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Prerequisite Skills (p. 150) y x
Chapter 3
Prerequisite Skills (p. 150)
7. A system of linear equations can have either none, one,
1. The linear inequality that represents the graph shown at
3
the right is y < 2}4 x 1 3.
2. The graph of a linear inequality in two variables is the set
of all points in a coordinate plane that are solutions of the
inequality.
3. x 1 y 5 4
or infinitely many solutions.
3.1 Guided Practice (pp. 153 –155)
1. 3x 1 2y 5 24
2y 5 23x 2 4
3y 5 1 2 x
3
y 5 2}2 x 2 2
4. y 5 3x 2 3
y
y
x 1 3y 5 1
1
y
1
1
x
21
1
y 5 }3 2 }3 x
x
From the graph, the lines
appear to intersect at
(22, 1).
22
1
x
21
5. 22x 1 3y 5 212
1
y
x
21
Check:
3x 1 2y 5 24
x 1 3y 5 1
3(22) 1 2(1) 0 24
22 1 3(1) 0 1
26 1 2 0 24
22 1 3 0 1
151
24 5 24 The solution is (22, 1).
6. 2x 2 12 5 16
7. 23x 2 7 5 12
2x 5 28
23x 5 19
2. 4x 2 5y 5 210
4
5
x 5 2}
3
8. 22x 1 5 5 2x 2 5
y
1
y
x
21
10 5 4x
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
2
7
5
2
}5x
4
7
}x 2 } 5 y
9. yq2x 1 2
5 5 4x 2 5
2x 2 4 5 7y
}x 1 2 5 y
19
x 5 14
2x 2 7y 5 4
4x 1 10 5 5y
From the graph, the lines
appear to intersect at
(25, 22).
1
x
21
10. x 1 4y < 216
1
y < 2}4 x 2 4
4x 2 5y 5 210
2x 2 7y 5 4
11. 3x 1 5y > 25
4(25) 2 5(22) 0 210
2(25) 2 7(22) 0 4
3
220 1 10 0 210
210 1 14 0 4
y > 2}5 x 2 1
1
22
Check:
y
y
The solution is (25, 22).
1
x
22
454
210 5 210 3. 8x 2 y 5 8
x
3x 1 2y 5 216
y 5 8x 2 8
2y 5 23x 2 16
3
y 5 2}2 x 2 8
Lesson 3.1
Investigating Algebra Activity 3.1 (p. 152)
4
y
22
x
From the graph, the lines
appear to intersect at
(0, 28).
1. y1 5 y2 5 3 when x 5 21. The solution is (21, 3).
2. no solution; There are no values of x where y1 5 y2.
3. y1 5 y2 5 5 when x 5 2. The solution is (2, 5).
4. no solution; There are no values of x where y1 5 y2.
Check:
8x 2 y 5 8
3x 1 2y 5 216
5. y1 5 y2 5 2 when x 5 21. The solution is (21, 2).
8(0) 2 (28) 0 8
3(0) 1 2(28) 0 216
01808
0 2 16 0 216
6. y1 5 y2 for all values of x. There are infinitely many
solutions.
858
216 5 216 The solution is (0, 28).
Algebra 2
Worked-Out Solution Key
101
Chapter 3,
4.
continued
2x 1 5y 5 6
3.1 Exercises (pp. 156–158)
y
4x 1 10y 5 12
Skill Practice
1
2x 1 5y 5 6
1
1. A consistent system that has exactly one solution is
called independent.
x
2. If the two lines intersect at one point, then the solution
4x 110 y 512
is the point of intersection. If the two lines are parallel,
there is no solution. If the two lines coincide then there
are infinitely many solutions.
Each point on the line is a solution, and the system has
infinitely many solutions. The system is consistent
and dependent.
5. 3x 2 2y 5 10
3.
