C H A P T E R 7

C H A P T E R 7
Systems of Equations and Inequalities
Section 7.1
Linear and Nonlinear Systems of Equations . . . . . . . . 611
Section 7.2
Two-Variable Linear Systems
. . . . . . . . . . . . . . . 625
Section 7.3
Multivariable Linear Systems
. . . . . . . . . . . . . . . 638
Section 7.4
Partial Fractions
Section 7.5
Systems of Inequalities . . . . . . . . . . . . . . . . . . . 674
Section 7.6
Linear Programming . . . . . . . . . . . . . . . . . . . . 685
Review Exercises
. . . . . . . . . . . . . . . . . . . . . . 661
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696
Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710
Practice Test
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716
C H A P T E R 7
Systems of Equations and Inequalities
Section 7.1
■
Linear and Nonlinear Systems of Equations
You should be able to solve systems of equations by the method of substitution.
1. Solve one of the equations for one of the variables.
2. Substitute this expression into the other equation and solve.
3. Back-substitute into the first equation to find the value of the other variable.
4. Check your answer in each of the original equations.
■
You should be able to find solutions graphically. (See Example 5 in textbook.)
Vocabulary Check
1. system of equations
2. solution
3. solving
4. substitution
5. point of intersection
6. break-even
1. 4x y 6x y 6
1
(a) 40 3 1
0, 3 is not a solution.
(b) 41 4 1
1, 4 is not a solution.
(c) 4
2 1
2 is not a solution.
4 12 3 1
6 12 3 6
12, 3 is a solution.
32
32,
(d)
2.
4x
x
y 11
2
y 3
?
(a) 422 13 3
16 13 3
?
2 13 11
2 13 11
2, 13 is a solution.
?
(b) 422 9 3
16 9 3
2, 9 is not a solution.
3 2
31 ?
(c) 4 2 3 3
36
4
31
3 3
32, 313 is not a solution.
7 2
37 ?
(d) 4 4 4 3
49
4
37
4 3
7
37 ?
4 4 11
7
4
37
4 11
74, 374 is a solution.
611
612
3.
Chapter 7
Systems of Equations and Inequalities
y 2ex
3x y 4. log x 3 y
1
28
9x y 9
2
(a) 0 2e2
37
(a) log 9 3 9
9, 379 is not a solution.
2, 0 is not a solution.
2 2e0
(b)
(b) log 10 3 2
30 2 2
1
9 10
0, 2 is a solution.
(c) 3 2 28
9
10, 2 is a solution.
2e0
(c) log1 3 3
0, 3 is not a solution.
1
9 1
(d) 2 2e1
3 28
9
1, 3 is a solution.
1, 2 is not a solution.
(d) log 2 3 4
2, 4 is not a solution.
5.
7.
2x y 6
x y 0
6. x y 4
x 2y 5
Equation 1
Equation 2
Equation 1
Equation 2
Solve for y in Equation 1: y 6 2x
Solve for x in Equation 1: x y 4
Substitute for y in Equation 2: x (6 2x) 0
Substitute for x in Equation 2: y 4 2y 5
Solve for x: 3x 6 0 ⇒ x 2
Solve for y: 3y 4 5 ⇒ y 3
Back-substitute x 2: y 6 2(2) 2
Back-substitute y 3: x 3 4 1
Solution: 2, 2
Solution: 1, 3
xx yy 4
2
2
Equation 1
Equation 2
Solve for y in Equation 1: y x 4
Substitute for y in Equation 2: x2 (x 4) 2
Solve for x: x2 x 2 0 ⇒ x 1x 2 0 ⇒ x 1, 2
Back-substitute x 1: y 1 4 3
Back-substitute x 2: y 2 4 6
Solutions: 1, 3, 2, 6
8. 3x
y2
x3 2 y 0
Equation 1
Equation 2
Solve for y in Equation 1: y 2 3x
Substitute for y in Equation 2: x 3 2 2 3x 0
x 3 3x 0
Solve for x: x 3 3x 0 ⇒ xx2 3 0 ⇒ x 0, ± 3
Back-substitute x 0: y 2 30 2
Back-substitute x 3: y 2 33
Back-substitute x 3: y 2 3 3 2 33
Solutions: 0, 2, 3, 2 33 , 3, 2 33 Section 7.1
9. 2x y 5
Equation 1
y 25
Equation 2
x2
2
10.
Linear and Nonlinear Systems of Equations
xy0
x 5x y 0
3
Equation 1
Equation 2
Solve for y in Equation 1: y 2x 5
Solve for y in Equation 1: y x
Substitute for y in Equation 2: x2 2x 52 25
Substitute for y in Equation 2: x 3 5x x 0
Solve for x:
Solve for x:
5x2
20x 0 ⇒ 5xx 4 0 ⇒ x 0, 4
613
x3 4x 0 ⇒ xx2 4 0 ⇒ x 0, ± 2
Back-substitute x 0: y 20 5 5
Back-substitute x 0: y 0 0
Back-substitute x 4: y 24 5 3
Back-substitute x 2: y 2
Solutions: 0, 5, 4, 3
Back-substitute x 2: y 2 2
Solutions: 0, 0, 2, 2, 2, 2
11. x2
y0
x2 4x y 0
Equation 1
12.
Equation 2
Solve for y in Equation 1: y x2
4
2x 2 1
Equation 2
2x 4 2x 2 1 2x 2 2
Solve for x: 2x2 4x 0 ⇒ 2xx 2 0 ⇒ x 0, 2
Back-substitute x 0: y y 2x
Equation 1
Substitute for y in Equation 1:
Substitute for y in Equation 2: x2 4x x2 0
02
y 2x2 2
Solve for x: x 4 2x 2 1 x 2 1 0
0
x4 x2 0
Back-substitute x 2: y 2 4
2
x 2x 2 1 0 ⇒ x 0, ± 1
Solutions: 0, 0, 2, 4
Back-substitute x 0: y 202 2 2
Back-substitute x 1: y 212 2 0
Back-substitute x 1: y 212 2 0
Solutions: 0, 2, 1, 0, 1, 0
13.
y x 3x 1
y x3 3x2 1
2
Equation 1
14.
Equation 2
Substitute for y in Equation 2:
y 2x 4
Equation 1
Equation 2
Substitute for y in Equation 1: 2x 4 x 3 3x 2 4
x3 3x2 1 x2 3x 1
Solve for x: 0 x 3 3x 2 2x
x3 4x2 3x 0
0 xx 2 3x 2
xx 1x 3 0 ⇒ x 0, 1, 3
Back-substitute x 0: y 15.
y x 3 3x 2 4
03
30 1 1
2
0 xx 2x 1 ⇒ x 0, 1, 2
Back-substitute x 0: y 20 4 4
Back-substitute x 1: y 13 312 1 1
Back-substitute x 1: y 21 4 2
Back-substitute x 3: y 33 332 1 1
Back-substitute x 2: y 22 4 0
Solutions: 0, 1, 1, 1, 3, 1
Solutions: 0, 4, 1, 2, 2, 0
x y 0
5x 3y 10
Equation 1
Equation 2
Solve for y in Equation 1: y x
Substitute for y in Equation 2: 5x 3x 10
Solve for x: 2x 10 ⇒ x 5
Back-substitute in Equation 1: y x 5
Solution: 5, 5
614
16.
18.
Chapter 7
x 2y Systems of Equations and Inequalities
1
5x 4y 23
17. 2x y 2 0
4x y 5 0
Equation 1
Equation 2
Equation 1
Equation 2
Solve for x in Equation 1: x 1 2y
Solve for y in Equation 1: y 2x 2
Substitute for x in Equation 2: 51 2y 4y 23
Substitute for y in Equation 2: 4x 2x 2 5 0
Solve for y: 14y 28 ⇒ y 2
1
Solve for x: 6x 3 0 ⇒ x 2
Back-substitute y 2: x 1 2y 1 22 3
1
1
Back-substitute x 2: y 2x 2 2 2 2 3
Solution: 3, 2
Solution:
6x 3y 4 0
Equation 1
x 2y 4 0
Equation 2
Solve for x in Equation 2: x 4 2y
Substitute for x in Equation 1: 64 2y 3y 4 0
4
Solve for y: 24 12y 3y 4 0 ⇒ 15y 20 ⇒ y 3
4
4
4
Back-substitute y 3: x 4 2y 4 23 3
Solution:
43, 43 19. 1.5x 0.8y 2.3
0.3x 0.2y 0.1
Equation 1
Equation 2
Multiply the equations by 10.
15x 8y 23
Revised Equation 1
3x 2y 1
Revised Equation 2
3
1
Solve for y in revised Equation 2: y 2 x 2
3
1
Substitute for y in revised Equation 1: 15x 82 x 2 23
Solve for x: 15x 12x 4 23 ⇒ 27x 27 ⇒ x 1
3
1
Back-substitute x 1: y 21 2 1
Solution: 1, 1
20. 0.5x 3.2y 9.0
0.2x 1.6y 3.6
Equation 1
Equation 2
Multiply the equations by 10.
5x 32y 90
2x 16y 36
Revised Equation 1
Revised Equation 2
Solve for x in revised Equation 2: x 8y 18
Substitute for x in revised Equation 1: 58y 18 32y 90
Solve for y: 40y 90 32y 90 ⇒ 72y 180
⇒ y 52
5
5
Back-substitute y 2: x 82 18 2
5
Solution: 2, 2 12, 3
Section 7.1
21.
xx yy 208
1
5
1
2
Equation 1
Equation 2
22.
Solve for x in Equation 2: x 20 y
Substitute for x in Equation 1:
Solve for y: 4 3
10 y
Back-substitute y 1
5 20
8 ⇒ y
40
3:
y 8
34 y 10
Equation 1
y 4
Equation 2
3 3
Substitute for y in Equation 1: 12x 44x 4 10
40
3
1
9
Solve for x: 2x 16x 3 10 ⇒
x 20 y 20 40
3
20
3
Back-substitute x Solution:
x6x 5yy 3
7
Equation 1
Equation 2
5
6
Solve for x in Equation 2: x 7 24.
5
6y
Substitute for x in Equation 1: 67 25.
1
2x
3
4x
5
6y
615
3
Solve for y in Equation 2: y 4x 4
1
2y
40
Solution: 20
3, 3
23.
Linear and Nonlinear Systems of Equations
208
17 :
13 ⇒ x 208
17
88
y 34208
17 4 17
88
208
17 , 17 y2
2 xx 3y 6
2
3
17
16 x
Equation 1
Equation 2
Solve for y in Equation 1: y 23x 2
5y 3
Substitute for y in Equation 2: 2x 323x 2 6
Solve for y: 42 5y 5y 3 ⇒ 42 3 (False)
Solve for x: 2x 2x 6 6 ⇒ 0 12 Inconsistent
No solution
No solution
x2 y 0
2x y 0
Equation 1
Equation 2
Solve for y in Equation 2: y 2x
Substitute for y in Equation 1: x2 2x 0
Solve for x: x2 2x 0 ⇒ xx 2 0 ⇒ x 0, 2
Back-substitute x 0: y 20 0
Back-substitute x 2: y 22 4
Solutions: 0, 0, 2, 4
26.
x 2y 0
3x y
2
0
Equation 1
27.
Equation 2
x y 1
Equation 1
y 4
Equation 2
x
2
Solve for x in Equation 1: x 2y
Solve for y in Equation 1: y x 1
Substitute for x in Equation 2: 32y y 2 0
Substitute for y in Equation 2: x 2 x 1 4
Solve for y: 6y y 2 0 ⇒ y6 y 0 ⇒ y 0, 6
Solve for x: x 2 x 1 4 ⇒ x 2 x 3 0
Back-substitute y 0: x 20 0
The Quadratic Formula yields no real solutions.
Back-substitute y 6: x 26 12
Solutions: 0, 0, 12, 6
28. y x
y x
3
Equation 1
3x2 2x
29.
Equation 2
Substitute for y in Equation 2: x x 3 3x2 2x
x 2y 2 ⇒ y 3x y 15 ⇒ y 3x 15
y
Solve for x: x 3 3x2 3x 0 ⇒ xx2 3x 3 0
⇒ x 0,
−x + 2y = 2
6
5
3 ± i3
2
Point of intersection: 4, 3
4
(4, 3)
3
2
Back-substitute x 0: y 0
The only real solution is 0, 0.
x2
2
−2 −1
−2
x
1
2
3
4
3x + y = 15
6
616
30.
