3-6
ALGEBRA
3-6
Lines in the Coordinate Plane
1. Plan
Objectives
1
2
To graph lines given their
equations
To write equations of lines
Check Skills You’ll Need
• To graph lines given their
Find the slope of the line that contains each pair of points.
equations
2
3
4
5
6
Graphing Lines in SlopeIntercept Form
Graphing Lines Using
Intercepts
Transforming to SlopeIntercept Form
Using Point-Slope Form
Writing an Equation of a Line
Given Two Points
Equations of Horizontal and
Vertical Lines
. . . And Why
To determine whether a
wheelchair ramp complies
with the law, as in Exercise 52
René Descartes published his
philosophical treatise Discours de
la méthode in 1637. Its appendix
Géométrie contained the first
published record of methods of
analytic geometry, permanently
sealing the partnership between
geometry and algebra. Within
30 years, the methods of analytic
geometry would lead to the
invention of calculus.
7. G(-8, -9), H(-3, -5) 45
8. L(7, -10), M(1, -4) 1
New Vocabulary • slope-intercept form
• standard form of a linear equation • point-slope form
1
1 1 Graphing Lines
Part
In algebra, you learned that the graph of a linear
equation is a line. The slope-intercept form
of a linear equation is y = mx + b, where m is
the slope of the line and b is the y-intercept.
Each line at the right has slope 2, but the lines
have y-intercepts of 3, -1, and -4.
Vocabulary Tip
Math Background
5. C(0, 1), D(3, 3) 32
undefined or
4. K(-3, 3), T(-3, 1) no slope
6. E(-1, 4), F(3, -2) 23
3. R(-3, -4), S(5, -4) 0
of lines
The y-intercept is the
y-coordinate of the point
where a line crosses the
y-axis. The x-intercept is
the x-coordinate of the
point where a line crosses
the x-axis.
By Postulate 1-1 (two points determine a line),
you need only two points to graph a line. The
y-intercept gives you one point. You can use
the slope to plot another.
1
EXAMPLE
8
4
O
y 2x 1
3
4
x
y 2x 4
6
y
Move right
4 units
y = 34 x + 2
Move up
3 units
See p. 124E for a list of the
resources that support this lesson.
1.
Start at
(0, 2)
y
O
PowerPoint
1
x
–6
O
–2
2
4
slope y-intercept
x
6
–2
Bell Ringer Practice
3
Check Skills You’ll Need
Quick Check
For intervention, direct students to:
166
1 Graph the line y = 2 1 x - 2. See left.
2
Chapter 3 Parallel and Perpendicular Lines
Special Needs
Below Level
L1
For Example 1, help students see that another point
on the line can be found by moving three units up
and four units right from (4, 5), or by moving three
units down and four units left from (0, 2).
166
1
Graph the line y = 34 x + 2.
Lesson Planning and
Resources
Algebra Review, page 165
y
y 2x 3
Graphing Lines in Slope-Intercept Form
More Math Background: p. 124D
Slope
page 165
2. P(3, 0), X(0, -5) 53
1. A(-2, 2), B(4, -2) 32
• To write equations
Examples
1
GO for Help
What You’ll Learn
learning style: visual
L2
Encourage students to plot three points when
graphing a linear equation. Because algebra errors
produce noncollinear points, students can check their
work.
learning style: visual
The standard form of a linear equation is Ax + By = C, where A, B, and C
are real numbers and A and B are not both zero. To graph an equation written in
standard form, you can readily find two points for the graph by finding the x- and
y-intercepts.
Vocabulary Tip
In the standard form, A, B,
and C are constants. In
Example 2, 6x + 3y = 12 is
in standard form with
A = 6, B = 3, and C = 12.
Guided Instruction
2
EXAMPLE
Graphing Lines Using Intercepts
Math Tip
Algebra Graph 6x + 3y = 12.
Step 1 To find the y-intercept,
substitute 0 for x; solve for y.
Step 2 To find the x-intercept,
substitute 0 for y; solve for x.
6x + 3y = 12
6x + 3y = 12
6(0) + 3y = 12
6x + 3(0) = 12
3y = 12
6x = 12
y=4
x=2
The y-intercept is 4.
A point on the line is (0, 4).
