ1. Because the y intercept is Ϫ1,
we can plot the point (0, Ϫ1). Then because the slope
2
is , let’s move 3 units to the right and 2 units up from
3
(0, Ϫ1) to locate the point (3, 1). The two points (0, Ϫ1)
and (3, 1) determine the line in Figure 8.39.
y = 23 x – 1
(3, 1)
(0, –1)
Figure 8.39
In general,
Classroom Example
Find the slope and y intercept of
each of the following lines, and
graph the lines:
(a) 12x ϩ 4y ϭ 8
1
(b) Ϫy ϭ Ϫ x ϩ 1
5
(c) x ϭ Ϫ3
If the equation of a nonvertical line is written in slope-intercept form, the coefficient of
x is the slope of the line, and the constant term is the y intercept.
(Remember that the concept of slope is not defined for a vertical line.) Let’s consider a
few more examples.
EXAMPLE 9
Find the slope and y intercept of each of the following lines and graph the lines:
(a) 5x Ϫ 4y ϭ 12
(b) Ϫy ϭ 3x Ϫ 4
(c) y ϭ 2
Solution
(a) We change 5x Ϫ 4y ϭ 12 to slope-intercept form to get
5x Ϫ 4y ϭ 12
Ϫ4y ϭ Ϫ5x ϩ 12
x
8.4 • Writing Equations of Lines
345
4y ϭ 5x Ϫ 12
5
yϭ xϪ3
4
5
(the coefficient of x) and the y intercept is Ϫ3 (the constant
4
5
term). To graph the line, we plot the y intercept, Ϫ3. Then because the slope is , we
4
can determine a second point, (4, 2), by moving 5 units up and 4 units to the right
from the y intercept. The graph is shown in Figure 8.40.
The slope of the line is
y
y = 54 x – 3
(4, 2)
x
(0, –3)
Figure 8.40
(b) We multiply both sides of the given equation by Ϫ1 to change it to slope-intercept
form.
Ϫy ϭ 3x Ϫ 4
y ϭ Ϫ3x ϩ 4
The slope of the line is Ϫ3, and the y intercept is 4. To graph the line, we plot the y interϪ3
cept, 4. Then because the slope is Ϫ3 ϭ
, we can find a second point, (1, 1), by
1
moving 3 units down and 1 unit to the right from the y intercept. The graph is shown
in Figure 8.41.
y
(0, 4)
y = −3x + 4
(1, 1)
x
Figure 8.41
(c) We can write the equation y ϭ 2 as
y ϭ 0(x) ϩ 2
346
Chapter 8 • Coordinate Geometry and Linear Systems
The slope of the line is 0, and the y intercept is 2. To graph the line, we plot the y intercept, 2. Then because a line with a slope of 0 is horizontal, we draw a horizontal line
through the y intercept. The graph is shown in Figure 8.42.
y
y=2
(0, 2)
x
Figure 8.42
Parallel and Perpendicular Lines
We can use two important relationships between lines and their slopes to solve certain kinds
of problems. It can be shown that nonvertical parallel lines have the same slope and that two
nonvertical lines are perpendicular if the product of their slopes is Ϫ1. (Details for verifying
these facts are left to another course.) In other words, if two lines have slopes m1 and m2,
respectively, then.
1. The two lines are parallel if and only if m1 ϭ m2.
2. The two lines are perpendicular if and only if (m1) (m2) ϭ Ϫ1.
The following examples demonstrate the use of these properties.
Classroom Example
(a) Verify that the graphs of
2x Ϫ 5y ϭ 7 and 6x Ϫ 15y ϭ 10
are parallel lines.
(b) Verify that the graphs of
3x ϩ 7y ϭ 4 and 14x Ϫ 6y ϭ
19 are perpendicular lines.
EXAMPLE 10
(a) Verify that the graphs of 2x ϩ 3y ϭ 7 and 4x ϩ 6y ϭ 11 are parallel lines.
(b) Verify that the graphs of 8x Ϫ 12y ϭ 3 and 3x ϩ 2y ϭ 2 are perpendicular lines.
Solution
(a) Let’s change each equation to slope-intercept form.
2x ϩ 3y ϭ 7
S
4x ϩ 6y ϭ 11
S
3y ϭ Ϫ2x ϩ 7
2
7
yϭϪ xϩ
3
3
6y ϭ Ϫ4x ϩ 11
4
11
yϭϪ xϩ
6
6
2
11
yϭϪ xϩ
3
6
2
Both lines have a slope of Ϫ , but they have different y intercepts. Therefore, the two
3
lines are parallel.