Chapter 7
Systems of Equations and Inequalities
x y 0
3x 2y 10
31.
x 3y 2 ⇒ y 13x 2
5x 3y 17 ⇒ y 135x 17
y
y
5x + 3y = 17
1
4
x
−1
1
2
3
4
(2, −2)
−2
1
x+y=0
−3
−4
( 52 , 32 )
x − 3y = −2 2
−1
−2 −1
x
1
−1
2
3
3x − 2y = 10
−2
Point of intersection: 2, 2
Point of intersection:
32. x 2y 1
x y2
33.
x y
2
y
4
52, 32 x y 4 ⇒ y x 4
2
4x 0 ⇒ x 22 y 2 4
y
x+y=4
6
6
4
4
(2, 2)
−x + 2y = 1
2
(5, 3)
2
(4, 0)
−2
4
6
−4
x−y=2
x 6x x27 y
y3
2 0
2
x 2 + y 2 − 4x = 0
35.
xy30
⇒ yx3
y x 4x 7 ⇒ y x 2 3
2
2
y
y
12
8
10
(1, 4)
(4, 7)
6
8
(− 3, 0)
x
8
Points of intersection: 2, 2, 4, 0
Point of intersection: 5, 3
34.
6
−2
x
2
2
(3, 6)
6
x−y+3=0 4
y = x 2 − 4x + 7
x
2 4
6
−6
8 10 12
x
−2
2
4
6
−2
−4
−6
Points of intersection: 3, 0, 3, 6
36. y2 4x 11 0
12x y 12
y
Points of intersection: 1, 4, 4, 7
37.
7x 8y 24 ⇒ y 78x 3
x 8y − 1x + y = − 1
2
2
9
1
8x
8 ⇒ y
y
(15, 7)
7x + 8y = 24
6
3
2
(3, 1)
(4, − 21 )
x
6
9
12
15
18
−3
−2
−6
−9
−2
y2 −
4x + 11 = 0
Points of intersection: 3, 1, 15, 7
x
x − 8y = 8
−4
1
Point of intersection: 4, 2 1
Section 7.1
38.
x y0
5x 2y 6
39.
y
Linear and Nonlinear Systems of Equations
617
3
3x 2y 0 ⇒ y x
2
x2 y2 4 ⇒
x2 y2
1
4
4
4
y
3x − 2y = 0
3
2
(2, 2)
1
−2
−1
2
1
x
2
3
4
x
−1
−4 −3
−1
−2
1
3
4
−2
−3
Point of intersection: 2, 2
−4
x 2 − y2 = 4
No points of intersection ⇒ No solution
40.
x 2xy y4x3 00
2
y
2
5
4
No points of intersection
so, no solution
3
1
−3
x
−1
1
3
2
5
−2
−3
41.
3xx 16yy 250⇒ y 2
2
2
Algebraically we have:
3 2
16 x
x2 25 y2
y
16
3y
6
−4
16y = 0
(4, 3)
3y2 16y 75 0
2
−2
x
2
4
3y 25y 3 0
6
−2
400
2
y 25
3 ⇒ x 9 ,
−4
−6
x 2 + y 2 = 25
x 2 y 2 25
x 82 y 2 41
Solutions: ± 4, 3
y
43.
12
10
8
6
Points of intersection:
3, 4, 3, 4
6 8 10 12
y 3x 8y 04e
x
16
−7
6
0
8
Point of intersection:
0.49, 6.53
− 10
6
−2
(3, − 4)
−6
−8
−10
−12
x
−6
x
−2
x y 1y 0e ⇒ y x 1
Point of intersection: 0, 1
(3, 4)
2
−6
44.
No real solution
y 3 ⇒ x2 16
Points of intersection: 4, 3 and 4, 3
42.
25 y2
16y 75 3y2
4
(−4, 3)
−6
3x 2 −
618
45.
Chapter 7
Systems of Equations and Inequalities
1
⇒ y x4
2
ln x
y log2 x ⇒ y ln 2
x 2y 8
46.
y 2 lnx 1
3y 2x 9
4
5
−1
−2
10
14
−6
Point of intersection: 5.31, 0.54
−3
Point of intersection: 4, 2
47. x2 y2 169 ⇒ y1 169 x2 and y2 169 x2
x2 8y 104 ⇒ y3 18x2 13
16
Points of intersection: 0, 13, ± 12, 5
−24
24
−16
48.
x2 y2 4 ⇒ y1 4 x2, y2 4 x2
2x2 y 2 ⇒ y3 2x2 2
Points of intersection:
49.
−6
6
2
Equation 1
1
Equation 2
Solve for x: x2 2x 1 x 12 0 ⇒ x 1
Back-substitute x 1 in Equation 1: y 2x 2
Solution: 1, 2
−4
xx yy 24
2
Substitute for y in Equation 2: 2x x2 1
4
0, 2, 1.32, 1.5, 1.32, 1.5
50.
y 2x
y x
Equation 1
Equation 2
Solve for y in Equation 1: y 4 x
Substitute for y in Equation 2: x2 4 x 2
Solve for x: x2 x 2 0
No real solutions because the discriminant in the Quadratic Formula is negative.
Inconsistent; no solution
51. 3x 7y 6 0
Equation 1
x2 y2 4
Equation 2
Solve for y in Equation 1: y 3x 6
7
Substitute for y in Equation 2: x2 Solve for x:
x2 9x
2
3x 6
7
2
4
36x 36
4
49
Back-substitute x 49x2 9x2 36x 36 196
Back-substitute x 2: y 40x2 36x 232 0
410x 29x 2 0 ⇒ x 3x 6 3(2910) 6 21
29
:y
10
7
7
10
29
, 2
10
Solutions:
10, 10, 2, 0
29 21
3x 6 32 6
0
7
7
Section 7.1
52.
x2x yy 2510
2
2
Linear and Nonlinear Systems of Equations
619
Equation 1
Equation 2
Solve for y in Equation 2: y 10 2x
Substitute for y in Equation 1: x2 10 2x2 25
Solve for x: x2 100 40x 4x2 25 ⇒ x2 8x 15 0
⇒ x 5x 3 0 ⇒ x 3, 5
Back-substitute x 3: y 10 23 4
Back-substitute x 5: y 10 25 0
Solutions: 3, 4, 5, 0
53. x 2y 4
x y 0
Equation 1
2
Equation 2
Solve for y in Equation 2: y x2
Substitute for y in Equation 1: x 2x2 4
Solve for x: 0 2x2 x 4 ⇒ x 1 ± 1 424
1 ± 31
⇒ x
22
4
The discriminant in the Quadratic Formula is negative.
No real solution
54. y x 13
y x 1
55.
y e x 1 ⇒ y e x 1
y ln x 3
y
⇒ y ln x 3
y
3
6
2
5
1
4
x
−3
−1
1
−1
2
3
2
1
−2
x
–2 –1
−3
2
3
4
5
Point of intersection: approximately 0.287, 1.751
No points of intersection, so no solution
56. x2 y 4 ⇒ y 4 x2
ex y 0 ⇒ y ex
1
57.
y x4 2x2 1
y 1 x
2
5
Solve for x: x4 x2 0 ⇒ x2x2 1 0
⇒ x 0, ± 1
3
Back-substitute x 0: 1 x2 1 02 1
2
x
−1
Equation 2
Substitute for y in Equation 1: 1 x2 x4 2x2 1
y
−3
Equation 1
1
3
−1
Points of intersection (solutions):
approximately 1.96, 0.14, 1.06, 2.88
Back-substitute x 1: 1 x2 1 12 0
Back-substitute x 1: 1 x2 1 12 0
Solutions: 0, 1, ± 1, 0
620
Chapter 7
Systems of Equations and Inequalities
58. y x3 2x2 x 1
y x2 3x 1
Equation 1
Equation 2
59.
Substitute for y in Equation 1:
x2
3x 1 x3
2x2
xy 1 0
2x 4y 7 0
x1
1
⇒ x , 4
2
y 02 30 1 1
Back-substitute x 2 in Equation 2:
1
1
Back-substitute x : y 2
2
12
y 22 32 1 1
Back-substitute x 4: y Back-substitute x 1 in Equation 2:
y 12 31 1 5
Solutions:
Solutions: 0, 1, 2, 1, 1, 5
x 1 2x 1
x 12 4x 1
x2 2x 1 4x 4
x2 6x 5 0
x 1x 5 0 ⇒ x 1, 5
Back-substitute x 1: y 1 1 0
Back-substitute x 5: y 5 1 2
Solutions: 1, 0, 5, 2
62. C 5.5x 10,000, R 3.29x
RC
3.29x 5.5x 10,000
3.29x 5.5x 10,000 0
Let u x.
3.29u2 5.5u 10,000 0
u
5.5 ± 5.52 43.2910,000
23.29
u
5.5 ± 131,630.25
6.58
Choosing the positive value for u, we have
x u2 ⇒ x 55.9742 3133 units.
1
1
4
4
2, 2, 4, 4
1
1
61. C 8650x 250,000, R 9950x
Substitute for y in Equation 1: x 2x 1 1
u 55.974, 54.302
1
2x2 4 7x 0 ⇒ 2x 1x 4 0
Back-substitute x 0 in Equation 2:
Solve for x:
x 7 0
Solve for x:
0 xx 2x 1 ⇒ x 0, 2, 1
Equation 1
Equation 2
1
x
Substitute for y in Equation 2: 2x 4
0 xx2 x 2
Equation 2
Solve for y in Equation 1: y Solve for x: 0 x3 x2 2x
60. x 2y 1
y x 1
Equation 1
RC
9950x 8650x 250,000
1300x 250,000
x 192 units
Section 7.1
63. C 35.45x 16,000, R 55.95x
(a)
Linear and Nonlinear Systems of Equations
64. C 2.16x 5000, R 3.49x
RC
RC
(a)
55.95x 35.45x 16,000
2.16x 5000 3.49x
20.50x 16,000
5000 1.33x
x 781 units
(b)
621
x 3760
PRC
3760 items must be sold to break even.
60,000 55.95x 35.45x 16,000
(b) P R C
60,000 20.50x 16,000
8500 3.49x 2.16x 5000
76,000 20.50x
8500 1.33x 5000
13,500 1.33x
x 3708 units
10,151 x
10,151 items must be sold to make a profit of $8500.
65. R 360 24x
R 24 18x
Equation 1
Equation 2
(a) Substitute for R in Equation 2: 360 24x 24 18x
Solve for x: 336 42x ⇒ x 8 weeks
(b)
66. (a)
Weeks
1
2
3
4
5
6
7
8
9
10
R 360 24x
336
312
288
264
240
216
192
168
144
120
R 24 18x
42
60
78
96
114
132
150
168
186
204
S
25x 100
S 50x 475
The rentals are equal when
x 8 weeks.
Rock CD
Rap CD
25x 100 50x 475
75x 100 475
75x 375
x5
Conclusion: It takes 5 weeks for the sales of the two CDs to become equal.
(b)
Number of weeks, x
0
1
2
3
4
5
6
Sales, S (rock)
100
125
150
175
200
225
250
Sales, S (rap)
475
425
375
325
275
225
175
67. 0.06x 0.03x 350
0.03x 350
x $11,666.67
To make the straight commission offer the better offer,
you would have to sell more than $11,666.67 per week.
By inspecting the table, we can see that the
two sales figures are equal when x 5.
68. p 1.45 0.00014x 2
10
p 2.388 0.007x2
The market equilibrium
(point of intersection) is
approximately 99.99, 2.85.
0
150
0
622
Chapter 7
Systems of Equations and Inequalities
69. (a)
0.06x 0.085y 2000
(c) The point of intersection occurs when x 5000, so
the most that can be invested at 6% and still earn
$2000 per year in interest is $5000.
x y 25,000
(b) y1 25,000 x
27,000
2000 0.06x
y2 0.085
As the amount at 6% increases, the amount at 8.5%
decreases. The amount of interest is fixed at $2000.
70.
D 4 ,
VV 0.79D
2D 4,
2
2
(a)
5 ≤ D ≤ 40
5 ≤ D ≤ 40
0
12,000
10,000
Doyle Log Rule
Scribner Log Rule
(b) The graphs intersect when D 24.7 inches.