1
y
3x 2 2y 5 2
From the graph, the lines
appear to intersect at
(1, 21).
y
x
21
1
1
x
3x 2 2y 5 2
3x 2 2y 5 10
y 5 2x 2 3
21 0 2(1)23
21 0 23 1 2
21 0 2 2 3
21 5 21 4.
1
y
y
x
21
22x 1 y 5 5
y 5 2x 1 2
21 5 21 The solution is (1, 21).
The two lines have no point of intersection, so the system
has no solution. The system is inconsistent.
6. 22x 1 y 5 5
y 5 23x 1 2
21 0 23(1) 1 2
From the graph, the lines
appear to intersect at
(21, 23).
(21, 3)
y 5 2x 1 2
1
x
y 5 5x 1 2
The lines intersect at (21, 3), so (21, 3) is the solution.
The system is consistent and independent.
7. y 5 0.75 + x 1 35
Total cost (dollars)
y 5 0.75x 1 35
y52+x
5.
y 5 2x
102
21
x
y 5 2x 1 3
2x 2 3y 5 21
21 0 2(4) 1 3
21 0 24 1 3
21 5 21 y 5 2x
56 5 2(28)
56 5 56 56 5 56 The total costs are equal after 28 rides. If the monthly
pass is increased to $35, it will take longer for both
options to cost the same.
Algebra 2
Worked-Out Solution Key
From the graph, the lines
appear to intersect at
(4, 21).
y
1
Check:
56 5 21 1 35
23 5 23 The solution is (21, 23).
The lines appear to intersect at (28, 56).
56 5 0.75(28) 1 35
23 0 3(21)
23 5 23 y
70
y 5 0.75x 1 35
60
(28, 56)
50
40
30
20
y 5 2x
10
0
0 5 10 15 20 25 30 35 40 x
Number of rides
y 5 0.75x 1 35
y 5 3x
23 0 5(21) 1 2
23 0 25 1 2
The solution is (4, 21).
2(4) 2 3(21) 0 21
24 1 3 0 21
21 5 21 Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
1
Chapter 3,
6.
continued
From the graph, the lines
appear to intersect at
(6, 22).
y
2
10.
From the graph, the lines
appear to intersect at
(26, 5).
y
x
21
1
21
x 1 2y 5 2
6 1 2(22) 0 2
x 2 4y 5 14
6 2 4(22) 0 14
62402
6 1 8 0 14
2 5 2 The solution is (6, 22).
7.
From the graph, the lines
appear to intersect at
(5, 0).
y
1
y 5 23x 2 13
5 0 23(26) 2 13
555
11.
x
x 2 7y 5 6
1
6
12.
y
3
x
22
22
2x 1 6y 5 212
x 1 6y 5 12
2(12) 1 6(0) 0 212
212 1 0 0 212
12 1 6(0) 0 12
y 5 4x 1 3
y 5 4x 1 3
12 5 12 From the graph, the lines
appear to intersect at
(22, 4).
y
The graphs of the equations are the same line.
Each point on the line is a solution, so there are infinitely
many solutions.
13.
y 5 23x 2 2
4 0 23(22) 2 2
1
x
21
x
5x 1 2y 5 22
5(22) 1 2(4) 0 22
40622
4 5 4 From the graph, the lines
appear to intersect at (3, 3).
y
1
21
20x 2 5y 5 215
25y 5 215 2 20x
The solution is (12, 0).
9.
x
12 1 0 0 12
212 5 212 x
y 5 }7 2 }7
The graphs of the equations are the same line. Each
point on the line is a solution. There are infinitely many
solutions.
The solution is (5, 0).
From the graph, the lines
appear to intersect at
(12, 0).
21y 5 3x 2 18
6
x
2 }7
y5}
7
555
y
23x 1 21y 5 218
7y 5 6 2 x
52005
050
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
y
1
x 2 4y 5 5
5 2 4(0) 0 5
0 0 10 2 10
8.