8.4 • Writing Equations of Lines
347
(b) Solving each equation for y in terms of x, we obtain
8x Ϫ 12y ϭ 3
S
3x ϩ 2y ϭ 2
S
Ϫ12y ϭ Ϫ8x ϩ 3
8
3
yϭ xϪ
12
12
2
1
yϭ xϪ
3
4
2y ϭ Ϫ3x ϩ 2
3
yϭϪ xϩ1
2
2
3
Because a b aϪ b ϭ Ϫ1 (the product of the two slopes is Ϫ1), the lines are
3
2
perpendicular.
Remark: The statement “the product of two slopes is Ϫ1” is the same as saying that the two
slopes are negative reciprocals of each other; that is, m1 ϭ Ϫ
Classroom Example
Find the equation of the line that
contains the point (Ϫ2, 2) and is
parallel to the line dertermined by
3x Ϫ y ϭ 9.
1
.
m2
EXAMPLE 11
Find the equation of the line that contains the point (1, 4) and is parallel to the line determined by x ϩ 2y ϭ 5.
Solution
First, let’s draw a figure to help in our analysis of the problem (Figure 8.43). Because the line
through (1, 4) is to be parallel to the line determined by x ϩ 2y ϭ 5, it must have the same
slope. Let’s find the slope by changing x ϩ 2y ϭ 5 to the slope-intercept form.
x ϩ 2y ϭ 5
2y ϭ Ϫx ϩ 5
1
5
yϭϪ xϩ
2
2
1
The slope of both lines is Ϫ . Now we can choose a variable point (x, y) on the line through
2
(1, 4) and proceed as we did in earlier examples.
yϪ4
1
ϭ
xϪ1
Ϫ2
1(x Ϫ 1) ϭ Ϫ2(y Ϫ 4)
x Ϫ 1 ϭ Ϫ2y ϩ 8
x ϩ 2y ϭ 9
y
(1, 4)
x + 2y = 5
(x, y)
(0, −25)
(5, 0)
Figure 8.43
x
348
Chapter 8 • Coordinate Geometry and Linear Systems
Classroom Example
Find the equation of the line that
contains the point (7, 5) and is
perpendicular to the line determined
by 4x Ϫ 3y ϭ 6.
EXAMPLE 12
Find the equation of the line that contains the point (Ϫ1, Ϫ2) and is perpendicular to the
line determined by 2x Ϫ y ϭ 6.
Solution
First, let’s draw a figure to help in our analysis of
the problem (Figure 8.44). Because the line through
(Ϫ1, Ϫ2) is to be perpendicular to the line determined
by 2x Ϫ y ϭ 6, its slope must be the negative reciprocal of the slope of 2x Ϫ y ϭ 6. Let’s find the slope of
2x Ϫ y ϭ 6 by changing it to the slope-intercept
form.
y
2x − y = 6
(3, 0)
(−1, −2)
2x Ϫ y ϭ 6
Ϫy ϭ Ϫ2x ϩ 6
y ϭ 2x Ϫ 6
The slope is 2
1
(the negative
2
reciprocal of 2), and we can proceed as before by
using a variable point (x, y).
The slope of the desired line is Ϫ
x
(x, y)
(0, −6)
Figure 8.44
yϩ2
1
ϭ
xϩ1
Ϫ2
1(x ϩ 1) ϭ Ϫ2(y ϩ 2)
x ϩ 1 ϭ Ϫ2y Ϫ 4
x ϩ 2y ϭ Ϫ5
We use two forms of equations of straight lines extensively. They are the standard form
and the slope-intercept form, and we describe them as follows.
Standard Form Ax ϩ By ϭ C, where B and C are integers, and A is a nonnegative integer
(A and B not both zero).
y ϭ mx ϩ b, where m is a real number representing the slope, and
b is a real number representing the y intercept.
Slope-Intercept Form
Concept Quiz 8.4
For Problems 1–10, answer true or false.
1. If two lines have the same slope, then the lines are parallel.
2. If the slopes of two lines are reciprocals, then the lines are perpendicular.
3. In the standard form of the equation of a line Ax ϩ By ϭ C, A can be a rational number
in fractional form.