1500
V1
(c) For large logs, the Doyle Log Rule gives a greater volume for a given diameter.
V2
0
40
0
71.
t
8
9
10
11
12
13
Solar
70
69
66
65
64
63
Wind
31
46
57
68
105
108
(a) Solar: C 0.1429t 2 4.46t 96.8
(d) 0.1429t 2 4.46t 96.8 16.371t 102.7
Wind: C 16.371t 102.7
(b)
0.1429t 2 20.831t 199.5 0
By the Quadratic Formula we obtain t 10.3 and
t 135.47.
150
8
13
0
(c) Point of intersection: 10.3, 66.01
(e) The results are the same for t 10.3. The other “solution”, t 135.47, is too large to consider
as a reasonable answer.
(f) Answers will vary.
During the year 2000, the consumption of solar
energy will equal the consumption of wind energy.
72. (a) For Alabama, P 17.4t 4273.2.
4800
For Colorado, P 84.9t 3467.9.
(b) The lines appear to intersect at (11.93, 4480.79).
Colorado’s population exceeded Alabama’s just after this point.
(c) Using the equations from part (a),
17.4t 4273.2 84.9t 3467.9
4273.2 67.5t 3467.9
805.3 67.5t
11.93 t.
9
4000
13
Section 7.1
73. 2l 2w 30
⇒
Linear and Nonlinear Systems of Equations
74. 2l 2w 280
l w 15
⇒
l w 140
l w 3 ⇒ w 3 w 15
w l 20 ⇒ l l 20 140
2w 12
2l 160
l 80
w6
w l 20 80 20 60
lw39
Dimensions: 60 80 centimeters
Dimensions: 6 9 meters
76. 2l 2w 210 ⇒
75. 2l 2w 42 ⇒ l w 21
w
3
4l
⇒ l
3
4l
21
7
4l
21
l
3
2w
⇒
l w 105
3
2w
w 105
5
2w
l 12
w
3
4l
105
w 42
9
l
Dimensions: 9 12 inches
77. 2l 2w 40 ⇒
623
3
2 42
63
Dimensions: 42 63 feet
l w 20 ⇒ w 20 l
78. A 12bh
lw 96 ⇒ l20 l 96
1 12a2
a2 2
20l l 2 96
a 2
0 l 2 20l 96
2
a
The dimensions are
2 2 2 inches.
0 l 8l 12
l 8 or l 12
a
If l 8, then w 12.
If l 12, then w 8.
Since the length is supposed to be greater than the width,
we have l 12 kilometers and w 8 kilometers.
Dimensions: 8 12 kilometers
79. False. To solve a system of equations by substitution, you
can solve for either variable in one of the two equations
and then back-substitute.
80. False. The system can have at most four solutions
because a parabola and a circle can intersect at
most four times.
81. To solve a system of equations by substitution, use the following steps.
1. Solve one of the equations for one variable in terms of the other.
2. Substitute this expression into the other equation to obtain an equation in one variable.
3. Solve this equation.
4. Back-substitute the value(s) found in Step 3 into the expression found in Step 1 to find the value(s) of the other variable.
5. Check your solution(s) in each of the original equations.
82. For a linear system the result will be a contradictory equation such as 0 N, where N is a nonzero real number.
For a nonlinear system there may be an equation with imaginary solutions.
83. y x2
(a) Line with two points of intersection
(b) Line with one point of intersection
y 2x
y0
0, 0 and 2, 4
0, 0
(c) Line with no points of intersection
yx2
624
Chapter 7
Systems of Equations and Inequalities
84. (a) b 1
b2
6
b4
b3
6
6
−6
6
−6
−2
6
6
−6
−6
6
6
−2
−2
−2
(b) Three
85. 2, 7, 5, 5
m
86. 3.5, 4, 10, 6
57
2
5 2
7
m
2
y 7 x 2
7
y6
7y 49 2x 4
2x 7y 45 0
2
x 10
6.5
6.5y 39 2x 20
2x 6.5y 19 0
87. 6, 3, 10, 3
m
64
2
10 3.5 6.5
88. 4, 2, 4, 5
33
0 ⇒ The line is horizontal.
10 6
x4
x40
y3
y30
89.
5, 0, 4, 6
m
y6
60
6
30
4 35 175 17
m
30
x 4
17
y
17y 102 30x 120
29y 0 30x 17y 18
30x 17y 18 0
91. f x 5
x6
Domain: All real numbers except x 6
Horizontal asymptote: y 0
Vertical asymptote: x 6
93. f x 7
5 1
90. , 8 , ,
3
2 2
3
x2 2
x2 16
8 12
152
45
73 52 296
29
1
45
5
x
2
29
2
29
225
45x 2
2
45x 29y 127 0
92. f x 2x 7
3x 2
Domain: All real numbers except x 2
3
2
Vertical asymptote: x 3
Horizontal asymptote: y 94. f x 3 2
x2
Domain: All real numbers except x ± 4.
Domain: All real numbers except x 0
Horizontal asymptote: y 1
Horizontal asymptote: y 3
Vertical asymptotes: x ± 4
Vertical asymptote: x 0
2
3
Section 7.2
Two-Variable Linear Systems
625
Two-Variable Linear Systems
Section 7.2
■
You should be able to solve a linear system by the method of elimination.
1. Obtain coefficients for either x or y that differ only in sign. This is done by multiplying all the terms of one or both
equations by appropriate constants.
2. Add the equations to eliminate one of the variables and then solve for the remaining variable.
3. Use back-substitution into either original equation and solve for the other variable.
4. Check your answer.
■
You should know that for a system of two linear equations, one of the following is true.
1. There are infinitely many solutions; the lines are identical. The system is consistent. The slopes are equal.
2. There is no solution; the lines are parallel. The system is inconsistent. The slopes are equal.
3. There is one solution; the lines intersect at one point. The system is consistent. The slopes are not equal.
Vocabulary Check
1. elimination
2. equivalent
3. consistent; inconsistent
4. equilibrium price
1. 2x y 5
xy1
Equation 1
Equation 2
Add to eliminate y: 3x 6 ⇒ x 2
2.
x 2y 4
x 3y 1
Equation 1
Equation 2
x 3y 1
Add to eliminate x:
x 2y 4
Substitute x 2 in Equation 2: 2 y 1 ⇒ y 1
Solution: 2, 1
y
5y 5 ⇒ y 1
Substitute y 1 in Equation 1: x 31 1 ⇒ x 2
x−y=1
Solution: 2, 1
4
3
y
2
1
−2 − 1
x
1
2
4
5
4
x + 3y = 1
6
− x + 2y = 4
2x + y = 5
−3
−4
−6
−4
x
−2
−2
3.
x y0
3x 2y 1
Equation 1
y
4
Equation 2
3
Multiply Equation 1 by 2: 2x 2y 0
Add this to Equation 2 to eliminate y: x 1
Substitute x 1 in Equation 1: 1 y 0 ⇒ y 1
Solution: 1, 1
2
3x + 2y = 1
x+y=0
−4 −3 − 2 − 1
−2
−3
−4
x
2
3
4
626
4.
Chapter 7
Systems of Equations and Inequalities
2x4x 3yy 213
Equation 1
Equation 2
y
2x − y = 3
6
Multiply Equation 1 by 3: 6x 3y 9
4
6x 3y 9
Add this to Equation 2 to eliminate y:
4x + 3y = 21
4x 3y 21
2
30
10x
x
2
⇒ x 3
4
−2
Substitute x 3 in Equation 1: 23 y 3 ⇒ y 3
Solution: 3, 3
5.
x y2
2x 2y 5
6. 3x 2y 3
6x 4y 14
Equation 1
Equation 2
Equation 1
Equation 2
Multiply Equation 1 by 2: 2x 2y 4
Multiply Equation 1 by 2: 6x 4y 6
Add this to Equation 2: 0 9
Add this to Equation 2: 6x 4y 6
6x 4y 14
There are no solutions.
0
y
There are no solutions.
4
8
y
− 2x + 2y = 5
6x + 4y = 14
1
−4
x
−2 −1
2
−2
3
4
x
−2
x−y=2
4
2
−2
−4
3x + 2y = 3
−4
7.
3x 2y 6x 4y 10
Equation 1
5
8.
Equation 2
3x9x 3yy 155
Equation 1
Equation 2
Multiply Equation 1 by 2 and add to Equation 2: 0 0
Multiply Equation 2 by 3: 9x 3y The equations are dependent. There are infinitely
many solutions.
Add this to Equation 1:
Let x a, then y Solution:
3a 5 3
5
a .
2
2
2
2
y
1
x
2
3
4
− 3x + y = 5
8
5
6
−2
15
0
0
Solution: a, 3a 5, where a is any real number.
3
−3 −2 −1
9x 3y 3a y 5 ⇒ y 3a 5
3x − 2y = 5
4
a is any real number.
9x 3y 15
There are infinitely many solutions. Let x a.
y
5
3
a, a where
2
2
15
− 6x + 4y = −10
9x − 3y = − 15
x
−8 −6 −4
2
−4
−6
−8
4
6
8
Section 7.2
9. 9x 3y 1
3x 6y 5
Equation 1
4
9x 3y 3
Add to eliminate x: 21y 14 ⇒ y 32
in Equation 1: 9x 3
23
Substitute y Solution:
10. 5x 3y 18
Equation 1
1
Equation 2
2x 6y 10x 6y 36
2x 6y 2
1
3
4
9x + 3y = 1
−2
−3
−4
y
5x + 3y = −18 4
Multiply Equation 1 by 2: 10x 6y 36
Add this to Equation 2 to eliminate y:
x
−4 −3 −2 − 1
x 13
13, 23 3x − 6y = 5
1
9x 18y 15
23
627
y
Equation 2
Multiply Equation 2 by 3:
Two-Variable Linear Systems
2
−6
2x − 6y = 1
x
−4
2
−2
1
35 ⇒ x 35
12
12x
35
Substitute x 12 in Equation 2:
41
2 35
12 6y 1 ⇒ y 36
35
41
Solution: 12 , 36 11. x 2y 4
x 2y 1
Equation 1
12. 3x 5y 2
2x 5y 13
Equation 2
Add to eliminate y:
2x 5y 13
5
2
Substitute x 3 in Equation 1: 33 5y 2 ⇒ y 57
Solution: 3, 75 2y 4 ⇒ y 43
Solution:
52, 34 13. 2x 3y 18
Equation 1
y 11
Equation 2
5x 15 ⇒ x 3
5x
5
Substitute x 2 in Equation 1:
5
2
Equation 2
Add to eliminate y: 3x 5y 2
2x 5
x
Equation 1
14.
x 7y 12
3x 5y 10
Equation 1
Equation 2
Multiply Equation 2 by 3: 15x 3y 33
Multiply Equation 1 by 3: 3x 21y 36
Add this to Equation 1 to eliminate y:
Add this to Equation 2 to eliminate x:
17x 51 ⇒ x 3
Substitute x 3 in Equation 1:
6 3y 18 ⇒ y 4
Solution: 3, 4
3x 21y 36
3x 5y 10
26y 26
⇒ y
1
Substitute y 1 in Equation 1: x 7 12 ⇒ x 5
Solution: 5, 1
628
Chapter 7
Systems of Equations and Inequalities
15. 3x 2y 10
2x 5y 13
16.
Equation 1
Equation 2
Multiply Equation 1 by 2 and
Equation 2 by 3:
2r 4s 5
Equation 1
16r 50s 55
Equation 2
Multiply Equation 1 by 8: 16r 32s 40
Add this to Equation 2 to eliminate r:
6x 4y 20
6x 15y 9
16r 32s 40
16r 50s 55
15
Substitute y 1 in Equation 1:
18s ⇒ s
5
6
3x 2 10 ⇒ x 4
5
6
Add to eliminate x: 11y 11 ⇒ y 1
Substitute s in Equation 1:
2r 456 5 ⇒ r 65
Solution: 4, 1
Solution:
17. 5u 6v 24
3u 5v 18
Equation 1
18.
