Download A. Graphing Linear Equations BEnChmArk 2

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Date ————————————
BEnChmArk 2
(Chapters 3 and 4)
A. Graphing Linear Equations
The x-axis and y-axis divide a coordinate plane into four equal parts called quadrants.
The quadrants are labeled with roman numerals I, II, III, and IV, moving counterclockwise from the upper right quadrant. Each point in a coordinate plane has a unique
ordered pair (x, y) that describes the point’s location with respect to the origin (0, 0).
The solution of an equation is the set of all ordered pairs (x, y) that make the equation
a true statement. The graph of an equation is a graph of all the ordered pairs that make
up the solution of the equation.
1. Plot Points in a Coordinate Plane
Plot each point and describe its location.
b. Q(3, 24)
c. R(22, 0)
d. S(0, 23)
Benchmark 2
a. P(24, 21)
Solution:
Another name for
the x-coordinate
is abscissa.
Another name for
the y-coordinate
is ordinate.
a. Start at the origin. Move 4 units left, then 1 unit
b. Start at the origin. Move 3 units right, then 4
units down. Point Q is in Quadrant IV.
R
c. Start at the origin. Move 2 units left. Point R
is on the x-axis.
d. Start at the origin. Move 3 units down. Point P
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
4
down. Point P is in Quadrant III.
y
3
2
1
23 22 21
1 2
21
P
22
23
S
24
is on the y-axis.
PrACTICE
3
4 x
Q
Plot each point and describe its location.
1. A(3, 5)
2.
B(24, 0)
3. C(21, 4)
4. D(0, 21)
5.
E(22, 23)
6. F(1, 24)
2. Identify Solutions to Equations in Two Variables
ExAmPLE
Tell whether the ordered pair is a solution of the equation.
a. x 1 2y 5 8; (24, 6)
b. 5x 2 2y 5 10; (2, 1)
Solution:
a.
x 1 2y 5 8
(24) 1 2(6) 0 8
858✓
Write original equation.
Substitute 24 for x and 6 for y.
Simplify.
(24, 6) is a solution.
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_a.indd 19
a. Graphing Linear equations
ExAmPLE
19
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Name ———————————————————————
Date ————————————
BEnChmArk 2
(Chapters 3 and 4)
5x 2 2y 5 10
b.
Write original equation.
5(2) 2 2(1) 0 10
Substitute 2 for x and 1 for y.
8 Þ 10 ✗
Simplify.
(2, 1) is not a solution.
PrACTICE
Tell whether the ordered pair is a solution of the equation.
4
7. 22x 1 3y 5 4; 0, }
8. 28 5 y; (25, 28)
3
2
9.
3x 2 4y 5 21; (23, 24)
10.
x 5 22; (21, 22)
11.
y 2 5x 5 23; (22, 213)
12.
24y 1 2x 5 0; 1 2}2 , }4 2
1 1
3. Graph an Equation Using a Table
ExAmPLE
Graph the equation 23x 1 y 5 1.
Solution:
Step 1: Solve the equation for y:
8
23x 1 y 5 1
6
4
2
y 5 3x 1 1
You can choose
any (x, y) pair from
the graph and
substitute it in the
equation to make
a true statement.
Step 2: Make a table by choosing a few values for x
and finding the values of y.
x
22
21
0
1
2
y
25
22
1
4
7
y
28 24 24 22
22
24
26
28
2 4
6
8 x
Step 3: Plot the points. Notice that the points appear to lie on a line.
Step 4: Connect the points by drawing a line through them. Use arrows to indicate
that the graph goes on without end.
PrACTICE
Graph the equation.
13. x 1 y 5 3
14. y 2 2x 5 21
15. 23x 1 2y 5 2
16. x 2 3y 5 3
17. 4y 2 3x 5 8
18. 2y 2 5x 5 0
4. Graph horizontal and Vertical Lines
Vocabulary
20
Linear equation An equation that can be written in the form Ax 1 By 5 C, where A,
B, and C are real numbers and A and B are not both equal to zero. The graph of a linear
equation is a straight line. When A 5 0, the graph of the linear equation is a horizontal
line. When B 5 0, the graph of the linear equation is a vertical line.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Benchmark 2
a. Graphing Linear equations
1
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_a.indd 20
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Name ———————————————————————
Date ————————————
BEnChmArk 2
(Chapters 3 and 4)
ExAmPLE
Graph the equation.
b. x 5 1
a. y 5 23
Solution:
All the solutions
of y 5 23 are
ordered pairs in
the form (x, 23).
a. Notice that x can be any real number, but that y
is always 23. The graph of the equation y 5 23
is a horizontal line 3 units below the x-axis.
1
4 x
1 2
3
4 x
y
3
2
1
24 23 22 21
21
22
23
24
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
3
Benchmark 2
4
any real number. The graph of the equation
x 5 1 is a vertical line 1 unit to the right of the
y-axis.
1 2
Graph the equation.
19. x 5 25
20.
y51
21. 2y 5 23
22. 2x 2 1 5 0
23.
x2350
24. y 1 2 5 0
Quiz
Plot each point and describe its location.
1. A(4, 27)
2.
B(29, 22)
3. C(0, 7)
4. D(1, 3)
5.
E(26, 0)
6. F(24, 8)
Tell whether the ordered pair is a solution of the equation.
7. 25 5 y; (5, 25)
1
8.
2
1
10. y 2 2x 5 26; }, 25 11.
2
2x 1 4y 5 4; (4, 5)
1
1
9. 28y 1 4x 5 0; 22, 2}
4
x 5 29; (1, 29)
12. 3x 2 7y 5 24; (1, 1)
2
Graph the equation.
13. x 2 y 5 22
14.
y 1 3x 5 24
15. 25x 1 3y 5 2
16. y 5 27
17.
23y 2 2x 5 9
18. x 5 8
19. 4y 2 6x 5 0
20.
3 5 2x
21. 2y 1 5 5 23
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_a.indd 21
a. Graphing Linear equations
24 23 22 21
21
22
23
24
25
26
27
b. Notice that x will always be 1, but that y can be
PrACTICE
y
21
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Name ———————————————————————
Date ————————————
Benchmark 2
(Chapters 3 and 4)
B. Slope-Intercept Form and Direct
Variation
1. Find the Intercepts of the Graph of an equation
Vocabulary
exampLe
x-intercept The x-coordinate of the point where a graph intersects the x-axis.
y-intercept The y-coordinate of the point where a graph intersects the y-axis.
Find the x-intercept and the y-intercept of the graph of 3x 1 4y 5 12.
Solution:
To find the x-intercept, substitute 0 for y and solve for x.