24 5 24 21
y 5 2x 2 10
0 0 2(5) 2 10
6 2 10 0 24
The solution is (26, 5).
x
21
2x 2 2y 5 24
2(26)22(5) 0 24
5 0 18 2 13
14 5 14 x
22(3) 1 3 5 23
210 1 8 0 22
22 5 22 3x 1 2y 5 15
22x 1 y 5 23
23 5 23 3(3) 1 2(3) 5 15
15 5 15 The solution is (3, 3).
The solution is (22, 4).
Algebra 2
Worked-Out Solution Key
103
Chapter 3,
14.
continued
y
1
x
21
From the graph, the lines
appear to intersect at
(22, 23).
18.
From the graph, the lines
appear to intersect at
(3, 2).
y
(3, 2)
1
1
x
x 2 2y 5 21
7x 1 y 5 217
7(22) 1 (23) 0 217
3x 2 10y 5 24
3(22) 2 10(23) 0 24
214 2 3 0 217
26 1 30 0 24
2x 2 y 5 4
24 5 24 217 5 217 2x 2 y 5 4
x 2 2y 5 21
2(3) 2 2 0 4
62204
3 2 2(2) 0 21
3 2 4 0 21
The solution is (22, 23).
15. C;
4 5 4 y
1
24x 2 y 5 2
1
x
From the graph, the lines
appear to intersect at
(1, 26).
21 5 21 The solution is (3, 2). The system is consistent and
independent.
19.
y
7x 1 2y 5 25
2
(1, 26)
y 5 3x 1 2
1
x
y 5 3x 2 2
24x 2 y 5 2
24(1) 2 (26) 0 2
7x 1 2y 5 25
7(1) 1 2(26) 0 25
24 1 6 0 2
7 2 12 0 25
2 5 2 y 5 3x 1 2
25 5 25 The graphs of the equations are two parallel lines.
The system has no solution. The system is inconsistent.
The solution is (1, 26).
16. The student did not check the solution in the second
y 5 3x 2 2
20.
y
equation.
x 1 2y 5 6
0 1 2(21) 0 6
01202
22 Þ 6
2 5 2 x
y
3x 1 y 5 5
From the graph, the lines
appear to intersect at
(2, 21).
1
y 5 21
(2, 21)
21 5 21 3x 1 y 5 5
3(2) 1 (21) 0 5
62105
555
The solution is (2, 21). The system is consistent and
independent.
Algebra 2
Worked-Out Solution Key
26x 1 3y 5 23
3y 5 6x 2 3
The graphs of the equations are the same line.
The system has infinitely many solutions.
The system is consistent and dependent.
x
y 5 21
y 5 2x 2 1
y 5 2x 2 1
1
104
1
y 5 2x 2 1
The solution is not (0, 21).
17.
26x 1 3y 5 23
1
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
3x 2 2y 5 2
3(0) 2 2(21) 0 2
Chapter 3,
21.
continued
24.
y
From the graph, the lines
appear to intersect at
(23, 3).
y
4x 1 5y 5 3
5x 2 3y 5 6
1
2
(23, 3)
220x 1 12y 5 224
12y 5 20x 2 24
23y 5 25x 1 6
5
y 5 }3 x 2 2
y 5 }3x 2 2
5
The graphs of the equations are the same line.
The system has infinitely many solutions.
The system is consistent and dependent.
3x 2 5y 5 25
(5, 4)
2
4x 1 5y 5 3
6x 1 9y 5 9
4(23) 1 5(3) 0 3
212 1 15 0 3
6(23) 1 9(3) 0 9
218 1 27 0 9
5x 2 3y 5 6
y
x
21
220x 1 12y 5 224
22.
2
6x 1 9y 5 9
x
From the graph, the lines
appear to intersect at
(5, 4).
3 5 3 959
The solution is (23, 3).
The system is consistent and independent.