Equation 2
2x3x 11y5y 49
Equation 1
Equation 2
Multiply Equation 1 by 2 and Equation 2 by 3:
Multiply Equation 1 by 5 and Equation 2 by 6:
25u 30v 56, 56 8
6x6x 22y
15y 27
18u 30v 108
120
Add to eliminate x:
Add to eliminate v: 7u 12 ⇒ u 12
7
Substitute u 12
7 in Equation 1:
7y 35 ⇒ y 5
108
18
512
7 6v 24 ⇒ 6v 7 ⇒ v 7
Solution:
6x 22y 8
6x 15y 27
Substitute y 5 in Equation 1: 3x 115 4
127, 187 ⇒ x 17
Solution: 17, 5
19.
9
5x
65 y 4
9x 6y 3
Equation 1
20.
Equation 2
x 3yy 3x
4
9
4
1
8
3
8
Equation 1
Equation 2
Multiply Equation 1 by 3:
Multiply Equation 1 by 10 and Equation 2 by 2:
18x 12y 40
18x 12y 6
94 x 3y 38
9
4x
3y 3
8
Add to eliminate x and y: 0 34
Add these two together to obtain 0 0.
Inconsistent
The original equations are dependent. They have infinitely
many solutions.
No solution
3
1
Set x a in 4x y 8 and solve for y.
1
3
The points on the line have the form a, 8 4a.
21.
x
y
1
4 6
xy3
Equation 1
Equation 2
Multiply Equation 1 by 6:
3
xy6
2
Add this to Equation 2 to eliminate y:
5
18
x9 ⇒ x
2
5
Substitute x 18
in Equation 2:
5
18
y3
5
3
y
5
18 3
Solution:
,
5 5
Section 7.2
22.
2
3x
16 y 23
4x y 4
Two-Variable Linear Systems
23. 5x 6y 3
20x 24y 12
Equation 1
Equation 2
Multiply Equation 1 by 6: 4x y 4
Add this to Equation 2: 4x y 4
4x y 4
0
0
Equation 1
Equation 2
Multiply Equation 1 by 4:
24y 12
20x
20x 24y 12
Add to eliminate x and y: 0 0
There are infinitely many solutions. Let x a.
4a y 4 ⇒ y 4 4a
Solution: a, 4 4a where a is any real number
The equations are dependent. There are infinitely many
solutions.
Let x a, then
5a 3 5
1
a .
6
6
2
5a 6y 3 ⇒ y Solution:
24.
7x 8y 14x 16y 12
6
a, 65 a 21 where a is any real number
25. 0.05x 0.03y 0.21
0.07x 0.02y 0.16
Equation 1
Equation 2
Multiply Equation 1 by 2:
6y 42
10x
21x 6y 48
12
Add to eliminate y: 31x 90
Add these two together to obtain 0 0.
The original equations are dependent. They have infinitely
many solutions.
x 90
31
90
Substitute x 31 in Equation 2:
0.0790
31 0.02y 0.16
Set x a in 7x 8y 6 and solve for y.
3
7
The points on the line have the form a, 4 8a.
y 67
31
Solution:
26.
0.3x 0.4y 0.2x 0.5y 27.8
68.7
3b 11m 13
0.8x 2y 111.2
Equation 1
Equation 2
Multiply Equation 1 by 3 and Equation 2 by 4:
12b 9m 12b 44m 52
343.5
Add these to eliminate y: 0.8x 2y 111.2
9
Add to eliminate b: 35m 43
1.5x 2y 343.5
m 43
35
232.3
Substitute m 43
35 in Equation 1:
⇒ x
101
2.3x
Substitute x 101 in Equation 1:
0.2101 0.5y 27.8 ⇒ y 96
Solution: 101, 96
9031, 6731 27. 4b 3m 3
Equation 1
Equation 2
Multiply Equation 1 by 4 and Equation 2 by 5:
1.5x 2y Equation 1
Equation 2
Multiply Equation 1 by 200 and Equation 2 by 300:
14x
14x
16y 12
16y 629
6
4b 343
35 3 ⇒ b 35
6 43
Solution: 35, 35 630
Chapter 7
Systems of Equations and Inequalities
28. 2x 5y 8
5x 8y 10
Equation 1
Equation 2
29.
Multiply Equation 1 by 5 and Equation 2 by 2:
40
3x 4y 7
Add to eliminate y: 11x 55 ⇒ x 5
40
Substitute x 5 into Equation 2:
10x 16y 20
9y Substitute y 20 ⇒ y 25 y 12 ⇒ y 2
20
9
Solution: 5, 2
20
20
8
in Equation 1: 2x 5
9
9
⇒ x
Solution:
30.
Equation 2
8x 4y 48
Add to eliminate x:
10x 25y Equation 1
Multiply Equation 1 by 12 and Equation 2 by 4:
10x 16y 20
10x 25y x3 y1
1
4
3
2x y 12
14
9
149, 209
x1 y2
4
2
3
Equation 1
x 2y 5
Equation 2
31. 2x 5y 0
x
Multiply Equation 2 by 5:
Multiply Equation 1 by 6:
2x 5y 0
5x 5y 15
3x 1 2 y 2 24 ⇒ 3x 2y 23
Add this to Equation 2 to eliminate y:
Add to eliminate y: 3x 15 ⇒ x 5
3x 2y 23
Matches graph (b).
x 2y 5
Number of solutions: One
28
4x
y3
Consistent
⇒x 7
Substitute x 7 in Equation 2: 7 2y 5 ⇒ y 1
Solution: 7, 1
33. 2x 5y 0
2x 3y 4
32. 7x 6y 4
14x 12y 8
7x 6y 4 ⇒ 6y 7x 4 ⇒ y 76x 23;
The graph contains 0,
23
Multiply Equation 1 by 1:
and 4, 4.
14x 12y 8 ⇒ 12y 14x 8 ⇒ y 2x 5y 0
7
6x
2
3;
2x 3y 4
The graph is the same as the previous graph.
Add to eliminate x: 2y 4 ⇒ y 2
The graph of the system matches (a).
Matches graph (c).
Number of solutions: Infinite
Number of solutions: One
Consistent
Consistent
Section 7.2
34.
7x 6y 6 ⇒
35. 3x 5y 7
Equation 1
y9
Equation 2
2x 7x7x 6y6y 6
4
6y 7x 6
⇒ y
7
6x
1;
The graph contains 0, 1 and 3, .
10x 5y 45
7x 6y 4 ⇒ 6y 7x 4 ⇒ y 76x 23;
The graph contains 0,
previous graph.
631
Multiply Equation 2 by 5:
9
2
2
3
Two-Variable Linear Systems
Add this to Equation 1:
and is parallel to the
13x 52 ⇒ x 4
Back-substitute x 4 into Equation 2:
The graph of the system matches (d).
24 y 9 ⇒ y 1
Number of solutions: None
Solution: 4, 1
Inconsistent
36. x 3y 17
Equation 1
7
Equation 2
4x 3y 37.
y 2x 51
y 5x 11
Equation 1
Equation 2
Since both equations are solved for y, set them
equal to one another and solve for x.
Subtract Equation 2 from Equation 1 to eliminate y:
x 3y 17
4x 3y 7
5x
10 ⇒ x 2
2x 5 5x 11
2x 6 3x
2x 2 x
Substitute x 2 in Equation 1:
2 3y 17 ⇒ y 5
Back-substitute x 2 into Equation 1:
Solution: 2, 5
y 22 5 1
Solution: 2, 1
38.
7x 3y 16
yx2
Equation 1
Equation 2
39.
x 5y 21
6x 5y 21
Equation 1
Equation 2
Substitute for y in Equation 1:
Add the equations: 7x 42 ⇒ x 6
7x 3x 2 16
Back-substitute x 6 into Equation 1:
6 5y 21 ⇒ 5y 15 ⇒ y 3
7x 3x 6 16
10x 10 ⇒ x 1
Solution: 6, 3
Substitute x 1 in Equation 2: y 1 2 3
Solution: 1, 3
40. y 3x 8
y 15 2x
Equation 1
Equation 2
Since both equations are solved for y, set them equal to
one another and solve for x:
3x 8 15 2x
41. 2x 8y 19
yx3
Equation 1
Equation 2
Substitute the expression for y from Equation 2
into Equation 1.
2x 8x 3 19 ⇒ 2x 8x 24 19
x 23
6x 43
x 23
Back-substitute x 23 into Equation 1:
y 323 8 61
Solution: 23, 61
x 43
6
Back-substitute x 43
6 into Equation 2:
25
y 43
6 3 ⇒ y 6
Solution:
436, 256 632
Chapter 7
42.
5x4x 3y7y 16
Systems of Equations and Inequalities
Equation 1
Equation 2
Multiply Equation 1 by 5 and Equation 2 by 4:
20x 15y 30
20x
28y 4
Add to eliminate x:
20x 15y 30
20x 28y 4
13y 26 ⇒ y 2
Back-substitute y 2 into Equation 1:
4x 32 6 ⇒ x 3
Solution: 3, 2
43. Let r1 the air speed of the plane
and r2 the wind air speed.
3.6r1 r2 1800
Equation 1
⇒
3r1 r2 1800
Equation 2
⇒
r1 r2 500
r1 r2 600
2r1
1100
r1
550
550 r2 600
r2 50
Add the equations.
The air speed of the plane is 550 mph and the speed of the wind is 50 mph.
44. Let x the speed of the plane that leaves first and y the speed of the plane that leaves second.
2xy x 80
3200
3
2y
Equation 1
Equation 2
2x 2y 160
2x 32 y 3200
7
2y
3360
y 960
960 x 80
x 880
Solution: First plane: 880 kilometers per hour; Second plane: 960 kilometers per hour
45.
50 0.5x 0.125x
50 0.625x
x 80 units
p $10
Solution: 80, 10
46.
Supply Demand
25 0.1x 100 0.05x
0.15x 75
x 500
p 75
Equilibrium point: 500, 75
Section 7.2
47. 140 0.00002x 80 0.00001x
Two-Variable Linear Systems
633
Supply Demand
48.
225 0.0005x 400 0.0002x
60 0.00003x
0.0007x 175
x 2,000,000 units
x 250,000
p $100.00
Solution: 2,000,000, 100
p 350
Equilibrium point: 250,000, 350
49. Let x number of calories in a cheeseburger
50. Let x Vitamin C in a glass of apple juice
y number of calories in a small order of french fries
y Vitamin C in a glass of orange juice.
3x 2y 1390
2xx 3yy 185
452
2x y 850
Equation 1
Equation 2
Multiply Equation 1 by 2:
Multiply Equation 1 by 2; then add the equations:
4x3x 2y2y 1700
1390
x
2x2x 2y3y 370
452
310
x
310
y
230
Equation 1
Equation 2
Add the equations.
y
82
Then x 185 82 103.
The point (103, 82) is the solution of the system.
Solution: The cheeseburger contains 310 calories and the
fries contain 230 calories.
Apple juice has 103mg of Vitamin C, while orange juice
has 82 mg.
51. Let x the number of liters at 20%
Let y the number of liters at 50%.
(a)
x
y 10
0.2x 0.5y 0.310
2 Equation 1:
10
(b)
2x 2y 20
Equation 2:
2x 5y 30
3y 10
y
10
3
x
10
3
10
x
20
3
12
−6
18
−4
As x increases, y decreases.
(c) In order to obtain the specified concentration of
the final mixture, 623 liters of the 20% solution and
313 liters of the 50% solution are required.
52. Let x the number of gallons of 87 octane gasoline; y the number of gallons of 92 octane gasoline.
(a)
(b)
87x 92y 44,500
x
y
500
Equation 1
Equation 2
(c) 87Equation 1:
Equation 2:
87x 87y 43,500
87x 92y 44,500
5y 1000
y
200
x 200 500
x
300
500
0
500
0
As the amount of 87 octane gasoline increases,
the amount of 92 octane gasoline decreases.
Solution: 87 octane: 300 gallons;
92 octane: 200 gallons
634
Chapter 7
Systems of Equations and Inequalities
53. Let x amount invested at 7.5%
y amount invested at 9%.
x
y 12,000
Equation 1
990
Equation 2
0.075x 0.09y Multiply Equation 1 by 9 and Equation 2 by 100.
9x 9y 108,000
7.5x 9y 99,000
9,000
x
$6000
y
$6000
1.5x
Add the equations.
The most that can be invested at 7.5% is $6000.