Write original equation.
3x 1 4y 5 12
3x 1 4(0) 5 12
Remember
that the x- and
y-intercepts are
numbers, NOT
ordered pairs.
12
x5}
54
3
Substitute 0 for y.
Solve for x.
To find the y-intercept, substitute 0 for x and solve for y.
Write original equation.
3x 1 4y 5 12
3(0) 1 4y 5 12
12
y5}
53
4
Substitute 0 for x.
Solve for y.
The x-intercept is 4. The y-intercept is 3.
practIce
Find the x-intercept and the y-intercept of the graph of the equation.
1.
x 1 y 5 26
4. 27y 5 14x
2.
23y 1 8 5 212x
3. 4.5x 1 0.5y 5 9
5.
215 1 10y 5 60x
6. 3 2 18x 5 26y
2. Find the Slope of a Line
Vocabulary
Slope Describes how quickly a line rises or falls as it moves from left to right. Slope
is the ratio m of the vertical change between two points on the line to the horizontal
change between the same two points.
y2 2 y1
For points (x1, y1) and (x2, y2), m 5 }
x 2x .
2
exampLe
22
1
Find the slope of the line that passes through the points.
a. (1, 5) and (4, 6)
b. (25, 7) and (3, 21)
c. (22, 7) and (8, 7)
d. (6, 28) and (6, 2)
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Benchmark 2
B. Slope-Intercept Form
For any two points, there is one and only one line that contains both points. This fact
can help you graph a linear equation. Many times, it will be convenient to use the
points where the line crosses the x-axis and y-axis. These points are the intercepts.
Knowing how steep the line is, or the slope of the line, also can help you graph a linear
equation. If the graph of a linear equation passes through the origin (0, 0), the relationship between x and y is called a direct variation.
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_b.indd 22
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Name ———————————————————————
Date ————————————
Benchmark 2
(Chapters 3 and 4)
Solution:
Think of (x1, y1) as
“the coordinates
of the first point”
and (x2, y2) as “the
coordinates of the
second point.” Be
sure to subtract
the x- and
y-coordinates in
the same order.
a. Let (x1, y1) 5 (1, 5) and (x2, y2) 5 (4, 6).
y2 2 y1
m5}
x 2x
2
Write formula for slope.
1
625
1
5}
5 }3
421
b.
Substitute and simplify.
Let (x1, y1) = (25, 7) and (x2, y2) 5 (3, 21).
y2 2 y1
m5}
x 2x
2
Write formula for slope.
1
28
Substitute and simplify.
Benchmark 2
B. Slope-Intercept Form
21 2 7
3 2 (25)
5}5}
5 21
8
c. Let (x1, y1) 5 (22, 7) and (x2, y2) 5 (8, 7).
y2 2 y1
m5}
x 2x
2
Write formula for slope.
1
727
8 2 (22)
0
5}5}
50
10
Substitute and simplify.
The slope is 0. The line is horizontal.
d. Let (x1, y1) 5 (6, 28) and (x2, y2) 5 (6, 2).
y2 2 y1
m5}
x 2x
2
Write formula for slope.
1
2 2 (28)
10
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
5}
5}
626
0
Substitute. Division by 0 is undefined.
The slope is undefined. The line is vertical.
practIce
Find the slope of the line that passes through the points.
7. (6, 29) and (29, 6)
9. (211, 8) and (13, 5)
11. (2.5, 25) and (5.5, 29)
8. (4, 2) and (4, 0)
10. (21, 27) and (1, 27)
12. (23, 25) and (22, 0)
3. Graph an equation Using Slope-Intercept Form
Vocabulary
exampLe
Slope-intercept form A linear equation in the form y 5 mx 1 b, where m is the slope
and b is the y-intercept of the graph of the equation.
Graph the equation 2x 1 2y 5 4.
Solution:
If you can
substitute the
coordinates of
the second point
in the original
equation and get
a true statement,
then your graph is
correct.
Step 1: Rewrite the equation in slope-intercept form.
1
y 5 }2 x 1 2
Step 2: Identify the slope and the y-intercept.
1
m 5 }2 and b 5 2.
Step 2: Plot the point that corresponds to the
y-intercept, (0, 2).
Step 4: Use the slope to find another point on the line.
Draw a line through the two points.
6
5
4
y
3
2
(0, 2)
1
24 23 22 21
21
22
1 2
(2, 3)
3
4 x
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_b.indd 23
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Name ———————————————————————
Date ————————————
Benchmark 2
(Chapters 3 and 4)
practIce
Graph the equation.
2
13. y 5 2} x 1 7
5
16. y 5 24
14.
23x 5 4y 1 8
15. 3x 2 3y 5 6
17.
214x 2 7y 5 21
18. 1.5y 2 6x 2 12 5 0
4. Identify Direct Variation equations
Benchmark 2
B. Slope-Intercept Form
Vocabulary
exampLe
Direct variation An equation in the form y 5 ax, where a Þ 0, represents direct
variation. The variable y varies directly with x.
Constant of variation The constant a in the direct variation equation y 5 ax.
tell whether the equation represents direct variation. If so, identify the
constant of variation.
a. 6x 2 4y 5 0
b. x 1 y 5 8
Solution:
Try to rewrite the equation in the form y 5 ax.
a. 6x 2 4y 5 0
Write original equation.
24y 5 26x
3
y 5 }2 x
Subtract 26x from each side.
Simplify.
Because the equation 6x 2 4y 5 0 can be rewritten in the form y 5 ax,
3
b. x 1 y 5 8
Write original equation.
y 5 2x 1 8
Subtract x from each side.
Because the equation x 1 y 5 8 cannot be rewritten in the form y 5 ax, it does
not represent direct variation.
practIce
tell whether the equation represents direct variation. If so, identify the
constant of variation.
7
19. y 5 2} x
20. x 1 4 5 16y
21. 9y 5 5x
8
22. x 5 247y
23.
23 1 x 1 7 5 2y 1 4 24. 13 5 26x
5. Write and Use a Direct Variation equation
exampLe
the graph of a direct variation equation is shown.
a. Write the direct variation equation.
b. Find the value of y when x 5 36.
5
4
(6, 5)
3
2
1
22 21
21
22
23
24
y
1 2
3
4
5
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
it represents direct variation. The constant of variation is }2 .
6 x
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_b.indd 24
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Name ———————————————————————
Date ————————————
Benchmark 2
(Chapters 3 and 4)
practIce
Solution:
a. Because y varies directly x, the equation has the form y 5 ax. Use the fact that
y 5 5 when x 5 6 to find a.
y 5 ax
Write direct variation equation.