PowerPoint
Additional Examples
2
(2, 0)
2 O
1 Use the slope and y-intercept
to graph the line y = -2x + 9.
Check that points (0, 9) and (1, 7)
are on students’ graphs.
x
1
2
2.
Quick Check
2 Graph -2x + 4y = -8.
O y
1
2
4
x
As an alternative, you can graph an equation in standard form by transforming it
into slope-intercept form. Knowing the slope and y-intercept beforehand can give
you a good mental image of what the graph should look like.
3
EXAMPLE
Transforming to Slope-Intercept Form
Algebra Graph 4x - 2y = 9.
Step 1 Transform the equation to
slope-intercept form.
4x - 2y = 9
2
y
O
Step 2 Use the y-intercept and the slope
to plot two points and draw the
line containing them.
22y
9
24x
22 = –2 + 22
y = 2x - 29
The y-intercept is 24 12 and
2
2
2 Use the x-intercept and
y-intercept to graph 5x - 6y = 30.
Check that points (6, 0) and
(0, –5) are on students’ graphs.
3 Transform the equation
-6x + 3y = 12 to slope-intercept
form, and then graph the
resulting equation. y ≠ 2x ± 4;
check that points (1, 6) and
(0, 4) are on students’ graphs.
y
-2y = -4x + 9
x
EXAMPLE
Students are accustomed to
solving for x and adding to each
side of the equation. Remind
them that the slope-intercept
form isolates y.
The x-intercept is 2.
A point on the line is (2, 0).
y
(0, 4)
4
For the standard form of a linear
equation, review the meaning
of real numbers, and consider
the case A = B = 0, where there
is no line.
3
Step 3 Plot (0, 4) and (2, 0). Draw the
line containing the two points.
3.
2. Teach
2
O
1
4
x
2
4
the slope is 2.
Quick Check
3 Graph -5x + y = -3. See left.
Lesson 3-6 Lines in the Coordinate Plane
Advanced Learners
167
English Language Learners ELL
L4
Have students find the slope and intercept of the line
in standard form ax + by = c.
learning style: verbal
Write the general forms of the point-slope, slopeintercept, and standard form of a linear equation on
the board. Point out that the names relate to what
information the respective form provides.
learning style: visual
167
Guided Instruction
5
EXAMPLE
2
1 2 Writing Equations of Lines
Part
Visual Learners
A third form for an equation of a line is point-slope form. The point-slope form
for a nonvertical line through point (x1, y1) with slope m is y - y1 = m(x - x1).
Have students use coordinate
graphs to see that slopes of -34,
-4 and 4 are the same.
-3
3,
6
EXAMPLE
4
EXAMPLE
Using Point-Slope Form
Algebra Write an equation of the line through point P(-1, 4) with slope 3.
Error Prevention
y - y1 = m(x - x1)
Students may think the equation
of a horizontal line begins with
“x =” because the x-axis is
horizontal. Point out that when
the value of x is constant, the
value of y changes to form a
vertical line like the y-axis, and
when the value of y is constant,
the value of x changes to form
a horizontal line like the x-axis.
Quick Check
Use point-slope form.
y - 4 = 3[x - (-1)]
Substitute 3 for m and (–1, 4) for (x1, y1).
y - 4 = 3(x + 1)
Simplify.
4 Write an equation of the line with slope -1 that contains point P(2, -4).
y 4 1(x 2)
By Postulate 1-1, you need only two points to write an equation of a line.
5
EXAMPLE
Writing an Equation of a Line Given Two Points
Algebra Write an equation of the line through A(-2, 3) and B(1, -1).
PowerPoint
Additional Examples
Step 1 Find the slope.
y 2y
m = x2 2 x1
2
1
4 Write an equation in point-
23
m = 121
2 (22)
slope form of the line with slope
-8 that contains P(3, –6).
y ± 6 ≠ –8(x – 3)
Substitute (–2, 3) for (x1, y1) and (1, –1) for (x2, y2).
m = 2 43
Simplify.
Step 2 Select one of the points. Write an equation in point-slope form.