25.
y
5x 2 2y 5 17
1
1
From the graph, the lines
appear to intersect at
(3, 21).
x
(3, 21)
4x 2 5y 5 0
1
x
8x 1 9y 5 15
4x 2 5y 5 0
4(5) 2 5(4) 0 0
3x 2 5y 5 25
3(5) 2 5(4) 0 25
20 2 20 0 0
15 2 20 0 25
0 5 0 Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
25 5 25 The solution is (5, 4).
From the graph, the lines
appear to intersect at
(2, 0).
y
2
5x 2 2y 5 17
5(3) 2 2(21) 0 17
24 2 9 0 15
15 1 2 0 17
15 5 15 26.
2 1 x 1 2y 5 22
4
x
8
1
(8, 22)
3x 1 7y 5 6
1
x
2
3x 1 7y 5 6
3(2) 1 7(0) 0 6
2x 1 9y 5 4
2(2) 1 9(0) 0 4
61006
41004
6 5 6 From the graph, the lines
appear to intersect at
(8, 22).
y
(2, 0) x
2x 1 9y 5 4
17 5 17 The solution is (3, 21).
The system is consistent and independent.
The system is consistent and independent.
23.
8x 1 9y 5 15
8(3) 1 9(21) 0 15
The solution is (2, 0).
The system is consistent and independent.
2 3y 5 10
1
2
1
4
} x 2 3y 5 10
} x 1 2y 5 22
} (8) 2 3(22) 0 10
} (8) 1 2(22) 0 22
4 1 6 0 10
2 2 4 0 22
1
2
454
10 5 10 1
4
22 5 22 The solution is (8, 22).
The system is consistent and independent.
Algebra 2
Worked-Out Solution Key
105
Chapter 3,
27.
continued
31.
y
y
1
3x 2 2y 5 215
x
22
2
5 25
2
x
y 5 {x 1 2{
x 2 }3 y 5 25
32.
2
2}3 y 5 2x 2 5
3
15
y 5 }2 x 1 }
2
22y 5 23x 2 15
3
There is no solution.
2
3x 2 2y 5 215
15
y 5 }2 x 1 }
2
1
y
5
x
2
2 y 5 24
1
5x 2 2y 5
1
4
1
x
From the graph, the lines
appear to intersect at
(2.5, 1.5).
y
x
21
The graphs of the equations are the same line.
The system has infinitely many solutions.
The system is consistent and dependent.
28.
y5x
y 5 {x 2 1{
y 5 2x 1 4
1.5 0 {2.5 2 1{
1.5 0 {1.5{
1.5 0 22.5 1 4
1.5 5 1.5 1.5 5 1.5 The solution is (2.5, 1.5).
33.
From the graph, the lines
appear to intersect at
(24, 2) and (4, 2).
y
1
22
1
5
2
x
5x 2 2y 5 }4
} x 2 y 5 24
5
1
2y 5 2}2 x 2 4
22y 5 25x 1 }4
5
5
y 5 }2 x 1 4
1
y 5 }2 x 2 }8
The graphs of the equations are parallel lines.
The system has no solution, so it is inconsistent.
29. A;
y
y 5 {x{ 22
2 0 {24{ 2 2
y52
252
20422
252
2 0 {4{ 2 2
20422
252
3x 1 4y 5 26
1
The solutions are (24, 2) and (4, 2).
1
x
34. a. The system is consistent and independent when a Þ c.
b. The system is consistent and dependent when a 5 c
and b 5 d.
212x 1 16y 5 10
c. The system is inconsistent when a 5 c and b Þ d.
3x 1 4y 5 26
There is exactly one solution of the system.
The system is consistent and independent.
30. a. Sample answer:
3x 1 2y 5 9
Problem Solving
35. Let x 5 hrs as lifeguard.
Let y 5 hrs as cashier.
2x 1 y 5 5
b. Sample answer:
3x 2 y 5 2
26x 1 2y 5 10
c. Sample answer:
8x 1 6y 5 96
2x 2 3y 5 4
24x 1 6y 5 28
106
x 1 y 5 14
Algebra 2
Worked-Out Solution Key
You worked 6 hours as a
lifeguard and 8 hours as a
cashier last week.