5 5 a(6)
Substitute.
5
6
Solve for a.
}5a
5
A direct variation equation that relates x and y is y 5 }6 x.
5
b. When x 5 36, y 5 } (36) 5 30.
6
Benchmark 2
B. Slope-Intercept Form
Check the sign of
the constant of
variation in your
equation. If the
graph of y 5 ax
passes through
Quadrants I and
III, the constant
should be positive.
If the graph of
y 5 ax passes
through Quadrants
II and IV, the
constant should
be negative.
Write the direct variation equation that passes through the given point.
then find the value of y for the given x.
25. (3, 21); x 5 12
26.
(24, 28); x 5 32
27. (26, 3); x 5 18
28. (9, 2); x 5 27
29.
(25, 7); x 5 100
30. (22, 21); x 5 74
Quiz
Find the x-intercept and the y-intercept of the graph of the equation.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
1. 221 1 14y 5 84x
2.
23 1 x 5 3y
3. 3.2x 1 0.8y 5 4
Find the slope of the line that passes through the points.
4. (8, 25) and (23, 4)
5.
(1, 7) and (22, 7)
6. (29, 7) and (3, 25)
8.
y 5 22
9. 4x 2 6y 5 12
Graph the equation.
7. y 5 x 1 1
Does the equation represent direct variation? If so, find the constant of
variation.
4
10. y 5 2} x
11. x 1 3 5 9y
12. 4y 5 7x
5
Write the direct variation equation that passes through the given point.
then find the value of y for the given x.
13. (2, 25); x 5 20
14.
(23, 29); x 5 43
15. (24, 6); x 5 64
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_b.indd 25
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Name ———————————————————————
Date ————————————
Benchmark 2
(Chapters 3 and 4)
c. Writing Linear equations
You can describe a line with equations in three different forms. You can write these
equations if you know the slope and y-intercept of the line, if you know the slope and
a point on the line, or if you know two points on the line. The following examples
illustrate these three different forms of the equation of a line and show how to find them.
1. Write an equation in Slope-Intercept Form
examPLe
Slope-intercept form The equation y 5 mx 1 b, for a line with slope m and y-intercept b.
1
Write an equation of the line with a slope of }
and a y-intercept of 22.
3
Solution:
y 5 mx 1 b
1
y 5 }3 x 2 2
y
Write slope-intercept form.
1
1
3
Substitute } for m and 22 for b.
22 21
21
1
13 2
3
4
6 x
5
(0, 22)
23
24
25
26
27
PractIce
Write an equation of the line with the given slope and y-intercept.
1. Slope is 6; y-intercept is 24.
2. Slope is 21; y-intercept is 3.
3
3. Slope is } ; y-intercept is 25.
5
2
4. Slope is } ; y-intercept is 23.
5
1
6. Slope is 2}; y-intercept is 22.
3
5. Slope is 24; y-intercept is 5.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
BENCHMARK 2
C. Writing Linear Equations
Vocabulary
2. Write an equation of a Line Given the Slope and a
Point
examPLe
Write an equation of the line that passes through (4, 23) and has a
slope of 22.
Solution:
Step 1: Identify the slope. The slope is 22.
Make sure you
don’t switch
the x and y
values when you
substitute.
Step 2: Find the y-intercept. Substitute the slope
and the coordinates of the given point in
y 5 mx 1 b. Solve for b.
y 5 mx 1 b
23 5 22(4) 1 b
55b
26
Write slope-intercept form.
Substitute 22 for m,
4 for x, and 23 for y.
Solve for b.
y
8
1
7
6
2
5 (0, 5)
4
3
2
1
24 23 22 21
1
2
3
4 x
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_c.indd 26
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Name ———————————————————————
Date ————————————
Benchmark 2
(Chapters 3 and 4)
Step 3: Write an equation of the line.
Write slope-intercept form.
y 5 mx 1 b
y 5 22x 1 5
Substitute 2 for m and 5 for b.
PractIce
3. Write an equation of a Line Given two Points
examPLe
Write an equation of the line shown.
(25, 6)
6
5
4
Solution:
Step 1: Calculate the slope using the formula.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
y2 2 y1
226
24
m5}
5}5}
5 22
2
23 2 (25)
x2 2 x1
3
2
1
(23, 2)
Step 2: Find the y-intercept. Use the point (25, 6).
Write slope-intercept
y 5 mx 1 b
You also could find
b by substituting
the x and y values
from the other
known point,
(23, 2).
y
26 25 24 23 22 21
21
22
1
2 x
form.
6 5 22(25) 1 b
6 2 10 5 b
24 5 b
Substitute 6 for y, 22 for m, and 25 for x.
Solve for b.
Step 3: Write an equation of the line.
Write slope-intercept form.
y 5 mx 1 b
y 5 22x 2 4
Substitute 22 for m and 24 for b.
PractIce
Write an equation of the line shown.
13.
7
6
14.
y
5
4
3
2
(6, 1)
1
21
21
1
2
3
4
5
(3, 21)
6
7 x
y
1
21
22
23
24
25
26
27
(6, 12 )
1
2
3
4 5
6
(8, 0)
7
8 x
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_c.indd 27
BENCHMARK 2
C. Writing Linear Equations
Write an equation of the line that passes through the given point and
has the given slope.
1
4
7. (26, 22); m 5 }
8. (21, 3); m 5 2}
4
3
3
9. (3, 4); m 5 26
10. (5, 23); m 5 }
2
2
11. (23, 6); m 5 2}
12. (21, 24); m 5 2
3
27
4/19/11 1:58:19 PM
Name ———————————————————————
Date ————————————
Benchmark 2
(Chapters 3 and 4)
15.
2
1
23 22 21
21
22
(21, 23)
23
24
25
26
BENCHMARK 2
C. Writing Linear Equations
16.
y
1
2
17.
3 4
(22, 0)
26 25 24 23 22
2
5 x
1
(3, 21)
(1, 1)
1 2
24 23 22 21
21
22
23
24
4 x
18.
y
5
4
3
8
6
(24, 2) 4
2
y
28 26 24 22
22
24
2
1
1
2
4
6
8 x
(5, 21)
26
28
2 x
4. Write an equation in Point-Slope Form
examPLe
Point-slope form The equation y 2 y1 5 m(x 2 x1), for the nonvertical line through a
given point (x1, y1) with slope m.
Write an equation in point-slope form of the
line that passes through the point (22, 1) and
has a slope of 2.
Solution:
Notice that (x1, y1) is
a point of the line,
and that m is the
slope of the line.
y 2 y1 5 m(x 2 x1)
Write point-slope form.
y 2 1 5 2(x 1 2)
Substitute 1 for y1, 2 for
m, and 22 for x1.