5 Write an equation in pointslope form of the line that
contains the points G(4, -9)
and H(-1, 1).
y ± 9 ≠ –2(x – 4) or
y – 1 ≠ –2(x ± 1)
y - y1 = m(x - x1)
y - 3 = 2 43 fx 2 (22)g
y - 3 = 2 43 (x 1 2)
Quick Check
Substitute (-2, 3) for (x1, y1) and 2 43 for the slope.
Simplify.
5 Write an equation of the line that contains the points P(5, 0) and Q(7, -3).
6 Write equations for the
horizontal line and the vertical
line that contain A(-7, -5).
horizontal line: y ≠ –5; vertical
line: x ≠ –7
y 0 23 (x 5) or y 3 32 (x 7)
Recall that the slope of a horizontal line is 0 and the slope of a vertical line is
undefined. Thus, horizontal and vertical lines have easily recognized equations.
6
Resources
• Daily Notetaking Guide 3-6 L3
• Daily Notetaking Guide 3-6—
L1
Adapted Instruction
EXAMPLE
Equations of Horizontal and Vertical Lines
Write equations for the horizontal line
and the vertical line that contain P(3, 2).
3
2 O
y
P (3, 2)
2
4
x
Closure
Every point on the horizontal line through P(3, 2) has a y-coordinate of 2.
The equation of the line is y = 2. It crosses the y-axis at (0, 2).
Write the equation of the line
containing the points (-3, 2) and
(3, 14) in point-slope form, slopeintercept form, and standard
form. y – 2 ≠ 2(x ± 3) or
y – 14 ≠ 2(x – 3); y ≠ 2x ± 8;
2x – y ≠ –8
Every point on the vertical line through P(3, 2) has an x-coordinate of 3.
The equation of the line is x = 3. It crosses the x-axis at (3, 0).
Quick Check
168
6 Write equations of the horizontal and vertical lines that contain the point P(5, -1).
y 1; x 5
Chapter 3 Parallel and Perpendicular Lines
23 – 28. Equations may vary
from the pt. chosen.
Samples are given.
168
26. y 4 1(x 4)
23. y 5 3
5(x 0)
24. y 2 12 (x 6)
27. y 0 21 (x 1)
25. y 6 1(x 2)
28. y 10 23 (x 8)
EXERCISES
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
3. Practice
Practice and Problem Solving
Assignment Guide
Practice by Example
x 2 Algebra Graph each line. 1–4. See back of book.
1. y = x + 2
Examples 1, 2
2. y = 3x + 4
(pages 166, 167)
GO for
Help
3. y = 12 x - 1
4. y = 2 53 x + 2
6. 3x + y = 15
8. 6x - y = 3
9. 10x + 5y = 40
7. 5x - 2y = 20
10. 1.2x + 2.4y = 2.4
2
Example 3 x Algebra Write each equation in slope-intercept form and graph the line.
11–16. See back of book.
(page 167)
11. y = 2x + 1
12. y - 1 = x
13. y + 2x = 4
14. 8x + 4y = 16
16. 34 x - 21 y = 18
15. 2x + 6y = 6
Example 4 x 2 Algebra Write an equation in point-slope form of the line that contains the given
point and has the given slope.
(page 168)
y 1 3(x 4)
y 3 2(x 2)
y 5 1(x 3)
17. P(2, 3), slope 2
18. X(4, -1), slope 3
19. R(-3, 5), slope -1
22. y 4 1(x 0) or
y4x
Example 5
(page 168)
21. V(6, 1), slope 12
1
20. A(-2, -6), slope -4
22. C(0, 4), slope 1
See left.
y 6 4(x 2)
y 1 2 (x 6)
Write an equation in point-slope form of the line that contains the given points.
23–28. See margin p. 168.
23. D(0, 5), E(5, 8)
24. F(6, 2), G(2, 4)
25. H(2, 6), K(-1, 3)
26. A(-4, 4), B(2, 10)
Example 6
(page 168)
B
Apply Your Skills
27. L(-1, 0), M(-3, -1)
Graph each line. 33–37. See back of book.
34. y = -2
35. x = 9
37. y = 6
NASA’s Advanced
Communications Technology
Satellite has a capacity for
250,000 phone calls.
Homework Quick Check
To check students’ understanding
of key skills and concepts, go over
Exercises 8, 20, 38, 53, 56.