54. Let x the amount invested at 5.75%; y the amount invested at 6.25%.
x
y 32,000
0.0575x 0.0625y 1900
Equation 1 ⇒ 5.75Equation 1:
Equation 2 ⇒ 100Equation 2:
5.75x 5.75y 184,000
5.75x 6.25y 190,000
0.5y 6000
y
12,000
x 12,000 32,000
x
20,000
The amount that should be invested in the bond that pays 5.75% interest is $20,000.
55. Let x number of student tickets
y number of adult tickets.
x
y 1435
1.50x 5.00y 3552.50
Equation 1
Equation 2
Multiply Equation 1 by 1.50.
1.50x 1.50y 2152.50
1.50x 5.00y 3552.50
3.50y 1400.00
y
400
x
1035
Add the equations.
Solution: 1035 student tickets and 400 adult tickets were sold.
56. Let x the number of jackets sold before noon; y the number of jackets sold after noon.
x
y 214
31.95x 18.95y 5108.30
Equation 1 ⇒ 31.95Equation 1:
Equation 2 ⇒
Equation 2:
31.95x 31.95y 6837.30
31.95x 18.95y 5108.30
13y 1729
So, 81 jackets were sold before noon and 133 jackets were sold after noon.
y
133
x 133 214
x
81
Section 7.2
57.
5b 10a 20.2 ⇒ 10b 20a 40.4
10b 30a 50.1 ⇒
10b 30a 50.1
10a 9.7
a
b
58.
Two-Variable Linear Systems
5b 10a 11.7 ⇒ 10b 20a 23.4
10b 30a 25.6
10b 30a 25.6 ⇒
10a 2.2
0.97
a
2.10
5b 100.22 11.7
b
1.9
Least squares regression line: y 0.97x 2.10
0.22
Least squares regression line: y 0.22x 1.9
59.
7b 21a 35.1 ⇒ 21b 63a 105.3
21b 91a 114.2 ⇒
21b 91a 114.2
28a a
60.
6b 15a 23.6 ⇒ 15b 37.5a 59
15b 55a 48.8
15b 55a 48.8 ⇒
17.5a 10.2
8.9
a 0.583
89
280
b
b 1137
280
5.390
Least squares regression line: y 0.583x 5.390
Least squares regression line: y 28089x 1137
1
y 0.32x 4.1
61. 0, 4, 1, 3, 1, 1, 2, 0
n 4,
4
62.
4
4
x 4, y 8, x
i
2
i
i
i1
i1
4b 4a 8 ⇒
6,
i1
4
x y 4
8b 28a 8 ⇒ 224b 784a 224
28b 116a 37 ⇒
224b 928a 296
144a 72
a
1
2
i i
i1
4b 4a 8
2a 4
4b 6a 4 ⇒ 4b 6a 4
a 2
b
8b 2812 8
b
34
1
3
Least squares regression line: y 2 x 4
4
Least squares regression line: y 2x 4
63. 5, 66.65, 6, 70.93, 7, 75.31, 8, 78.62, 9, 81.33, 10, 85.89, 11, 88.27
(a) n 7,
7
7
x 56, x
2
i 7
476,
7
y 547, x y 4476.8
(c)
t
Actual room rate
Model approximation
56b 476a 4476.8
5
$66.65
$67.34
6
$70.93
$70.94
Multiply Equation 1 by 8.
7
$75.31
$74.54
56b 448a 4376
56b 476a 4476.8
8
$78.62
$78.14
9
$81.33
$81.74
10
$85.89
$85.34
11
$88.27
$88.94
i
i1
i1
7b 56a i
i1
i i
i1
547
28a 100.8
a 3.6
b 49.343
Least squares regression line: y 3.6t 49.343
(b) y 3.6t 49.343, This agrees with part (a).
The model is a good fit to the data.
(d) When t 12: y $92.54
This is a little off from the actual rate.
(e) 3.6t 49.343 100
3.6t 50.657
t 14.1
According to the model, room rates will average
$100.00 during the year 2004.
635
636
Chapter 7
Systems of Equations and Inequalities
64. (a) 1.0, 32, 1.5, 41, 2.0, 48, 2.5, 53
4b 7a 174 ⇒ 7b 12.25a 304.5
7b 13.5a 322 ⇒
7b 13.5a (b) When x 1.6: y 141.6 19 41.4 bushels
per acre.
322
1.25a 17.5
a
14
4b 98 174
b
19
Least squares regression line: y 14x 19
65. False. Two lines that coincide have infinitely many
points of intersection.
66. False. Solving a system of equations algebraically will
always give an exact solution.
67. No, it is not possible for a consistent system of linear
equations to have exactly two solutions. Either the lines
will intersect once or they will coincide and then the
system would have infinite solutions.
68. Answers will vary.
69. 100y x 200
99y x 198
Equation 1
Equation 2
(a) No solution
xx yy 1020
13x 12y 1200
70. 21x 20y 100y x 200
Equation 2
23 x 200 ⇒ x 300
398
Back-substitute x 300 in Equation 1:
Substitute y 398 into Equation 1:
21300 20y 0 ⇒ y 315
100398 x 200 ⇒ x 39,600
Solution: 300, 315
Solution: 39,600, 398
The lines are not parallel. It is necessary to change the
scale on the axes to see the point of intersection.
The lines are not parallel. The scale on the axes
must be changed to see the point of intersection.
Equation 1
Add this to Equation 1 to eliminate y:
99y x 198
71. 4x 8y 3
2x ky 16
2xx 2yy 36
Multiply Equation 2 by 53 : 65
3 x 20y 200
Subtract Equation 2 from Equation 1 to eliminate x:
y
(b) Infinite number of solutions
Equation 1
Equation 2
Multiply Equation 2 by 2: 4x 2ky 32
Add this to Equation 1:
4x 8y 3
4x 2ky 32
8y 2ky 35
The system is inconsistent if 8y 2ky 0.
This occurs when k 4.
72.
15x 3y 6 ⇒ 30x 6y 12
10x
ky 9 ⇒ 30x 3ky 27
(6 3k)y 39
If k 2, then we would have 0 39 and
the system would be inconsistent.
Section 7.2
73. 11 6x ≥ 33
3
x
x ≤
−9
−8
637
74. 2x 3 > 5x 1
− 22
6x ≥ 44
Two-Variable Linear Systems
−7
−6
x
0
2x 6 > 5x 1
−5
22
3
1
2
3
4
7x > 7
x > 1
75. 8x 15 ≤ 42x 1
76. 6 ≤ 3x 10 < 6
19
16
x
8x 15 ≤ 8x 4
−1
0
1
2
4
3
≤ x <
16
3
x
4 ≤ 3x < 16
3
16x ≤ 19
x ≤
4
3
1
2
3
4
5
6
7
16
3
19
16
77. x 8 < 10
x
10 < x 8 < 10
78. x 10 ≥ 3
−2
−3
0
3
6
9
x
−3 −2 −1
0
1
2
3
All real numbers x
12 15 18
2 < x < 18
79. 2x2 3x 35 < 0
80. 3x2 12x > 0
7
2
2x 7x 5 < 0
− 6 − 5 − 4 − 3 −2 −1 0 1 2 3 4
Critical numbers: x 5,
Test intervals: , 5, 5, , 7
2,
Solution: x < 4, x > 0
82. ln x 5 lnx 3 ln x lnx 35
ln
12x 4x 2y 12
2x y 4
Test: Is 3xx 4 > 0?
7
2
81. ln x ln 6 ln6x
85.
2
Test Intervals: , 4, 4, 0, 0, Test: Is 2x 7x 5 < 0?
83. log9 12 log9 x log9
0
Critical numbers: x 0, 4
7
2
7
2
Solution: 5 < x <
x
− 8 −6 − 4 − 2
3xx 4 > 0
x
4 ⇒ y 2x 4
84.
x
x 35
1
4 3x
log6 3x log6 4
86. 30x 40y 33 0
1
21
10x 20y 21 0 ⇒ y 2x 20
4x 22x 4 12
1
21
30x 40 2x 20 33 0
4x 4x 8 12
30x 20x 42 33 0
8 12
50x 75
x 32
There are no solutions.
6
3
y 1232 21
20 20 10
Solution:
87. Answers will vary.
32, 103 638
Chapter 7
Section 7.3
■
Systems of Equations and Inequalities
Multivariable Linear Systems
You should know the operations that lead to equivalent systems of equations:
(a) Interchange any two equations.
(b) Multiply all terms of an equation by a nonzero constant.
(c) Replace an equation by the sum of itself and a constant multiple of any other equation in the system.
■
You should be able to use the method of Gaussian elimination with back-substitution.
Vocabulary Check
1. row-echelon
2. ordered triple
3. Gaussian
4. row operation
5. nonsquare
6. position
1.
3x y z 2x
1
3z 14
5y 2z 8
(a) 32 0 3 1
2, 0, 3 is not a solution.
(b) 32 0 8 1
2, 0, 8 is not a solution.
(c) 30 1 3 1
0, 1, 3 is not a solution.
(d) 31 0 4 1
21
34 14
50 24 8
1, 0, 4) is a solution.
3.
4x y z 0
8x 6y z 74
3x y
94
1
3
7
(a) 42 4 4 0
(b)
(c)
(d)
12, 34, 74 is not a solution.
4 32 54 54 0
32, 54, 54 is not a solution.
4 12 34 54 0
8 12 634 54 74
3 12 34 94
12, 34, 54 is a solution.
4 12 16 34 0
12, 16, 34 is not a solution.
2.
3x 4y z 17
5x y 2z 2
2x 3y 7z 21
(a) 33 41 2 17
3, 1, 2 is not a solution.
(b) 31 43 2 17
51 3 22 2
21 33 72 21
1, 3, 2 is a solution.
(c) 34 41 3 17
4, 1, 3 is not a solution.
(d) 31 42 2 17
1, 2, 2 is not a solution.
Section 7.3
4.
4x y 8z 6
y z 0
4x 7y
6
5.
(a) 42 2 82 6
2x y 5z 24
Equation 1
y 2z 6
Equation 2
z 4
Equation 3
y 24 6
42 72 6
y 2
2, 2, 2 is a solution.
Back-substitute y 2 and z 4 into Equation 1:
10 810 6
10, 10 is not a solution.
418 12 812 6
18, 12, 12 is not a solution.
4 11
2 4 84 6
2x 2 54 24
33
2
33
2,
(c)
(d)
639
Back-substitute z 4 into Equation 2:
2 2 0
(b) 4
Multivariable Linear Systems
2x 22 24
x1
Solution: 1, 2, 4
4 4 0
4 11
2 74 6
112, 4, 4 is a solution.
6.
4x 3y 2z 21
6y 5z 8
z 2
Equation 1
Equation 2
Equation 3
7.
2x y 3z 10
y z 12
z 2
Equation 1
Equation 2
Equation 3
Substitute z 2 into Equation 2: y 2 12 ⇒ y 10
Back-substitute z 2 in Equation 2:
Substitute y 10 and z 2 into Equation 1:
6y 52 8
2x 10 32 10
y 3
2x 4 10
Back-substitute z 2 and y 3 in Equation 1:
4x 33 22 21
2x 6
4x 13 21
x3
Solution: 3, 10, 2
x2
Solution: 2, 3, 2
8.
x y 2z 22
3y 8z 9
z 3
Equation 1
Equation 2
Equation 3
Back-substitute z 3 in Equation 2:
3y 83 9
9.
4x 2y z 8
Equation 1
4x 2y y z 4
Equation 2
z2
Equation 3
Substitute z 2 into Equation 2:
y 2 4 ⇒ y 2
3y 24 9
Substitute y 2 and z 2 into Equation 1:
y 11
4x 22 2 8
Back-substitute z 3 and y 11 in Equation 1:
4x 6 8
x 11 23 22
4x 2
x 5 22
x 21
x 17
Solution: 17, 11, 3
Solution:
12, 2, 2
640
10.
Chapter 7
5x
Systems of Equations and Inequalities
8z 22
3y 5z 10
z 4
11.
Back-substitute z 4 in Equation 2:
Equation 2
2x 2y 3z 0
Equation 3
5x 84 22 ⇒ x 2
10
Solution: 2, 3 , 4
x 3y 5z 4
x 2y 3z 5
y 2z 9
2x
3z 0
Back-substitute z 4 in Equation 1:
x 2y 3z 5
x 3y 5z 4
2x
3z 0
Equation 1
Add Equation 1 to Equation 2:
3y 54 10 ⇒ y 10
3
12.