1
2
(22, 1)
24 23 22 21
PractIce
8
7
6
5
4
y
3
2
1
1
2
3
Write an equation in point-slope form of the line that passes through
the given point and has the given slope.
2
19. (3, 21); m 5 }
3
1
21. (23, 24); m 5 }
2
23. (25, 3); m 5 21
1
20. (4, 0); m 5 2}
4
3
22. (1, 1); m 5 }
4
1
24. (24, 2); m 5 2}
3
4 x
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Vocabulary
28
3
(23, 22)
8
7
6
(25, 3)
y
4
3
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_c.indd 28
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Name ———————————————————————
Date ————————————
Benchmark 2
(Chapters 3 and 4)
5. Write an equation in Standard Form
Vocabulary
examPLe
Standard form The equation Ax 1 By 5 C, where A, B, and C are real numbers and
A and B are not both zero.
Write an equation in standard form of the
line shown.
3
2
1
Solution:
Step 1: Calculate the slope.
y2 2 y1
21 2 (22)
1
24 23 22 21
21
22
23
24
25
1
1
2
3
4 x
(4, 21)
(2, 22)
BENCHMARK 2
C. Writing Linear Equations
5 }2
m5}
x 2x 5}
422
2
y
Step 2: Write an equation in point-slope form. Use (2, 22).
y 2 y1 5 m(x 2 x1) Write point-slope form.
1
y 2 (22) 5 }2 (x 2 2)
Substitute 22 for y1,
1
}2 for m, and 2 for x1.
Step 3: Rewrite the equation in standard form.
1
y 1 2 5 }2 x 2 1
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
2y 1 4 5 x 2 2
2x 1 2y 5 26
PractIce
Apply the distributive property.
Multiply each term by 2.
Simplify. Collect variable terms on one side,
constants on the other.
Write an equation in standard form of the line shown.
25.
27.
26.
y
1
26 25 24 23 22 21
21
(21, 21)
22
23
24
25
(22, 25)
26
27
28
4
23 22 21
21
22
23
24
25
26
28.
y
(1, 3)
3
2
1
26 25 24 23 22 21
21
22
23
(23, 23)
24
2
1
2 x
1
2 x
(24, 5)
y
1
2
3 4
5 x
(1, 21)
(2, 24)
5
4
y
3
2
1
26 25 24 23 22 21
21
22
23
1
2 x
(2, 23)
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_c.indd 29
29
4/19/11 1:58:20 PM
Name ———————————————————————
Date ————————————
Benchmark 2
(Chapters 3 and 4)
29.
6
5
4
(23, 3)
(25, 4)
4
y
3
2
1
3
2
1
(22, 1)
1
26 25 24 23 22 21
21
22
BENCHMARK 2
C. Writing Linear Equations
30.
y
25 24 23 22 21
21
22
23
24
2 x
1
2
3 x
(3, 22)
Quiz
Write an equation in slope-intercept form of the line with the given
slope and y-intercept.
1. Slope is 4; y-intercept is 3.
2. Slope is 22; y-intercept is 1.
5
3. Slope is }; y-intercept is 24.
2
1
4. Slope is 2}; y-intercept is 25.
3
Write an equation in the given form of the line that passes through the
given point and has the given slope.
3
1
5. (23, 24); m 5 }
6. (22, 7); m 5 2 }
7. (1, 5); m 5 24
5
4
point-slope form
point-slope form
Write equations in slope-intercept form and standard form of the line
shown.
8.
21
22
23
24
25
26
27
28
30
1
2
3
4 5
y
9.
(4, 0)
y
6
7
(8, 22)
7
6
8 x
(26, 5)
(23, 1)
5
4
3
2
1
27 26 25 24 23 22 21
21
1 x
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
slope-intercept form
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_c.indd 30
4/19/11 1:58:20 PM
Name ———————————————————————
Date ————————————
BEnchmark 2
(Chapters 3 and 4)
D. Parallel and Perpendicular Lines
If two non-vertical lines in the same plane have the same slope, then they are parallel.
If their slopes are negative reciprocals, then they are perpendicular. The converse is
also true. If two non-vertical lines in the same plane are parallel, then they have the
same slope. If they are perpendicular, then their slopes are negative reciprocals.
1. Determine Whether Lines are Parallel or
Perpendicular
Vocabulary
Determine which lines, if any, are parallel or perpendicular.
Line a: y 5 4x 2 1
Line b: 24x 1 y 5 3
Line c: 2x 1 8y 5 4
Solution:
Step 1: Write each equation in slope-intercept form. Find the slopes of the lines.
Line a: The equation is in slope-intercept form. The slope is 4.
The product of a
non-zero slope m
and its negative
reciprocal is 21:
Line b: 24x 1 y 5 3
y 5 4x 1 3
Line c: x 1 4y 5 4
1
y 5 2}4 x 1 1
1
m 1 2}
m 2 5 21
Benchmark 2
D. Parallel and Perpendicular
ExamPLE
Perpendicular lines Lines in a plane that intersect to form a right (90°) angle.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Step 2: Compare the slopes. Line a and line b have slopes of 4,
so they are parallel. Line c has a slope of 2}4. 41 2}4 2 5 21,
so it is perpendicular to lines a and b.
1
PracticE
1
Determine which lines, if any, are parallel or perpendicular.
3
1. Line a: y 5 } x 1 2 Line b: 4x 2 3y 5 23 Line c: 3x 2 4y 5 20
4
2. Line d: x 2 2y 5 4 Line e: 2x 1 y 5 0
Line f: x 1 2y 5 3
7
3. Line g: 5x 1 7y 5 7 Line h: y 5 } x 1 3
5
Line j: 7x 2 5y 5 2
2. Write an Equation of a Parallel Line
ExamPLE
Write an equation of the line that passes through (1, 22) and is parallel
to the line y 5 5x 1 2.
Solution:
Step 1: Identify the slope. The graph of the given equation has a slope of 5.
So, the parallel line through (1, 22) will also have a slope of 5.
Step 2: Find the y-intercept. Use the slope and the given point.
y 5 mx 1 b
Write slope-intercept form.
22 5 5(1) 1 b
Substitute 22 for y, 5 for m, and 1 for x.
27 5 b
Solve for b.
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_d.indd 31
31
4/19/11 1:58:32 PM
Name ———————————————————————
Date ————————————
BEnchmark 2
(Chapters 3 and 4)
You can graph
both lines to
check your
answer.
y 5 mx 1 b
Write slope-intercept form.
y 5 5x 2 7
Substitute 5 for m and 27 for b.