Exercise 38 Ask: In which
quadrant of the coordinate plane
is the graph of this equation
relevant? Quadrant 1
GPS Guided Problem Solving
41. a. What is the slope of the y-axis? Explain. Undefined; it is a vertical line.
b. Write an equation for the y-axis. x 0
L4
Identify the form of each equation. To graph the line, would you use the given form
or change to another form? Explain.
42–44. See margin.
42. -5x - y = 2
43. y = 41 x - 27
44. y + 2 = -(x - 4)
Lesson 3-6 Lines in the Coordinate Plane
42. The eq. is in standard
form; change to slopeintercept form, because
it is easy to graph the
eq. from that form.
43. The eq. is in slope-int.
form; use slope-int.
form, because the eq. is
already in that form.
L2
L1
Adapted Practice
Practice
Name
Class
L3
Date
Practice 3-6
Slopes of Parallel and Perpendicular Lines
Are the lines parallel, perpendicular, or neither? Explain.
2. y = 12 x + 1
1. y = 3x – 2
y = 1x + 2
3
3. 32 x + y = 4
y = -2 x + 8
3
-4y = 8x + 3
5. y = 2
6. 3x + 6y = 30
x=0
4. -x - y = -1
y+x=7
8. 31 x + 12 y = 1
3y + 1x = 1
4
2
7. y = x
4y + 2x = 9
8y - x = 8
Are lines l1 and l2 parallel, perpendicular, or neither? Explain.
9.
y
ᐍ1
–6 –4 –2
–2
–4
–6
12.
10.
ᐍ2
6
4
13.
y
ᐍ1
–6 –4 –2
–2
–4
–6
–2
–4
–6
ᐍ1
ᐍ1
ᐍ2
–6
4 6 x
y ᐍ
2
ᐍ1
6
4
2
ᐍ2
6
4
2
–6 –4
y
11.
y
ᐍ2
2 4 6 x
x
–6 –4 –2
–2
–4
–6
6
4
–2
–2
–4
–6
14.
2 4 6 x
y
6
6
4
2
2
ᐍ1
2 4 6 x
2
–6 –4 –2
–2
–4
–6
ᐍ2
2 4 6 x
* )
Write an equation for the line perpendicular to XY that contains point Z.
*
)
15. XY : 3x + 2y = -6, Z(3, 2)
* )
16. XY : y = 34 x + 22, Z(12, 8)
*
)
17. XY : -x + y = 0, Z(-2, -1)
* )
Write an equation for the line parallel to XY that contains point Z.
*
)
18. XY : 6x - 10y + 5 = 0, Z(-5, 3)
42 – 44. Answers may vary.
Samples are given.
L3
Reteaching
39. Error Analysis A classmate claims that having no slope and having a slope of 0
are the same. Is your classmate correct? Explain. No; a line with no slope is
a vertical line. 0 slope is a horizontal line.
40. a. What is the slope of the x-axis? Explain. m 0; it is a horizontal line.
b. Write an equation for the x-axis. y 0
Connection
64-69
70-80
Enrichment
36. y = 4
38. Telephone Rates The equation C = $.05m + $4.95 represents the cost (C) of
a long distance telephone call of m minutes.
a. What is the slope of the line? 0.05
b. What does the slope represent in this situation? the cost per minute
c. What is the y-intercept (C-intercept)? 4.95
the initial charge for
d. What does the y-intercept represent in this situation? a call
Real-World
Test Prep
Mixed Review
28. P(8, 10), Q(-4, 2)
Write equations for (a) the horizontal line and (b) the vertical line that contain the
given point.
a. y 2
a. y 4
a. y 1
a. y 7
29. A(4, 7)
30. Y(3, -2)
31. N(0, -1)
32. E(6, 4)
b. x 3
b. x 6
b. x 0
b. x 4
33. x = 3
2 A B
17-32, 39-44, 47-52,
54-56
C Challenge
57-63
x 2 Algebra Graph each line using intercepts. 5–10. See back of book.
5. 2x + 6y = 12
1 A B 1-16, 33-38, 45, 46, 53
*
)
19. XY : y = -1, Z(0, 0)
21. Aviation Two planes are flying side by side at the same altitude. It is
important that their paths do not intersect. One plane is flying along the
path given by the line 4x - 2y = 10. What is the slope-intercept form
of the line that must be the path of another plane passing through the
point L(-1, -2) so that the planes do not collide? Graph the paths of
the two planes.