2x 2y 3z 5
This is the first step in putting the system in
row-echelon form.
Equation 1
Equation 2
Equation 3
13.
Add 2 times Equation 1 to Equation 3:
x 2y 3z 5
x 3y 5z 4
4y 9z 10
This is the first step in putting the system
in row-echelon form.
x
2x 3x
y
y
x
y
3y 3y z
6
z 9
4z 18
y
3y z
6
z 9
3z 9
x
x
z
z
z
y
3y Equation 1
Equation 2
Equation 3
6
3
0
2Eq.1 Eq.2
3Eq.1 Eq.3
Eq.2 Eq.3
z
6
z 9
z
3
13Eq.3
3y 3 9 ⇒ y 2
x23
6 ⇒ x1
Solution: 1, 2, 3
14.
y z
3
Equation 1
x 2y 4z 3y 4z 5
5
Equation 2
Equation 3
x y z
3y 3z 3y 4z 3
2
5
x y z
3y 3z 7z 3
2
7
x
x
y z 3
y z 23
z 1
y1
23
1
3
x 1
Solution:
5 1
3, 3,
1Eq.1 Eq.2
Eq.2 Eq.3
13 Eq.2
17 Eq.3
⇒ y
1
3
3 ⇒ x
5
3
1
15.
2x
2z 5x 3y
3y 4z 2
4
4
x
z
5x 3y
3y 4z 1
4
4
x
x
z 1
3y 5z 1
3y 4z 4
z 1
3y 5z 1
z 5
3y 55 1 ⇒ y x5
1
2 Eq.1
5Eq.1 Eq.2
Eq.2 Eq.3
8
1 ⇒ x 4
Solution: 4, 8, 5
16.
x y z 1
2x 4y z 1
x 2y 3z 2
x
y z 1
2y 3z 3
3y 2z 3
x
x
x
y z 1
2y 3z 3
6y 4z 6
y z 1
2y 3z 3
5z 15
y z 1
y 32 z 32
z
3
Interchange equations.
17.
2Eq.1 Eq.2
1Eq.1 Eq.3
2 Eq.3
3Eq.2 Eq.3
12 Eq.2
15 Eq.3
Section 7.3
Multivariable Linear Systems
9
3z 10
4z 12
Interchange equations.
1
3 Eq.1
6y 3
3z 10
4z 12
y
2y 6y 3
3z 4
4z 12
3x 2x
3y
x
2x
y
x
x
x
6y y
2y y
2y 2Eq.1 Eq.2
3z 5z 3
4
0
3Eq.2 Eq.3
3z z
3
4
0
15Eq.3
2y 30 4 ⇒ y 2
x23 ⇒ x
y 32 3 32 ⇒ y 3
x 3 3 1 ⇒ x 5
5
Solution: 5, 2, 0
Solution: 5, 3, 3
18.
x 4y z 0
2x 4y z 7
2x 4y 2z 6
x 4y z 0
4y 3z 7
12y
6
x 4y z 0
4y 3z 7
9z 27
x 4y y
z
0
74
z 3
3
4z
Interchange equations.
2Eq.1 Eq.2
2Eq.1 Eq.3
3Eq.2 Eq.3
14 Eq.2
19 Eq.3
y 34 3 74 ⇒ y x 412 3 1
2
0 ⇒ x1
1
Solution: 1, 2, 3
20.
x 11y 4z 3
5x 3y 2z 3
2x 4y z 7
Interchange equations.
x 11y 4z 3
52y 18z 12
26y 9z 1
5Eq.1 Eq.2
2Eq.1 Eq.3
x 11y 4z 3
52y 18z 12
0
7
12 Eq.2 Eq.3
Inconsistent, no solution
19.
x 2y 2z 9
2x y z 7
3x y z 5
Interchange equations.
x 2y 2z 9
5y 5z 25
5y 5z 32
2Eq.1 Eq.2
3Eq.1 Eq.3
x 2y 2z 9
5y 5z 25
0 7
Eq.2 Eq.3
Inconsistent, no solution
641
642
21.
Chapter 7
Systems of Equations and Inequalities
3x 5y 5z 5x 2y 3z 7x y 3z 1
0
0
6x 10y 10z 5x 2y 3z 7x y 3z 2
0
0
x 8y 7z 5x 2y 3z 7x y 3z 22.
2x y 3z 1
2x 6y 8z 3
6x 8y 18z 5
Equation 1
Equation 2
Equation 3
2x y 3z 1
5y 5z 2
5y 9z 2
1Eq.1 Eq.2
3Eq.1 Eq.3
x
1
2y
y
3
2z
1
2
z
2
5
z0
2
5
1Eq.2 Eq.3
Eq.1
15 Eq.2
14 Eq.3
2
5
1 2
3
x 25 20 2 ⇒ x 3
10
1
Solution:
38y 3232 10 ⇒ y 1
1
2, 1, 32 3
2
2 ⇒ x 12
x
2x 3x 2y 7z 4
y z 13
9y 36z 33
x
2y 7z 4
3y 15z 21
3y 15z 21
2Eq.1 Eq.2
3Eq.1 Eq.3
2y 7z 4
3y 15z 21
0
0
Eq.2 Eq.3
x
x
x
1
2
⇒ y
y0
12z 18 ⇒ z Solution:
23.
Eq.2 Eq.3
3
2
x 81 7
5Eq.1 Eq.2
7Eq.1 Eq.3
55Eq.2
38Eq.3
8y 7z 2
2090y 1760z 550
12z 18
Eq.2 Eq.1
x 8y 7z 2
38y 32z 10
55y 46z 14
2x y 3z 1
5y 5z 2
4z 0
8y 7z 2
2090y 1760z 550
2090y 1748z 532
x
2Eq.1
2
0
0
x
2y 7z 4
y 5z 7
3z 10
y 5z 7
13Eq.2
2Eq.2 Eq.1
Let z a, then:
y
103 , 25, 0
5a 7
x 3a 10
Solution: 3a 10, 5a 7, a
24.
2x 4x
2x y 3z 4
2z 10
3y 13z 8
2x y 3z 4
2y 8z 2
4y 16z 4
2x y 3z 4
2y 8z 2
0 0
2x
x
z
2y 8z 5
2
z2 52
y 4z 1
za
y 4a 1 ⇒ y 4a 1
1
5
1
5
x 2a 2 ⇒ x 2a 2
Solution: 12a 52, 4a 1, a
Equation 1
Equation 2
Equation 3
2Eq.1 Eq.2
Eq.1 Eq.3
2Eq.2 Eq.3
12 Eq.2 Eq.1
12 Eq.1
12 Eq. 2
25.
3x x
5x 3y 6z 2y z 8y 13z 6
5
7
x
x
5x y 2z 2y z 8y 13z 2
5
7
x
y 2z 2
3y 3z 3
3y 3z 3
x
x
y 2z y z
0
2
1
0
y
3
1
z
z
Let z a, then:
y
a1
x a 3
Solution: a 3, a 1, a
1
3 Eq.1
Eq.1 Eq.2
5Eq.1 Eq.3
1
3 Eq.2
Eq.2 Eq.3
Eq.2 Eq.1
Section 7.3
26.
x
2z 5
3x y z 1
6x y 5z 16
x
x
Equation 1
Equation 2
Equation 3
2z 5
y 7z 14
y 7z 14
3Eq.1 Eq.2
6Eq.1 Eq.3
2z 5
y 7z 14
0
0
1Eq.2 Eq.3
2z 5
y 7z 14
x
27.
1
4a
2y 5a 2
a 8y 20a 8
8y 21a 8
21
y 8a 1
Answer:
1Eq.2
2a 2y 58a 2
2y 42a 2
y 21a 1
Solution: 2a, 21a 1, 8a
Solution: 2a 5, 7a 14, a
x
18
80
3y 2z 18
2y 2z 10
x 3y 2z 18
y z 5
x
5z 3
y z 5
12 Eq.2
4x2x 3y9y z 27
2x 3y3y 2zz 23
2x 3y 3z2z 13
3Eq.2 Eq.1
Let z a, then:
Equation 1
Equation 2
5Eq.1 Eq.2
Solution: 5a 3, a 5, a
Equation 1
Equation 2
2Eq.1 Eq.2
2x 3y 3z 7
12y 9z 30
2x
34z 12
12y 9z 30
x
y
3
8z
3
4z
14
5
2
14 Eq.2 Eq.1
12 Eq.1
121 Eq.2
Let z a, then:
y 34a 5
2
29.
2Eq.1 Eq.2
Eq.2 Eq.1
y 23a 1
Let z a, then: y a 5 ⇒ y a 5
x 5a 3 ⇒ x 5a 3
30. 2x 3y 3z 7
4x 18y 15z 44
14a, 218a 1, a
To avoid fractions, we could go back and let
z 8a, then 4x 8a 0 ⇒ x 2a.
y 7 a 14 ⇒ y 7a 14
x 2a 5 ⇒ x 2a 5
5xx 13y3y 12z2z 4xx 2y 5zz 20
Let z a, then x 14a.
za
28.
Multivariable Linear Systems
⇒ y 34a 5
2
3
1
3
x 8a 4 ⇒ x 8a 1
4
Solution: 38a 14, 34a 52, a
x 32a 12
3
1
2
Solution: 2a 2, 3a 1, a
643
644
31.
32.
Chapter 7
Systems of Equations and Inequalities
3w 2y z w 2w 3y
2x y 4z
x
x
x
x
x
3w 4
2y z w 0
2w 1
3y
y 4z 6w 3
3w
y 4z 6w
2y z w
3y
2w
3w
y 4z 6w
7z 13w
12z 20w
x y
2x 3y
3x 4y x 2y x
2Eq.1 Eq.4
z w
w
z 2w z w
3w
y 4z 6w
z 3w
12z 20w
4
3
2
8
3w 4
y 4z 6w 3
z 3w 2
16w 16
12Eq.4 Eq.3
12Eq.3 Eq.4
16w 16 ⇒ w 1
4
3
0
1
Eq.4 and interchange
the equations.
4
3
6
8
z 31 2 ⇒ z 1
6
0
4
0
6
12
106
6
x y z w
6
y 2z 3w 12
53
z 13
9w 9
w
2
Equation 1
Equation 2
Equation 3
Equation 4
y 41 61 3 ⇒ y1
x 31 4 ⇒ x1
Solution: 1, 1, 1, 1
Eq.2 Eq.3
3Eq.2 Eq.4
x y z w
6
y 2z 3w 12
7y 4z 5w 22
y 2z
6
x y z w
y 2z 3w
18z 26w
3w
4
0
1
5
33.
2Eq.1 Eq.2
3Eq.1 Eq.3
1Eq.1 Eq.4
x
4z 1
x y 10z 10
2x y 2z 5
x
x
4z 1
y 6z 9
y 6z 7
4z y 6z 0
Eq.1 Eq.2
2Eq.1 Eq.3
1
9
2
Eq.2 Eq.3
No solution, inconsistent
7Eq.2 Eq.3
1Eq.2 Eq.4
181 Eq.3
13 Eq.4
53
z 13
9 2 9 ⇒ z 3
y 23 32 12 ⇒ y 0
6 ⇒ x1
x032
Solution: 1, 0, 3, 2
34.
2x 2y 6z 4
3x 2y 6z 1
x y 5z 3
Equation 1
Equation 2
Equation 3
x y 5z 3
3x 2y 6z 1
2x 2y 6z 4
Interchange equations.
x y 5z 3
y 9z 8
4z 2
3Eq.1 Eq.2
2Eq.1 Eq.3
x y 5z 3
y 9z 8
z 12
y 912 1Eq.2
14 Eq.3
8 ⇒ y 72
x 72 512 3 ⇒ x 3
Solution: 3, 72, 12 Section 7.3
35.
2x 4x 8x 3y
0
3y z 0
3y 3z 0
2x 3y
0
3y z 0
9y 3z 0
2x 36.