Hours as a cashier
212x 1 16y 5 10
y
16
14
12
10
8
6
4
2
0
(6, 8)
0 2 4 6 8 10 12 14 x
Hours as a lifeguard
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
x2
2
y
3
continued
36. Let x 5 warnings.
y 1 37 5 x
contained in the data set, but eventually the swimming
times will start leveling off as swimmers approach
the maximum swimming speed for a human. So, it is
not reasonable to assume that the winning times will
continue to decrease indefinitely at the given rates.
x 1 y 5 375
320
Number of tickets
x 1 y 5 375
d. No. The lines may be good models for the years
y
Let y 5 speeding tickets.
240
(206, 169)
120
80
40. a. Distance
y 1 37 5 x
The state trooper issued
206 warnings and 169
speeding tickets.
0
0
80
160 240 320
Number of warnings
from
park(ft)
d
x
37. Let y 5 total cost (in dollars).
Option A: y 5 121 1 x
Option B: y 5 12x
Total cost (dollars)
Let x 5 days.
y
175
150
125
100
75
50
25
0
y 5 12x
25t 5 5000
y 5 121 1 x
t 5 200
x
Cost (dollars)
2000
1000
d 5 3000 2 15t
0
50
(200, 0)
100 150 200
Time (seconds)
t
0 5 3000 2 r (200)
r 5 15
c. An equation for your friend’s distance is
d 5 3000 2 15t.
2000
B
Mixed Review for TAKS
A
1000
0
41. C;
20
40
60
80 x
Number of years
In 60 years the total costs of owning the refrigerators
will be equal.
c. No. It is not likely that the refrigerators would be in
use for 60 years. You can conclude that, for the life of
the refrigerators, the cost of owning refrigerator A will
always be less than the cost of owning refrigerator B.
Total salary 5 Base salary 1 Commission +
Real estate value
s 5 31,000 1 0.025x
42. F;
mŽM 1 mŽN 1 mŽP 5 1808
40 1 4x 1 x 5 180
5x 5 140
x 5 28
39. a. m 5 20.09583x 1 50.84 (Men)
b. w 5 20.1241x 1 57.08 (Women)
Winning times (seconds)
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
d 5 5000 2 25t
3000
Your friend travels at a speed of 15 feet per second.
(60, 3600)
3000
women
men
The lines appear
to intersect at
(220.7, 29.7).
0
4000
Use the verbal model from part (a) to write the
equation d 5 3000 2 rt for your friend’s distance d
from the park after t seconds, where r represents your
friend’s speed. Let d 5 0 and t 5 200 and solve for r.
Refrigerator B: y 5 1200 1 40x
y
64
56
48
40
32
24
16
8
0
t
d
5000
0
Let x 5 years since purchase.
c.
+
25
2
0 5 5000 2 25t
Refrigerator A: y 5 600 1 50x
0
5000
b. Let d 5 0 to find when you reach the park.
(11, 132)
38. a. Let y 5 total cost (in dollars).
y
4000
5
An equation is d 5 5000 2 25t.
0 2 4 6 8 10 12 14 16
The plans are equal
Number of days
when used for 11 days.
If the daily cost of Option B increases, the plans
will be equal in fewer days. The graph of the new
option B equation will be the rotation of y 5 12x
counterclockwise and the x-intersection with the
graph of the option A equation will be less than 11.
b.
Total
Time
Speed
+
5 distance 2
(sec)
(ft/sec)
(ft)
Distance from park (feet)
Chapter 3,
80
160 240 320 x
Years since 1972
You can predict that the
women’s performance
will catch up to the men’s
performance about 221
years after 1972, or in
2193.
mŽP is 288.