* )
20. XY : x = 12 y + 1, Z(1, -2)
© Pearson Education, Inc. All rights reserved.
A
169
44. The eq. is in point-slope
form; use point-slope
form, because the eq. is
already in that form.
169
4. Assess & Reteach
GO
PowerPoint
Critical Thinking Graph three different lines having the given property. Describe
how the equations of these lines are alike and how they are different.
nline
Homework Help
Visit: PHSchool.com
Web Code: aue-0306
Lesson Quiz
1. Find the x-intercept and the
y-intercept of the line
5x + 4y = -80.
x-intercept: –16,
y-intercept: –20
Three points are on a coordinate
plane: A(1, 5), B(-2, -4), and
C(6, -4).
3
1
52. 10
0.3, 12
0.083;
3
1
10 S 12 ; it is possible
only if the ramp
zigzags.
2. Write an equation in pointslope form of the line with
slope -1 that contains point C.
y ± 4 ≠ –1(x – 6)
Real-World
3. Write an equation in pointslope form of the line that
contains points A and B.
y – 5 ≠ 3(x – 1) or
y ± 4 ≠ 3(x ± 2)
45. The lines have slope 2.
See back of book.
46. The lines have y-intercept 2.
See margin.
47. Graphing Calculator Graphing calculators use slope-intercept form (rather
than standard form or point-slope form) to graph lines. Choose either Exercise
45 or Exercise 46 and write three equations for the lines you graphed. Use the
Y= window of your graphing calculator to enter your equations. Press
.
Do the graphs on the screen confirm the description you wrote previously?
Check students’ work.
Graph each pair of lines. Then find their point of intersection.
48–51. See margin pp. 170–171.
48. y = -4, x = 6 49. x = 0, y = 0
50. x = -1, y = 3 51. y = 5, x = 4
52. Building Access By law, the maximum slope of a ramp in new construction is
1
12. The plan for the new library shows a 3-ft height from the ground to the
main entrance. The distance from the sidewalk to the building is 10 ft. Can you
design a ramp for the library that complies with the law? Explain. See left.
Connection
1
To visualize a slope of 12 ,
think “one foot over, one
inch up.”
4. Write an equation of the line
that contains B and C. y ≠ –4
5. Graph and label the equations
of the lines in Exercises 2–4
above.
y
8
6
y 4 1(x 6)
4
y 5 3(x 1)
53. Writing Describe the similarities of and the differences between the graphs
of the equations y = 5x - 2 and y = -5x - 2. See margin p. 171.
2
8 6 4 2 O
2
y 4
2
4
6
8 x
4
6
8
Alternative Assessment
Have students work in pairs to
explain in writing how to write
the equation of a line in three
different forms. They should give
an example of a question when
each form would be used.
Exercises 57–60 Suggest that
students calculate and compare
slopes.
56c. The abs. value of
the slopes is the
same, but one slope
is pos. and the other
is neg. One y-int. is
at (0, 0) and the
other is at (0, 10).
54. Open-Ended Write equations for three different lines that contain the
point (5, 6). Answers may vary. Sample: x 5, y 6 2(x 5),
yx1
55. Critical Thinking The x-intercept of a line is 2 and the y-intercept is 4. Use
this information to write an equation for the line. See margin p. 171.
5
56. The vertices of a triangle are A(0, 0), B(2, 5), and C(4, 0). a. y 0 2(x 0)
5
or y 2 x
GPS a. Write an equation for the line through A and B.
b. Write an equation for the line through B and C. b. y 5 25(x 2) or
c. Compare the slopes and y-intercepts of the two lines.
y 5 x 10
2
C
Challenge
57. Yes; the slope of AB
equals the slope of BC.
58. No; the slope of DE
does not equal the
slope of EF .
60.Yes; the slope of JK
equals the slope of KL.
170
Do the three points lie on one line? Justify your answer. 57–58. See left.
57. A(5, 6), B(3, 2), C(6, 8)
59. G(5, -4), H(2, 3), I(-1, 10)
60. J(-2, 9), K(1, -1), L(4, -11)
See left.