2Eq.1 Eq.2
4Eq.1 Eq.3
3y
0
3y z 0
6z 0
3Eq.2 Eq.3
6z 0 ⇒ z 0
3y 0 0 ⇒ y 0
2x 30 0 ⇒ x 0
Solution: 0, 0, 0
Multivariable Linear Systems
4x 3y 17z 0
5x 4y 22z 0
4x 2y 19z 0
5x 4x 4x 4y 22z 0
3y 17z 0
2y 19z 0
x
4x 4x y 5z 0
3y 17z 0
2y 19z 0
x
y 5z 0
y 3z 0
2y z 0
4Eq.1 Eq.2
4Eq.1 Eq.3
y 5z 0
y 3z 0
5z 0
1Eq.2
2Eq.2 Eq.3
y 5z 0
y 3z 0
z0
15 Eq.3
x
x
Interchange equations.
1Eq.2 Eq.1
y 30 0 ⇒ y 0
x 0 50 0 ⇒ x 0
Solution: 0, 0, 0
5y z 0
4y z 0
10y 2z 0
24x
23x 4y z 0
23xx 6y4y 3zz 00
x 134y6y 70z3z 00
37. 12x 23x x
6y 3z 0
67y 35z 0
38.
2Eq.1
Eq.2 Eq.1
23Eq.1 Eq.2
x
1
2y
1
2z
Equation 1
Equation 2
Eq.1 Eq.2
12 Eq.1
15 Eq.2
0
3
y 5z 2
5
Let z a, then:
y 35a 2
5
⇒ y 35a x 12 35a 25 12a 0 ⇒ x 1
2 Eq.2
To avoid fractions, let z 67a, then:
Solution:
67y 3567a 0
y 35a
x 635a 367a 0
x 9a
Solution: 9a, 35a, 67a
39. s 12at2 v0t s0
1, 128, 2, 80, 3, 0
128 12a 2v0 s0 ⇒ a 2v0 2s0 256
180 2a 2v0 s0 ⇒ 2a 2v0 2s0 280
120 92a 3v0 s0 ⇒ 9a 6v0 2s0 2x2x 6yy 4zz 02
2x 5yy 3zz 02
0
Solving this system yields a 32, v0 0, s0 144.
Thus, s 1232t2 0t 144 16t2 144.
15a 15, 35a 25, a
1
5a
2
5
1
5
645
646
Chapter 7
Systems of Equations and Inequalities
40. s 12at 2 v0 t s0
1
41. s 2at2 v0t s0
1, 48, 2, 64, 3, 48
1, 452, 2, 372, 3, 260
v0 s0
452 12a 2v0 s0 ⇒ a 2v0 2s0 904
64 2a 2v0 s0
372 2a 2v0 s0 ⇒ 2a 2v0 2s0 372
48 48 1
2a
9
2a
a
2a 9a a
3v0 s0
2v0 2s0 2v0 s0 6v0 2s0 260 92a 3v0 s0 ⇒ 9a 6v0 2s0 520
96
64
96
2v0 2s0 96
2v0 3s0 128
12v0 16s0 768
a
2v0 2v0 a
2s0 96
3s0 128
2s0 0
2v0 2s0 v0 1.5s0 s0 96
64
0
2Eq. 1
2Eq. 3
Solving this system yields a 32, v0 32, s0 500.
Thus, s 1232t 2 32t 500
hus, s 16t 2 32t 500.
2Eq.1 Eq.2
9Eq.1 Eq.3
6Eq.2 Eq.3
0.5Eq.2
0.5Eq.3
v0 1.50 64 ⇒ v0 64
a 264 20 96 ⇒ a 32
Thus, s 1232t 2 64t 0
16t 2 64t.
1
42. s 2at 2 v0 t s0
43. y ax2 bx c passing through 0, 0, 2, 2, 4, 0
1, 132, 2, 100, 3, 36
0, 0: 0 c
2, 2: 2 14a 2b c ⇒ 1 2a b
132 1
2a
v0 s0
100 2a 2v0 s0
36 a
2a 9a a
a
a
9
2a
4, 0: 0 16a 4b c ⇒ 0 4a b
3v0 s0
2v0 2s0 2v0 s0 6v0 2s0 Solution: a 12, b 2, c 0
264
100
72
2Eq. 1
5
2Eq. 3
2v0 2s0 264
2v0 3s0 428
12v0 16s0 2304
2Eq.1 Eq.2
9Eq.1 Eq.3
2v0 2s0 264
2v0 3s0 428
2s0 264
6Eq.2 Eq.3
2v0 2s0 v0 1.5s0 s0 264
214
132
0.5Eq.2
0.5Eq.3
v0 1.5132 214 ⇒ v0 16
a 216 2132 264 ⇒ a 32
Thus, s 1232t 2 16t 132
16t 2 16t 132.
1
The equation of the parabola is y 2x2 2x.
−4
8
−3
Section 7.3
44. y ax2 bx c passing through 0, 3, 1, 4, 2, 3
Multivariable Linear Systems
45. y ax2 bx c passing through 2, 0, 3, 1, 4, 0
0, 3: 3 c
2, 0: 0 4a 2b c
1, 4: 4 a b c ⇒ 1 a b
3, 1: 1 9a 3b c
2, 3: 3 4a 2b c ⇒ 0 2a b
4, 0: 0 16a 4b c
Solution: a 1, b 2, c 3
The equation of the parabola is y x2 2x 3.
5
−5
7
0 4a 2b c
1 5a b
0 12a 2b
Eq.1 Eq.2
Eq.1 Eq.3
0 4a 2b c
1 5a b
2 2a
2Eq.2 Eq.3
Solution: a 1, b 6, c 8
−3
The equation of the parabola is y x2 6x 8.
10
−6
12
−2
46. y ax2 bx c passing through 1, 3, 2, 2, 3, 3
1, 3: 3 a b c
3, 3: 3 9a 3b c
a bc 3
3a b
1
8a 2b
6
1Eq.1 Eq.2
1Eq.1 Eq.3
a bc 3
3a b
1
2a
4
47. x2 y2 Dx Ey F 0 passing through
0, 0, 2, 2, 4, 0
0, 0: F 0
2, 2: 2 4a 2b c
2, 2: 8 2D 2E F 0 ⇒ D E 4
4, 0: 16 4D F 0 ⇒ D 4 and E 0
The equation of the circle is x2 y2 4x 0.
To graph, let y1 4x x2 and y2 4x x2.
3
2Eq.2 Eq.3
−3
6
Solution: a 2, b 5, c 0
The equation of the parabola is y 2x2 5x.
4
−4
8
−4
647
−3
648
Chapter 7
Systems of Equations and Inequalities
48. x2 y2 Dx Ey F 0 passing through 0, 0, 0, 6, 3, 3
7
0, 0: F 0
0, 6: 36 6E F 0 ⇒ E 6
3, 3: 18 3D 3E F 0 ⇒ D 0
−6
6
−1
The equation of the circle is x2 y2 6y 0. To graph, complete the square first, then solve for y.
x2 y2 6y 9 9
x2 y 32 9
y 32 9 x2
y 3 ± 9 x2
y 3 ± 9 x2
Let y1 3 9 x2 and y2 3 9 x2.
49. x2 y2 Dx Ey F 0 passing through 3, 1, 2, 4, 6, 8
10
3, 1: 10 3D E F 0 ⇒ 10 3D E F
2, 4: 20 2D 4E F 0 ⇒ 20 2D 4E F
6, 8: 100 6D 8E F 0 ⇒ 100 6D 8E F
−12
6
−2
Solution: D 6, E 8, F 0
The equation of the circle is x2 y2 6x 8y 0. To graph, complete the squares first, then solve for y.
x2 6x 9 y2 8y 16 0 9 16
x 32 y 42 25
y 42 25 x 32
y 4 ± 25 x 32
y 4 ± 25 x 32
Let y1 4 25 x 32 and y2 4 25 x 32 .
50. x2 y2 Dx Ey F 0 passing through 0, 0, 0, 2, 3, 0
0, 0: F 0
1
−2
4
0, 2: 4 2E F 0 ⇒ E 2
3, 0: 9 3D F 0 ⇒ D 3
The equation of the circle is
x
x2
y2
3x 2y 0. To graph, complete the squares first, then solve for y.
y 2y 1 1
x 32 2 y 12 134
3 2
y 12 13
4 x 2 3 2
y 1 ± 13
4 x 2 3 2
y 1 ± 13
4 x 2 3 2
13
3 2
Let y1 1 13
4 x 2 and y2 1 4 x 2 .
2
3x 9
4
2
9
4
−3
Section 7.3
51. Let x number of touchdowns.
Let y number of extra-point kicks.
Let z number of field goals.
x y z 13
6x y 3z 45
xy
0
x
6z 0
x y z 13
5y 3z 33
2y z 13
y 7z 13
x y z 13
y 7z 13
2y z 13
5y 3z 33
x y z 13
y 7z 13
2y z 13
5y 3z 33
x y z 13
y 7z 13
13z 13
32z 32
6Eq.1 Eq.2
Eq.1 Eq.3
Eq.1 Eq.4
Interchange Eq.2 and Eq.4.
Eq.2
2Eq.2 Eq.3
5Eq.2 Eq.4
z1
y 71 13 ⇒ y 6
x 6 1 13 ⇒ x 6
Thus, 6 touchdowns, 6 extra-point kicks, and 1 field goal were scored.
52. Let x number of 2-point baskets.
Let y number of 3-point baskets.
Let z number of free throws.
2x 3y z 70
x
z 2
2y z 1
Add Equation 2 to Equation 3, and then add Equation 1 to Equation 2:
2x 3y z 70
3x 3y
72
x 2y
3
Divide Equation 2 by 3:
2x 3y z 70
x y
24
x 2y
3
Subtract Equation 3 from Equation 2: 3y 21 ⇒ y 7
Back-substitute into Equation 2: x 24 7 17
Back-substitute into Equation 1: z 70 217 37 15
There were 17 two-point baskets, 7 three-pointers, and 15 free-throws.
Multivariable Linear Systems
649
650
Chapter 7
Systems of Equations and Inequalities
53. Let x amount at 8%.
54. Let x amount at 8%.
Let y amount at 9%.
Let y amount at 9%.
Let z amount at 10%.
Let z amount at 10%.
x y z 775,000
0.08x 0.09y 0.10z 67,500
x
4z
y 5z 775,000
x
y
z 800,000
0.08x 0.09y 0.10z 67,000
x 5z
y
6z 800,000
67,000
z 125,000
y 800,000 6125,000 50,000
x 5125,000 625,000
0.09y 0.5z 0.09y 0.42z 67,500
z 75,000
y 775,000 5z 400,000
Solution: x $625,000 at 8%
x 4z 300,000
$300,000 was borrowed at 8%.
y $50,000 at 9%
$400,000 was borrowed at 9%.
z $125,000 at 10%
$75,000 was borrowed at 10%.
55. Let C amount in certificates of deposit.
56. Let C amount in certificates of deposit.
Let M amount in municipal bonds.
Let M amount in municipal bonds.
Let B amount in blue-chip stocks.
Let B amount in blue-chip stocks.
Let G amount in growth or speculative stocks.
Let G amount in growth or speculative stocks.
This system has infinitely many solutions.
This system has infinitely many solutions.
Let G s, then B 125,000 s
Let G s, then B 125,000 s
1
2s
M 12s 31,250
1
C 406,250 12s.
C M B G 500,000
0.10C 0.08M 0.12B 0.13G 0.10500,000
B G 14500,000
M 125,000 C M B G 500,000
0.09C 0.05M 0.12B 0.14G 0.10500,000
B G 14 500,000
C 250,000 2s
One possible solution is to let s 50,000.
Solution:
Certificates of deposit: $225,000
406,250 12 s in certificates of deposit,
Municipal bonds: $150,000
31,250 2 s in municipal bonds,
Blue-chip stocks: $75,000
125,000 s in blue-chip stocks,
Growth or speculative stocks: $50,000
s in growth stocks
1
One possible solution is to let s $100,000.
Certificates of deposit: $356,250
Municipal bonds: $18,750
Blue-chip stocks: $25,000
Growth or speculative stocks: $100,000
Section 7.3
Multivariable Linear Systems
57. Let x pounds of brand X.
58. Let x liters of spray X.
Let y pounds of brand Y.
Let y liters of spray Y.
Let z pounds of brand Z.
Let z liters of spray Z.
Fertilizer A:
Fertilizer B:
Fertilizer C:
1
2x
1
2x
2
9z
5
9z
2
9z
5
13
4
Interchange Eq.1 and Eq.2.