Write an equation of the line that passes through the given point and is
parallel to the given line.
1
4
4. (23, 21); y 5 } x 1 1
5. (28, 5); y 5 2} x 2 2
4
3
3
6. (2, 3); y 5 26x 1 4
7. (2, 0); y 5 } x 2 7
2
2
}
8. (26, 4); y 5 2 x 1 3
9. (25, 22); y 5 2x 2 9
3
3. Write an Equation of a Perpendicular Line
ExamPLE
Write an equation of the line that passes through (4, 3) and is
perpendicular to the line y 5 2x 2 3.
Solution:
Step 1: Identify the slope. The graph of the given equation has a slope of 2.
So, the slope of the perpendicular line through (4, 3) will be the
1
negative reciprocal of 2, which is 2}2 .
Step 2: Find the y-intercept. Use the slope and the given point.
y 5 mx 1 b
Write slope-intercept form.
1
3 5 2}2 (4) 1 b
Substitute 3 for y, 2}2 for m, and 4 for x.
55b
Solve for b.
1
Step 3: Write an equation of the line in slope-intercept form.
PracticE
32
y 5 mx 1 b
Write slope-intercept form.
1
y 5 2}2 x 1 5
Substitute 2}2 for m and 5 for b.
1
Write an equation of the line that passes through the given point and is
perpendicular to the given line.
3
3
10. (23, 22); y 5 } x 1 2
11. (26, 1); y 5 2} x 2 1
4
2
1
}
12. (2, 5); y 5 28x 1 3
13. (4, 0); y 5 x 2 4
3
2
14. (4, 6); y 5 2} x 1 3
15. (28, 22); y 5 2x 2 6
3
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Benchmark 2
D. Parallel and Perpendicular
PracticE
Step 3: Write an equation of the line in slope-intercept form.
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_d.indd 32
4/19/11 1:58:32 PM
Name ———————————————————————
Date ————————————
BEnchmark 2
(Chapters 3 and 4)
Quiz
Determine which lines, if any, are parallel or perpendicular.
3
1. Line a: y 5 2} x 1 4 Line b: 3x 1 2y 5 2 Line c: 2x 2 3y 5 3
2
2. Line d: x 1 3y 5 9
Line e: y 5 3x 2 2
Line f: 3x 1 y 5 2
3. Line g: x 1 4y 5 2
Line h: x 2 4y 5 0
Line j: y 5 }4 x 1 1
1
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Write an equation of the line that passes through the given point and is
perpendicular to the given line.
2
1
10. (1, 7); y 5 } x 2 2
11. (6, 4); y 5 2} x 1 6 12. (24, 23); y 5 2x 2 7
3
3
3
3
13. (26, 2); y 5 } x 1 5 14. (3, 21); y 5 2} x 2 8 15. (8, 2); y 5 24x 1 1
2
4
Benchmark 2
D. Parallel and Perpendicular
Write an equation of the line that passes through the given point and is
parallel to the given line.
3
2
4. (8, 1); y 5 } x
5. (23, 3); y 5 2} x 2 5 6. (25, 22); y 5 2x 1 2
8
3
4
1
7. (26, 2); y 5 } x 1 4
8. (28, 0); y 5 2} x 2 3 9. (3, 2); y 5 25x 1 1
3
4
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_d.indd 33
33
4/19/11 1:58:32 PM
Name ———————————————————————
Date ————————————
Benchmark 2
(Chapters 3 and 4)
e. Linear models
Paired data graphed in a scatter plot may show a positive correlation, a negative
correlation, or no correlation. If there is a positive or negative correlation, the data can
be modeled by a line of fit drawn close to the points on the scatter plot. The equation
of this line will be in the form y 5 mx 1 b. Using linear regression, you can find
the line that best fits the data. This best-fitting line or its equation can be used to
approximate data points between or beyond known data points.
1. Describe the correlation of Data
exampLe
Correlation The relationship between paired data; If the value of y tends to increase
as the value of x increases, the correlation is positive. If the value of y tends to decrease
as the value of x increases, the correlation is negative.
Scatter plot A graph that shows the relationship, if any, between paired data.
Describe the correlation, if any, of the data graphed in the scatter plot.
a.
b.
y
4
3
2
2
1
1
24 23 22 21
21
22
23
24
c.
y
4
3
1 2
3
4 x
2
3
4 x
24 23 22 21
21
22
23
24
1 2
3
4 x
y
4
3
2
1
24 23 22 21
21
22
23
24
Solution:
a. The value of y
decreases as the
value of x increases:
negative correlation.
34
b.
There is no apparent
relationship between
the value of y and x:
no correlation.
c. The value of y
increases as the
value of x increases:
positive correlation.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Benchmark 2
E. Linear Models
Vocabulary
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_e.indd 34
4/19/11 2:01:12 PM
Name ———————————————————————
Date ————————————
Benchmark 2
(Chapters 3 and 4)
pracTice
Describe the correlation, if any, of the data graphed in the scatter plot.
1.
2.
y
4
3
2
2
1
1
24 23 22 21
21
22
23
24
3.
y
4
3
1
2
3
4 x
1
2
3
4 x
1 2
24 23 22 21
21
22
23
24
3
4 x
Benchmark 2
E. Linear Models
y
4
3
2
1
24 23 22 21
21
22
23
24
exampLe
Tracy is training for a swim race. The table shows her fastest time from
each practice session for six days.
practice day
Fastest time (min)
1
2
3
4
5
6
6.6
6.5
6.5
6.3
6.2
6.0
a. Make a scatter plot of the data.
b. Describe the correlation of the data.
Solution:
Notice that
a negative
correlation is
not always an
“undesirable”
outcome.
a. Treat the data as ordered pairs. Let x represent
the training day and let y represent the fastest
time each day. Plot the ordered pairs as points
in a coordinate plane.
b. The scatter plot shows a negative correlation.
The more Tracy trains, the less time she takes
to finish the race.
Fastest time (min)
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
2. make a Scatter plot
y
6.6
6.5
6.4
6.3
6.2
6.1
6.0
0
0
1
2
3 4 5 6
Training day
7 x
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_e.indd 35
35
4/19/11 2:01:12 PM
Name ———————————————————————
Date ————————————
Benchmark 2
(Chapters 3 and 4)
pracTice
make a scatter plot of the data in the table. Describe the correlation,
if any.
4.
x
y
2
2
3
5
5
6
21
0
2
3
6
6
5.
x
y
0
23 22 21 21
1
0
2
22 23 23 24
3. Draw a Line of Fit to Data
Benchmark 2
E. Linear Models
exampLe
Line of fit A line on a scatter plot that appears to fit the data closely.