Yes; the slope of GH equals the slope of HI.
A line passes through the given points. Write an equation for the line in pointslope form. Then, rewrite the equation in standard form with integer coefficients.
61. R(-2, 2), S(0, 8)
y 2 3(x 2);
3x y 8
Chapter 3 Parallel and Perpendicular Lines
46.
4
y
1
2 O
170
58. D(-2, -2), E(4, -4), F(0, 0)
2
x
2
62. T(5, 5), W(7, 6)
y 5 12 (x 5);
x 2y 5
The slopes are all
different, and the
y-intercepts are the
same.
63. X(2, 6), Y(5, 8)
y 6 23 (x 2);
2x 3y 14
48.
O y
1
3
5
1
3
5
x6
(6, 4)
y 4
7x
Test Prep
Test Prep
Standardized
Test Prep
Multiple Choice
Resources
64. Which equation is equivalent to 15x + 3y = 10? D
A. y = 5x + 10
3
B. y = -5x - 10
3
D. y = -5x + 10
3
C. y = 5x - 10
3
65. Which pair of points A(-2, 5), B(-1, -2), C(4, -5), and D(7, 0), lie on the
line with y-intercept closest to the origin? G
F. A and B
G. A and C
H. B and C
J. B and D
66. What is the y-intercept of the line whose equation is y 1 9 5 2(x 2 3) ?
A. 15
C. -15
B. 9
y
(0, 6) 6
67. What is the slope of the line? J
G. -51
H. 15
J. 5
C
D. -9
Use the graph at the right for Exercises 67–68.
F. -5
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 193
• Test-Taking Strategies, p. 188
• Test-Taking Strategies with
Transparencies
4
(1, 1)
68. Which equation is the equation for the line? C
A. 5y 5 x 2 6
4 2
B. y 5 5x 2 6
2
O
4x
2
2
C. 25x 1 y 5 6
D. x 1 5y 5 6
Short Response
69. The slope of line a is 32 and its y-intercept is 12. Line b passes through
(4, 1) and (7, -3).
a. Write an equation for each line. a–b. See margin.
b. Graph both lines on the same coordinate plane. From the graph, what is
their point of intersection?
53. The y-intercepts are the
same, and the lines have
the same steepness.
One line rises from left
to right while the other
falls from left to right.
Mixed Review
GO for
Help
Lesson 3-5
Find the sum of the measures of the angles of each polygon.
Lesson 2-2
70. a nonagon
71. a pentagon
72. an 11-gon
73. a 14-gon
1260
540
1620
2160
Is each statement a good definition? If not, find a counterexample.
74. A quadrilateral is a polygon with four sides. yes
75. Skew lines are lines that don’t intersect. No; parallel lines never intersect,
but they are not skew.
76. An acute triangle is a triangle with an acute angle.
No; all obtuse > have two acute '.
)
2
Lesson 1-7 x Algebra For Exercises 77–80, PQ is the bisector of &MPR. Solve for a and find
the missing angle measure.
77. m&MPQ = 3a, m&QPR = 2a + 5, m&MPR = 7 a 5; mlMPR 30
78. m&MPQ = 7a, m&QPR = 4a + 12, m&MPR = 7 a 4; mlMPR 56
4
55. (2, 0), (0, 4); m 02 2
20 24
2 2
y 0 2(x 2),
2x y 4 or
y 2x 4
69. [2] a. Line a: y 32 x 12
OR equivalent
equation; Line b:
3y 4x 19 OR
equivalent
equation
b.
79. m&MPQ = 8a - 8, m&QPR = 5a - 2, m&QPR = 7 a 2; mlQPR 8
y
80. m&MPQ = 2a + 9, m&QPR = 4a - 3, m&MPQ = 7a 6; mlMPQ 21
4
171
Lesson 3-6 Lines in the Coordinate Plane
lesson quiz, PHSchool.com, Web Code: aue-0306
x
4 O
49.
2
y
x0
y0
x
2 O (0, 0)
2
50.
4
(1, 3)
51.
y
y3
(4, 5)
4
2
x
4 2 O
x 1
y
6 y5
2
2
O
x4
2
x
6
2 4
point of intersection:
(2, 9)
[1] at least one correct
eq. or graph
171