4. In the slope-intercept form of an equation of a line y ϭ mx ϩ b, m is the slope.
5. In the standard form of the equation of a line Ax ϩ By ϭ C, A is the slope.
3
6. The slope of the line determined by the equation 3x Ϫ 2y ϭ Ϫ4 is .
2
8.4 • Writing Equations of Lines
7.
8.
9.
10.
349
The concept of slope is not defined for the line y ϭ 2.
The concept of slope is not defined for the line x ϭ 2.
The lines determined by the equations x Ϫ 3y ϭ 4 and 2x Ϫ 6y ϭ 11 are parallel lines.
The lines determined by the equations x Ϫ 3y ϭ 4 and x ϩ 3y ϭ 4 are perpendicular
lines.
Problem Set 8.4
For Problems 1–12, find the equation of the line that contains
the given point and has the given slope. Express equations
in the form Ax ϩ By ϭ C, where A, B, and C are integers.
(Objective 1a)
1. (2, 3), m ϭ
2
3
2. (5, 2), m ϭ
1
2
4. (5, Ϫ6), m ϭ
5. (Ϫ4, 8), m ϭ Ϫ
1
3
6. (Ϫ2, Ϫ4), m ϭ Ϫ
9. (0, 0), m ϭ Ϫ
4
9
11. (Ϫ6, Ϫ2), m ϭ 3
25. m ϭ 2 and b ϭ Ϫ1
1
27. m ϭ Ϫ and b ϭ Ϫ4
6
3
5
5
6
8. (Ϫ3, 9), m ϭ 0
10. (0, 0), m ϭ
5
and b ϭ 4
9
26. m ϭ 4 and b ϭ Ϫ3
3
7
3. (Ϫ3, Ϫ5), m ϭ
7. (3, Ϫ7), m ϭ 0
24. m ϭ
5
11
12. (2, Ϫ10), m ϭ Ϫ2
5
28. m ϭ Ϫ and b ϭ Ϫ1
7
29. m ϭ Ϫ1 and b ϭ
5
2
30. m ϭ Ϫ2 and b ϭ
7
3
5
1
31. m ϭ Ϫ and b ϭ Ϫ
9
2
For Problems 13–22, find the equation of the line that contains the two given points. Express equations in the form
Ax ϩ By ϭ C, where A, B, and C are integers. (Objective 1b)
32. m ϭ Ϫ
13. (2, 3) and (7, 10)
For Problems 33–44, determine the slope and y intercept of
the line represented by the given equation, and graph the line.
14. (1, 4) and (9, 10)
15. (3, Ϫ2) and (Ϫ1, 4)
16. (Ϫ2, 8) and (4, Ϫ2)
7
2
and b ϭ Ϫ
12
3
(Objective 2)
33. y ϭ Ϫ2x Ϫ 5
17. (Ϫ1, Ϫ2) and (Ϫ6, Ϫ7)
2
34. y ϭ x ϩ 4
3
18. (Ϫ8, Ϫ7) and (Ϫ3, Ϫ1)
35. 3x Ϫ 5y ϭ 15
19. (0, 0) and (Ϫ3, Ϫ5)
36. 7x ϩ 5y ϭ 35
20. (5, Ϫ8) and (0, 0)
37. Ϫ4x ϩ 9y ϭ 18
21. (0, 4) and (7, 0)
38. Ϫ6x ϩ 7y ϭ Ϫ14
22. (Ϫ2, 0) and (0, Ϫ9)
3
39. Ϫy ϭ Ϫ x ϩ 4
4
For Problems 23–32, find the equation of the line with the
given slope and y intercept. Leave your answers in slope-intercept form. (Objective 1a)
23. m ϭ
3
and b ϭ 2
5
40. 5x Ϫ 2y ϭ 0
41. Ϫ2x Ϫ 11y ϭ 11
2
11
42. Ϫy ϭ x ϩ
3
2
350
Chapter 8 • Coordinate Geometry and Linear Systems
43. 9x ϩ 7y ϭ 0
52. Contains the point (Ϫ4, 7) and is perpendicular to the
x axis
44. Ϫ5x Ϫ 13y ϭ 26
For Problems 45–60, write the equation of the line that satisfies the given conditions. Express final equations in standard form. (Objectives 1a, 1c, and 3)
45. x intercept of 2 and y intercept of Ϫ4
53. Contains the point (1, 3) and is parallel to the line
x ϩ 5y ϭ 9
54. Contains the point (Ϫ1, 4) and is parallel to the line
x Ϫ 2y ϭ 6
55. Contains the origin and is parallel to the line 4x Ϫ 7y ϭ 3
46. x intercept of Ϫ1 and y intercept of Ϫ3
56. Contains the origin and is parallel to the line
Ϫ2x Ϫ 9y ϭ 4
5
47. x intercept of Ϫ3 and slope of Ϫ
8
57. Contains the point (Ϫ1, 3) and is perpendicular to the
line 2x Ϫy ϭ 4
3
48. x intercept of 5 and slope of Ϫ
10