3.1 Graphing Calculator Activity (p. 159)
1. The solution is about (2.3,20.3).
2. The solution is about (2.71, 9.57).
3. The solution is about (11.08, 16.25).
4. The solution is (24,243).
5. The solution is about (251.43, 26.14).
6. The solution is about (212.21, 1.97).
Algebra 2
Worked-Out Solution Key
107
Chapter 3,
continued
3. 3x 2 6y 5 9 l x 5 2y 1 3
7. Let x 5 days in San Antonio.
24x 1 7y 5 216
Let y 5 say in Dallas.
When x 5 2y 1 3:
x1y57
24x 1 7y 5 216
275x 1 400y 5 2300
Using a graphing calculator, the solution is (4, 3).
You should spend 4 days in San Antonio and 3 days
in Dallas.
24(2y 1 3) 1 7y 5 216
28y 2 12 1 7y 5 216
2y 5 24
8. Let x 5 number of adult tickets sold.
y54
Let y 5 number of child tickets sold.
When y 5 4:
x 1 y 5 800
3x 2 6y 5 9
7x 1 5y 5 4600
Using a graphing calculator, the solution is (300, 500).
The movie theater admitted 300 adults and 500 children
that day.
3x 5 33
x 5 11
The solution is (11, 4).
Lesson 3.2
Check: 3x 2 6y 5 9
3(11) 2 6(4) 0 9
3.2 Guided Practice (pp. 161–163)
24x 1 7y 5 216
24(11) 1 7(4) 0 216
244 1 28 0 216
33 2 24 0 9
1. 4x 1 3y 5 22
9 5 9 x 1 5y 5 29 l x 5 29 2 5y
236 2 20y 1 3y 5 22
217y 5 34
Short
Long
Short
Long
Total
Sleeve
Sleeve
Sleeve
Sleeve
5 revenue
Selling + Shirts 1 Selling +
Shirts
($)
Price
Price
(shirts)
(shirts)
($/shirt)
($/shirt)
11
+
x
1
16
+
y
5 8335
y 5 22
When y 5 22:
x 5 29 2 5y
x 5 29 2 5(22)
x 5 29 1 10
x51
The solution is (1, 22).
Check: 4x 1 3y 5 22
4(1) 1 3(22) 0 22
x 1 5y 5 29
1 1 5(22) 0 29
4 2 6 0 22
1 2 10 0 29
2. 3x 1 3y 5 215
5x 2 9y 5 3
33
29 5 29 9x 1 9y 5 245
5x 2 9y 5 3
14x
5 242
x 5 23
When x 5 23:
38
288x 2 110y 5 261,820
88x 1 128y 5
18y 5
y5
4860
270
8x 1 10y 5 5620
8x 1 10(270) 5 5620
8x 5 2920
x 5 365
The school sold 365 short sleeve T-shirts and 270 long
sleeve T-shirts.
12x 2 3(4x 1 3) 5 29
5x 2 9y 5 3
12x 2 12x 2 9 5 29
5(23) 2 9(22) 0 3
215 1 18 0 3
29 5 29
353
66,680
When y 5 270:
12x 2 3y 5 29
y 5 22
The solution is (23, 22).
Algebra 2
Worked-Out Solution Key
11x 1 16y 5 8335
When y 5 4x 1 3:
29 1 3y 5 215
215 5 215 3 211
24x 1 y 5 3 l y 5 4x 1 3
3(23) 1 3y 5 215
29 2 6 0 215
8x 1 10y 5 5620
5. 12x 2 3y 5 29
3x 1 3y 5 215
Check: 3x 1 3y 5 215
3(23) 1 3(22) 0 215
216 5 216 There are infinitely many solutions.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
4x 1 3y 5 22
4(29 2 5y) 1 3y 5 22
22 5 22 4.
Short
Long
Long
Short
Total
Sleeve
Sleeve
Sleeve + Sleeve
5 Cost
+
1
Shirts
Shirts
Cost
Cost
($)
(shirts)
(shirts)
($/shirt)
($/shirt)
8
+
x
1
10
+
y
5 5620
When x 5 29 2 5y:
108
3x 2 6(4) 5 9
Fly UP