The table shows the shoe size and height for nine customers of a men’s
shoe store.
1
1
Shoe Size
9
9
10
10}2
11
11
11}2
12
12
height (in.)
67
69
68
70
72
73
75
74
76
Write an equation that models the height of
a customer as a function of his shoe size.
Height (in.)
Solution:
Step 1: Make a scatter plot of the data.
Let x represent shoe size. Let y
represent height.
y
78
76
74
72
70
68
66
0
Step 2: Decide whether the data can be modeled
0 8 9 10 11 12 13 14 x
by a line. As shoe size increases, height
Shoe size
tends to increase, so the scatter plot shows
a positive correlation. You can fit a line to the data.
Step 3: Draw a line of fit. The line should be close to the data points,
with about the same number of points above and below the line.
A line of fit and its
equation model
the trend in the
data.
Step 4: Write an equation using two points on the line. Use (9, 68) and (12, 75).
y2 2 y1
75 2 68
7
5 }3
Find the slope of the line. m 5 }
x 2x 5}
12 2 9
2
1
Find the y-intercept of the line. Use the point (9, 68).
y 5 mx 1 b
Write slope-intercept form.
7
68 5 }3 (9) 1 b
Substitute 68 for y, } for m, and 9 for x.
47 5 b
Solve for b.
7
3
The height in inches y of a customer can be modeled by the function
7
y 5 }3 x 1 47, where x is the customer’s shoe size.
pracTice
Write an equation that models y as a function of x.
6.
36
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Vocabulary
x
y
5
4
0
0
3
2
1
0
22 23 24 23
7.
x
y
0
0
1
1
25 23 23 21
2
2
0
2
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_e.indd 36
4/19/11 2:01:13 PM
Name ———————————————————————
Date ————————————
Benchmark 2
(Chapters 3 and 4)
4. interpolate Using an equation
Vocabulary
exampLe
Linear interpolation To use a line or its equation to estimate a value between two
known values.
Use the data about shoe store customer’s sizes and heights to find the
equation of the best-fitting line for the data. Then approximate the
1
height of a customer who wears shoe size 9}
.
2
Solution:
Step 2: Perform linear regression using the
paired data. The equation of the bestfitting line is approximately y 5 2.6x 1 44.
76
74
72
70
68
66
8
9
10
11
12
13
8
9
10
11
12
13
Benchmark 2
E. Linear Models
Step 1: Enter the data into lists on a graphing
calculator. Make a scatter plot of the
data. Let the x-values be shoe size and
the y-values be height.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Step 3: Find the value of y when x 5 9.5.
y 5 2.6x 1 44
Equation of bestfitting line
y 5 2.6(9.5) 1 44
Substitute 9.5 for x.
y ø 68.7
Simplify.
76
74
72
70
68
66
1
A customer who wears a size 9}2 shoe is probably
about 68.7 inches tall.
pracTice
make a scatter plot of the data. Find the equation of the best-fitting
line. approximate the value of y for x 5 4.
8.
x
y
25 23 21
4
3
1
1
0
3
5
22 23
9.
x
y
0
1
22 21
3
6
7
8
0
1
2
4
5. extrapolate Using an equation
Vocabulary
exampLe
Linear extrapolation To use a line or its equation to estimate a value beyond the
range of known values.
Use the data about shoe store customer’s sizes and heights to estimate
the height of a customer who wears size 13.
Solution:
Use the equation of the best-fitting line to find the value of y when x 5 13.
y 5 2.6x 1 44
y 5 2.6(13) 1 44 5 77.8
A customer who wears a size 13 shoe is probably about 77.8 inches tall.
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_e.indd 37
37
4/19/11 2:01:13 PM
Name ———————————————————————
Date ————————————
Benchmark 2
(Chapters 3 and 4)
pracTice
make a scatter plot of the data. Find the equation of the best-fitting
line. approximate the value of y for x 5 7.
10.
x
y
24 23
2
2
0
0
1
0
2
11.
5
21 24
x
y
1
22 21
25 24 21
2
4
5
0
0
3
Quiz
Describe the correlation, if any, of the data graphed in the scatter plot.
1.
2.
y
4
3
y
7
6
5
4
3
1
1
24 23 22 21
21
22
23
24
3.
2
1
21
22
23
24
25
26
2
3
4 x
2
1
24 23 22 21
21
1
2
3
4 x
y
1
2
3
4 5
6
7
8 x
4. A bookstore is interested in the relationship between the number of rainy days
in a month and the number of sales. The table shows data for six months.
rainy days
Sales
apr
may
Jun
Jul
aug
Sep
8
6
5
5
6
3
105
91
85
90
92
75
a. Make a scatter plot of the data.
b. Describe the correlation.
c. Draw a line of fit. Write the equation of the line.
d. Perform linear regression to find the equation of the best-fitting line.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Benchmark 2
E. Linear Models
2
e. Estimate the number of sales during a month with 4 rainy days. Estimate the
number of sales during a month with 10 rainy days.
38
Algebra 1
Benchmark 2 Chapters 3 and 4
CC_n1rm_bm_02_e.indd 38
4/19/11 2:01:13 PM
Answers
16.
14
continued
25
8
13. y 5 }x 1 16 14. y 5 2}x 1 75
2
3
3
1
15. y 5 2}x 2 }
2
2
y
12
10
8
6
4
Quiz
2
x
2
4
6 8 10 12 14
D. Solving Equations in One Variable
}
}
3 2 }
1. 1.23, }, 1}, Ï 3 2. 21.9, 2Ï 0.04 , 0.08, Ï 2
2 3
}
1
3. Ï 6 , 6.01, 6.1, 6} 4. 21 5. 11 6. 4
6
7. 20
8. 8 9. 227 10. 2 11. 23 12. 224
10
13. 2} 14. 9 15. 25 16. 3 17. 26
7
2
18. } 19. All real numbers 20. 220
13
21. No solution
P
V
1. l 5 } 2. h 5 }2 3. a 5 2s 2 b 2 c
4
πr
S
h2c
4. l 5 } 2 r 5. v 5 16t 1 }
t
πr
S 2 2lh
6. w 5 }
7. 3 ohms
2l 1 2h
s
1
8. n 5 } 1 2; 15 sides 9. c 5 }2 ; 16
180
(1 2 e)
3
1
10. y 5 }x 2 2 11. y 5 2}x 1 2
7
2
4
7
9
12. y 5 27x 1 } 13. y 5 2} x 2 }
5
5
3
8
2
14. y 5 2}x 2 } 15. y 5 6x 1 9
3
3
Benchmark 2
A. Graphing Linear Equations
1–6.