49. Contains the point (2, Ϫ4) and is parallel to the y axis
58. Contains the point (Ϫ2, Ϫ3) and is perpendicular to the
line x ϩ4y ϭ 6
50. Contains the point (Ϫ3, Ϫ7) and is parallel to the
x axis
59. Contains the origin and is perpendicular to the line
Ϫ2x ϩ 3y ϭ 8
51. Contains the point (5, 6) and is perpendicular to the
y axis
60. Contains the origin and is perpendicular to the line
y ϭ Ϫ5x
Thoughts Into Words
61. Explain the importance of the slope-intercept form
(y ϭ mx ϩ b) of the equation of a line.
63. How would you describe coordinate geometry to a group
of elementary algebra students?
62. What does it mean to say that two points “determine” a
line?
64. How can you tell by inspection that y ϭ 2x Ϫ 4 and
y ϭ Ϫ3x Ϫ 1 are not parallel lines?
Answers to the Concept Quiz
1. True
2. False
3. False
4. True
9. True
10. False
8.5
5. False
6. True
7. False
8. True
Systems of Two Linear Equations
OBJECTIVES
1
Solve linear systems of two equations by graphing
2
Solve linear systems of two equations by the substitution method
3
Recognize consistent and inconsistent systems of equations
4
Recognize dependent equations
5
Use a system of equations to solve word problems
Suppose we graph x Ϫ 2y ϭ 4 and x ϩ 2y ϭ 8 on the same set of axes, as shown in Figure
8.45. The ordered pair (6, 1), which is associated with the point of intersection of the two
lines, satisfies both equations. That is to say, (6, 1) is the solution for x Ϫ 2y ϭ 4 and x ϩ
2y ϭ 8.
8.5 • Systems of Two Linear Equations
351
To check this, we can substitute 6 for x and 1 for y in both equations.
x Ϫ 2y ϭ 4 becomes 6 Ϫ 2(1) ϭ 4
x ϩ 2y ϭ 8 becomes 6 ϩ 2(1) ϭ 8
A true statement
A true statement
y
x + 2y = 8
(6, 1)
x
x − 2y = 4
Figure 8.45
Thus we say that {(6, 1)} is the solution set of the system
a
x Ϫ 2y ϭ 4
b
x ϩ 2y ϭ 8
Two or more linear equations in two variables considered together are called a system of
linear equations. Here are three systems of linear equations:
a
x Ϫ 2y ϭ 4
b
x ϩ 2y ϭ 8
a
4x Ϫ y ϭ 5
° 2x ϩ y ϭ 9 ¢
7x Ϫ 2y ϭ 13
5x Ϫ 3y ϭ 9
b
3x ϩ 7y ϭ 12
To solve a system of linear equations means to find all of the ordered pairs that are solutions of all of the equations in the system. There are several techniques for solving systems
of linear equations. We will use three of them in this chapter: two methods are presented in
this section, and a third method is presented in Section 8.6.
To solve a system of linear equations by graphing, we proceed as in the opening discussion of this section. We graph the equations on the same set of axes, and then the ordered pairs
associated with any points of intersection are the solutions to the system. Let’s consider
another example.
Classroom Example
x ϩ 2y ϭ 5
Solve the system a
b.
x Ϫ y ϭ Ϫ4
EXAMPLE 1
Solve the system a
xϩ yϭ 5
b.
x Ϫ 2y ϭ Ϫ4
Solution
We can find the intercepts and a check point for each of the lines.
x ؊ 2y ؍؊4
x؉y؍5
x
y
0
5
2
5
0
3
Intercepts
Check point
x
y
0
Ϫ4
Ϫ2
2
0
1
Intercepts
Check point