Quiz
Quiz
3
2
1. } 2. }
5
4
7. 52
1
1
3. } 4. 15}
14
3
8. 163 9. 9
5. 30
6. 25
F. Rewriting Equations in Two or More
Variables
}
I
A
}
1. P 5 rt 2. r 5 }
 3. V 5 E 2 F 1 2
π 
V
3V
2A
4. w 5 } 5. h 5 }2 6. b1 5 } 2 b2
h
lh
πr
40
m
3
7. V 5 }; 64 cm
8. d 5 }
} ; 4 miles
d
Ïs
2
n
9. h 5 } 1 2r; 46 ft 10. y 5 2x 2 10
64
4
1
8
11. y 5 2}x 2 } 12. y 5 2}x 2 3
3
9
3
Ï
A
3
2
1
B
24 23 22 21
1 2
21
D
22
23
E
24
E. Proportion and Percent Problems
27
2
1
1
12
2
1. } 2. } 3. } 4. } 5. } 6. } 7. 14
5
3
2
37
3
10
8. 60 9. 10 10. 5 11. 6 12. 12 13. 57
14. 27 15. 180 16. 845 17. 15 18. 7
C 4 y
3
4 x
F
1. Quadrant I 2. x-axis 3. Quadrant II
4. y-axis 5. Quadrant III 6. Quadrant IV
7. Yes, it is a solution. 8. Yes, it is a solution.
9. No, it is not a solution. 10. No, it is not a
solution. 11. Yes, it is a solution. 12. No, it is
not a solution.
13.
7
6
5
4
y
3
2
1
24 23 22 21
21
1 2
3
4 x
Algebra 1
Benchmark Answer Key
CC_n1rm_bm_ans.indd 3
Answer Key
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
6
1. 5.3 2. 2.2 3. 215 4. 2} 5. 10.5
5
2
6. 7 7. } 8. 110 9. 162 10. 6 11. 15
9
5
2
12. 220 13. 211 14. } 15. }
3
2
16. 0 17. All real numbers 18. 210
A3
4/19/11 2:08:38 PM
14.
continued
1 2
24 23 22 21
21
22
23
24
25
15.
5
4
18.
y
3
2
1
3
4 x
19.
y
Answer Key
3
4
4 x
y
3
2
1
1 2
3
4 x
20.
y
1 2
24 23 22 21
21
22
23
24
25
6
5
4
3
y
24 23 22 21
21
22
23
24
21.
4
1 2
3
4 x
1 2
3
4 x
1 2
3
4 x
y
3
2
1
3
2
1
24 23 22 21
21
22
4
3
2
1
4 x
y
1 x
27 26 25 24 23 22 21
21
22
23
24
23
17.
1 2
24 23 22 21
21
22
23
24
24 23 22 21
21
22
3
2
1
y
3
2
1
3
2
1
16.
4
1 2
3
24 23 22 21
21
22
23
24
4 x
22.
y
4
3
2
1
24 23 22 21
21
22
23
24
A4
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Answers
Algebra 1
Benchmark Answer Key
CC_n1rm_bm_ans.indd 4
4/19/11 2:08:38 PM
Answers
continued
y
4
23.
14.
24 23 22 21
21
22
23
24
25
26
27
28
3
2
1
1 2
22 21
21
22
23
24
24.
3
4
5
6 x
15.
y
2
1
1 2
24 23 22 21
21
22
23
24
25
26
3
E
y
D
2 4
6
8 x
A
17.
1. Quadrant IV 2. Quadrant III 3. y-axis
4. Quadrant I 5. x-axis 6. Quadrant II
7. Yes, it is a solution. 8. No, it is not a solution.
9. No, it is not a solution. 10. Yes, it is a
solution. 11. No, it is not a solution.
12. Yes, it is a solution.
6
5
4
1
24 23 22 21
21
22
23
24
25
26
3
4 x
1 2
3
4 x
y
1 2
3
4 x
27
y
18. y
4
3
2
1
3
2
1
24 23 22 21
21
22
1 2
Answer Key
28 26 24 22
22
B
24
26
28
4 x
y
24 23 22 21
21
22
23
24
25
26
27
28
C
6
4
2
3
y
24 23 22 21
21
22
23
24
8
F
1 2
3
2
1
16.
1–6.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
4
4 x
Quiz
13.
y
1 2
3
4 x
21
22
23
24
1 2
3
4
5
6
7
8 x
Algebra 1
Benchmark Answer Key
CC_n1rm_bm_ans.indd 5
A5
4/19/11 2:08:39 PM
19.
continued
4
13.
y
8
7
6
5
4
3
2
1
1 2
24 23 22 21
21
22
23
24
3
4 x
3
2
1
21
20.
3
2
1
21.
2
1
Answer Key
15.
y
24 23 22 21
21
22
23
24
25
26
1 2
3
3
4 x
B. Slope-Intercept Form and Direct
Variation
1. x-intercept: 26, y-intercept: 26
2
8
2. x-intercept: 2}, y-intercept: }
3
3
3. x-intercept: 2, y-intercept: 18 4. x-intercept: 0,
1
3
y-intercept: 0 5. x-intercept: 2}4, y-intercept: }2
1
1
6. x-intercept: }, y-intercept: 2} 7. 21
6
2
1
4
8. Undefined 9. 2} 10. 0 11. 2} 12. 5
8
3
5
2
1
2
1
16.
2
1
2
1
7 x
y
1 2 x
1 2
3
4 x
1 2
3
4 x
1 2
3
4 x
y
24 23 22 21
21
22
23
24
25
26
17.
6
y
24 23 22 21
21
22
23
24
25
26
24 23 22 21
21
22
23
24
25
26
A6
4
26 25 24 23 22 21
21
22
23
24
25
26
1 x
27 26 25 24 23 22 21
21
22
23
24
1 2
14.
y
4
y
y
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Answers
Algebra 1
Benchmark Answer Key
CC_n1rm_bm_ans.indd 6
4/19/11 2:08:39 PM
Answers
continued
18.
8
7
6
5
4
9.
y
24 23 22 21
21
22
23
24
25
26
3
2
1
1 2
24 23 22 21
7
19. Yes; 2}
8
20. No
23. Yes; 21
24. No
3
4 x
5
21. Yes; }
9
1
22. Yes; 2}
47
1
25. y 5 2}x; 24
3
1
26. y 5 2x; 64 27. y 5 2}x; 2 9
2
7
2
28. y 5 }x; 6 29. y 5 2} x; 2140
5
9
1
30. y 5 }x; 37
2
Quiz
3
1
1. x-intercept: 2}, y-intercept: } 2. x-intercept:
4
2
3, y-intercept: 21 3. x-intercept: 1.25,
9
y-intercept: 5 4. 2}
5. 0 6. 21
11
y
7.
3
2
1
24 23 22 21
21
22
23
8.
3
2
1
24 23 22 21
21
22
23
24
25
1 2
3
4 x
1 2
3
4 x
y
y
1 2
3
4 x
7
4
10. Yes; 2} 11. No 12. Yes; }
5
4
5
13. y 5 2}x; 250 14. y 5 3x; 129
2
3
15. y 5 2}x; 296
2
C. Writing Linear Equations
3
1. y 5 6x 2 4 2. y 5 2x 1 3 3. y 5 }x 2 5
5
2
4. y 5 }x 2 3 5. y 5 24x 1 5
5
1
4
6. y 5 2}x 2 2 7. y 5 }x 1 6
3
3
1
11
8. y 5 2}x 1 } 9. y 5 26x 1 22
4
4
2
3
21
10. y 5 }x 2 } 11. y 5 2}x 1 4
3
2
2
2
12. y 5 2x 2 2 13. y 5 }x 2 3
3
1
5
1
14. y 5 2}x 1 2 15. y 5 }x 2 }
4
2
2
3
1
16. y 5 }x 1 } 17. y 5 2x 2 2
4
4
1
2
2
18. y 5 2}x 1 } 19. y 1 1 5 }(x 2 3)
3
3
3
1
1
20. y 2 0 5 2}(x 2 4) 21. y 1 4 5 }(x 1 3)
4
2
3
}
22. y 2 1 5 (x 2 1) 23. y 2 3 5 2(x 1 5)
4
1
24. y 2 2 5 2}(x 1 4) 25. 24x 1 y 5 3
3
26. 3x 1 y 5 2 27. 23x 1 2y 5 3
28. 4x 1 3y 5 21 29. 2x 1 y 5 23
30. 3x 1 4y 5 1
Answer Key
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
5
4
2
1
Quiz
5
1. y 5 4x 1 3 2. y 5 22x 1 1 3. y 5 }x 2 4
2
17
1
1
}
}
}
4. y 5 2 x 2 5 5. y 5 x 2
5
5
3
3
6. y 2 7 5 2}(x 1 2) 7. y 2 5 5 24(x 2 1)
4
1
4
8. y 5 2}x 1 2; x 1 2y 5 4 9. y 5 2}x 2 3;
2
3
4x 1 3y 5 29
Algebra 1
Benchmark Answer Key
CC_n1rm_bm_ans.indd 7
A7
4/19/11 2:08:40 PM
continued
D. Parallel and Perpendicular Lines
1. a and c are parallel. 2. d and e are
perpendicular. 3. h and j are parallel. g is
4
perpendicular to h and j. 4. y 5 }3 x 1 3
1
5. y 5 2}x 1 3 6. y 5 26x 1 15
4
3
2
7. y 5 }x 2 3 8. y 5 2}x 9. y 5 2x 1 8
2
3
2
4
10. y 5 2} x 2 4 11. y 5 }x 1 9
3
3
19
1
12. y 5 }x 1 } 13. y 5 23x 1 12
8
4
1
3
14. y 5 }x 15. y 5 2}x 2 6
2
2
Quiz
1. a and b are parallel. c is perpendicular to
a and b. 2. d and e are perpendicular.
3
3. h and j are parallel. 4. y 5 }x 2 2
8
2
5. y 5 2}x 1 1 6. y 5 2x 1 8
3
4
1
7. y 5 }x 1 10 8. y 5 2}x 2 2
3
4
9. y 5 25x 1 17 10. y 5 23x 1 10
1
3
11. y 5 }x 2 5 12. y 5 2}x 2 5
2
2
2
4
1
13. y 5 2}x 2 2 14. y 5 }x 2 5 15. y 5 }x
3
3
4
5.
Answer Key
3 x
5
7. Answers may vary. Sample answer: y 5 }x 2 4
2
8.
y
4
3
2
1
1 2
24 23 22 21
21
22
23
24
3
4 x
y 5 20.73x 1 0.5; 22.4
9.
4
3
2
1
21
22
23
24
y
4
3
2
1
21
21
22
2
negative correlation
4
6. Answers may vary. Sample answer: y 5 }x 2 4
5
1. No correlation 2. Negative correlation
3. positive correlation
6
5
1
25 24 23 22 21
21
22
23
24
25
26
E. Linear models
4.
y
2
1
y
1
2
3
4 5
6
7
8 x
y 5 0.63x 2 1.94; 0.58
1
2
3
4
5
6
7 x
10.
y
4
3
2
1
positive correlation
1 2
24 23 22 21
21
22
23
24
2
3
4 x
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Answers
2
y 5 2}3 x; 24}3
A8
Algebra 1
Benchmark Answer Key
CC_n1rm_bm_ans.indd 8
4/19/11 2:08:40 PM
Answers
11.
continued
11. 21 ≤ x < 6
y
3
2
1
�1
1
23 22 21
21
22
23
24
25
2
3 4
54
13.
1. No correlation 2. positive correlation
3. Negative correlation
Number of sales
55
56
6
5
4
57
58
59
60
61
y
1
2
3
4 x
1 2
3
4 x
1 2
3
4 x
1 2
3
4 x
1
0
2
4
24 23 22 21
21
6 8 10 12 14 t
Days of rain
22
23
24
25
15.
y
4
3
1.
1
71
72
73
74
75
76
77
24 23 22 21
21
2.
22
23
24
�31 �30 �29 �28 �27 �26 �25 �24
6
7
8
9
10
11
12
16.
160
162
164
166
5. x > 22 6. x ≤ 210
9. 23 < x < 0
�4 �3 �2 �1
7. x ≥ 6
8. x < 25
3
4
5
y
4
3
2
1
0
1
2
3
6
7
8
9
10. x ≤ 2 or x ≥ 9
2
6
y
2
4.
5
3
2
A. Graphing Inequalities
5
4
Answer Key
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
14.
Benchmark 3
3.
3
24 23 22 21
21
22
4b. positive correlation 4c. y 5 6x 1 57
4d. y 5 5.85x 1 57.5 4e. 81 sales; 116 sales
70
2
3
2
1
Quiz
y
130
120
110
100
90
80
70
0
1
12. x < 55 or x ≥ 60
5 x
y 5 x 2 2.7; 1.3
4a–c.
0
24 23 22 21
21
22
23
24
Algebra 1
Benchmark Answer Key
CC_n1rm_bm_ans.indd 9
A9
4/19/11 2:08:41 PM