29 z 5
Chemical A: 15x 12z 12
Chemical B:
Chemical C:
2
5x
2
5x
1
2z
16
y 26
⇒ x 20, z 16
⇒ y 18
20 liters of spray X, 18 liters of spray Y, and 16 liters of
spray Z are needed to get the desired mixture.
29 z 4
23 y 59 z 13
1
2x
2
5
3 y 9 z 13
1
3y
1
2x
1
2x
1
2x
1
3y
2
3y
651
1
3y
2
3y
29 z 1
3z
5
9
Eq.1 Eq.3
23 y 59 z 13
1
3y
29 z 5
1
9z
1
2Eq.2 Eq.3
z9
1
3y
299 5 ⇒ y 9
1
2x
239 599 13 ⇒ x 4
4 pounds of brand X, 9 pounds of brand Y, and 9 pounds
of brand Z are needed to obtain the desired mixture.
59. Let x pounds of Vanilla coffee.
60. Each centerpiece costs $30.
Let y pounds of Hazelnut coffee.
Let x number of roses in a centerpiece.
Let z pounds of French Roast coffee.
Let y number of lilies.
Let z number of irises.
x
y z 10
2x 2.50 y 3z 26
y z 0
x
x
y z 10
0.5 y z 6
yz 0
y z 10
0.5 y z 6
3z 12
x y z 12
2.5x 4 y 2z 30
x 2y 2z 0
2Eq.1 Eq.2
2Eq.2 Eq.3
z4
0.5y 4 6 ⇒ y 4
x 4 4 10 ⇒ x 2
2 pounds of Vanilla coffee, 4 pounds of Hazelnut coffee,
and 4 pounds of French Roast coffee are needed to obtain
the desired mixture.
x y z 12
3.5x 2 y
30
3x
24
Eq.3 Eq.2
2Eq.1 Eq.3
3x 24 ⇒ x 8
3.5x 2y 30 ⇒ y 1230 3.58
1230 28 122 1
x y z 12 ⇒ z 12 8 1 3
The point (8, 1, 3) is the solution of the system of equations.
Each centerpiece should contain 8 roses, 1 lily, and 3 irises.
652
Chapter 7
Systems of Equations and Inequalities
61. Let x number of television ads.
62. Let x number of rock songs.
Let y number of radio ads.
Let y number of dance songs.
Let z number of local newspaper ads.
Let z number of pop songs.
x
y
z
60
1000x 200 y 500z 42,000
x
y
z
0
x y z 32
x
2z 0
y z 4
x
y
z
60
800 y 500z 18,000
2y 2z 60
x
y
z
60
2 y 2z 60
800y 500z 18,000
1000Eq.1 Eq.2
Eq.1 Eq.3
x
y
2 y z 60
2z 60
300z 6000
Interchange
Eq.2 and Eq.3.
x
y z 32
y 3z 32
y z 4
x
y
y 1Eq.1 Eq.2
z 32
3z 32
4z 36
Eq.2 Eq.3
4z 36 ⇒ z 9
y 39 32 ⇒ y 5
400Eq.2 Eq.3
x 5 9 32 ⇒ x 18
Play 18 rock songs, 5 dance songs, and 9 pop songs.
z 20
2y 220 60 ⇒ y 10
x 10 20 60 ⇒ x 30
30 television ads, 10 radio ads, and 20 newspaper ads can
be run each month.
63. (a) To use 2 liters of the 50% solution:
Let x amount of 10% solution.
Let y amount of 20% solution.
xy8 ⇒ y8x
x0.10 y0.20 20.50 100.25
0.10x 0.208 x 1 2.5
0.10x 1.6 0.20x 1 2.5
0.10x 0.1
x 1 liter of 10% solution
y 7 liters of 20% solution
Given: 2 liters of 50% solution
(b) To use as little of the 50% solution as possible, the
chemist should use no 10% solution.
Let x amount of 20% solution.
Let y amount of 50% solution.
x y 10 ⇒ y 10 x
x0.20 y0.50 100.25
x0.20 10 x0.50 100.25
x0.20 5 0.50x 2.5
0.30x 2.5
1
x 83 liters of 20% solution
y 123 liters of 50% solution
(c) To use as much of the 50% solution as possible, the
chemist should use no 20% solution.
Let x amount of 10% solution.
Let y amount of 50% solution.
x y 10 ⇒ y 10 x
x0.10 y0.50 100.25
0.10x 0.5010 x 2.5
0.10x 5 0.50x 2.5
0.40x 2.5
x 614 liters of 10% solution
y 334 liters of 50% solution
Section 7.3
Multivariable Linear Systems
653
64. Let x amount of 10% solution.
Let y amount of 15% solution.
Let z amount of 25% solution.
0.10xx 0.15yy 0.25zz 0.20 1212
2xx 3yy 5zz 1248
20Eq.2
(a) If z 4,
2xx 3yy 204 1248
2xx 3yy 288
x yy 128
(b)
Minimize z while x ≥ 0, y ≥ 0, and z ≥ 0.
xx y 2zz 1212
Eq.2 2Eq.1
y 12 ⇒ x 8 12 4, but x ≥ 0.
There is no solution; 4 gallons of the 25% solution is
not enough.
(c)
2xx 3yy 5zz 1248
x yy 3zz 1224
2Eq.1 Eq.2
1
y 3z 24 ⇒ z 8 3 y ⇒ z is largest when
y 0.
y 0 and z 8 ⇒ x 12 0 8 4.
The 12-gallon mixture made with the largest portion
of the 25% solution contains 4 gallons of the 10%
solution, none of the 15% solution, and 8 gallons of
the 25% solution.
65.
I1 I2 I3 0
3I1 2I2
7
2I2 4I3 8
I1 I1 I1 I2 I3 0
5I2 3I3 7
2I2 4I3 8
2xx 3yy 5zz 1248
Equation 1
Equation 2
Equation 3
3Eq.1 Eq.2
I2 I3 0
10I2 6I3 14
10I2 20I3 40
2Eq.2
5Eq.3
I2 I3 0
10I2 6I3 14
26I3 26
1Eq.2 Eq.3
26I3 26 ⇒ I3 1
10I2 61 14 ⇒ I2 2
I1 2 1 0 ⇒ I1 1
Solution: I1 1, I2 2, I3 1
x 2z 12 ⇒ z 6 x 0.
3Eq.1 Eq.2
1
2x
⇒ z is smallest when
x 0 and z 6 ⇒ y 6
The 12-gallon mixture using the least amount of the
25% solution is made using none of the 10% solution
and 6 gallons each of the 15% and 25% solution.
654
Chapter 7
66. (a)
Systems of Equations and Inequalities
t1 2t2
0
t1
2a 128 ⇒ 2t2 2a 128
t2 a 32 ⇒ 2t2 2a 64
(b)
4a 64
a 16
t2 48
t1 96
t1 2t2
0
t1
2a 128
t2 2a 64
Equation 1
Equation 2
Equation 3
0
t1 2t2
2t2 2a 128
t2 2a 64
1Eq.1 Eq.2
t1 2t2
0
2t2 2a 128
3a 0
So, t1 96 pounds
t2 48 pounds
3a 0 ⇒ a 0
2t2 20 128 ⇒ t2 64
t1 264 0 ⇒ t1 128
a 16 feet per second squared.
Solution: a 0 ftsec2
t1 128 lb
t2 64 lb
The system is stable.
67. 4, 5, 2, 6, 2, 6, 4, 2
n 4,
4
xi 0,
i1
4
xi2 40,
i1
4
xi3 0,
i1
4
xi4 544,
i1
4
yi 19,
i1
4c
40a 19
40b
12
40c
544a 160
4c
40a 19
40b
12
144a 30
10Eq.1 Eq.3
5
144a 30 ⇒ a 24
3
40b 12 ⇒ b 10
4c 40 24 5
19 ⇒ c 41
6
5 2
3
Least squares regression parabola: y 24
x 10
x 41
6
68.
5c
10a 8
10b
12
10c
34a 22
5c
10a 8
10b
12
14a 6
2Eq.1 Eq.3
14a 6 ⇒ a 3
7
10b 12 ⇒ b 6
5
5c 1037 8 ⇒ c 26
35
12 Eq.2 Eq.3
Least squares regression parabola: y 37 x2 65 x 26
35
4
i1
xi yi 12,
4
x
2
i yi
i1
160
Section 7.3
Multivariable Linear Systems
69. 0, 0, 2, 2, 3, 6, 4, 12
n 4,
4
4
x 9, x
i1
2
i i
4
29,
i1
x
3
i i1
9b 29a 29b 99a 99b 353a 20
70
254
9c 4c 29c 29b 99a 9b 29a 99b 353a 70
20
254
c
11b 41a 30
35b 135a 100
220b 836a 616
c
11b 41a 30
1540b 5940a 4400
1540b 5852a 4312
11b 41a 1540b 5940a 88a 88a x
4
i 4
353,
i1
4c 9c 29c c
4
99,
i
i1
Interchange equations.
2Eq.2 Eq.1
4Eq.1 Eq.2
29Eq.1 Eq.3
44Eq.2
7Eq.3
30
4400
88
Eq.2 Eq.3
88 ⇒ a 1
1540b 59401 4400 ⇒ b 1
c 111 411 30 ⇒ c 0
Least squares regression parabola: y x2 x
70.
4c 6c 14c 4c 4c 6b 14b 36b 14a 25
36a 21
98a 33
6b 14a 25
10b 30a 33
60b 196a 218
6b 10b 14a 25
30a 33
16a 20
4
3Eq.1 2Eq.2
14Eq.1 4Eq.3
6Eq.2 Eq.3
16a 20 ⇒ a 54
10b 30 54 33 ⇒ b 9
20
9
4c 620
14 54 25 ⇒ c 199
20
4
y 20, x y 70, x
9
Least squares regression parabola: y 54 x2 20
x 199
20
2
i yi
i i
i1
i1
254
655
656
Chapter 7
Systems of Equations and Inequalities
71. (a) 100, 75, 120, 68, 140, 55
3
3
x 360, n 3,
i
i1
3
xi2
i1
x
2
i yi
xi3
5,472,000
3
yi 198,
3
xi yi 23,360,
i1
Actual Percent y
Model Approximation y
100
75
75
120
68
68
140
55
55
The model is a good fit to the actual data.
The values are the same.
2,807,200
i1
3c 360b 44,000a 198
360c 44,000b 5,472,000a 23,360
44,000c 5,472,000b 691,520,000a 2,807,200
Solving this system yields a 0.0075, b 1.3
and c 20.
(d) For x 170:
y 0.00751702 1.3170 20
24.25%
(e) For y 40:
40 0.0075x2 1.3x 20
Least squares regression parabola:
y 0.0075x2 1.3x 20
(b)
x
i1
i1
3
3
44,000,
i1
xi4 691,520,000,
(c)
0.0075x2 1.3x 20 0
By the Quadratic Formula we have x 17 or x 156.
100
Choosing the value that fits with our data, we have
156 females.
75
175
0
72. 30, 55, 40, 105, 50, 188
(a)
3c 120b 5000a 348
120c 5000b 216,000a 15,250
5000c 216,000b 9,620,000a 687,500
3c 120b 5000a 348
200b 16,000a 1330
48,000b 3,860,000a 322,500
3c 120b 5000a 348
200b 16,000a 1330
20,000a 3300
40Eq.1 Eq.2
5000Eq.1 3Eq.3
240Eq.2 Eq.3
20,000a 3300 ⇒ a 0.165
200b 16,0000.165 1330 ⇒ b 6.55
3c 1206.55 50000.165 348 ⇒ c 103
Least-squares regression parabola: y 0.165x2 6.55x 103
y
Stopping distance
(in feet)
(b)
450
400
350
300
250
200
150
100
50
x
10 20 30 40 50 60 70
Speed
(in miles per hour)
(c) When x 70, y 453 feet.
Section 7.3
73. Let x number of touchdowns.
Multivariable Linear Systems
74. Let t number of touchdowns.
Let y number of extra-point kicks.
Let x number of extra-points.
Let z number of two-point conversions.
Let f number of field goals.
Let w number of field goals.
Let s number of safeties.
xy z w
6x y 2z 3w
x
4w
2z w
1
2w
4w y 16
32 29
0
⇒ x 4w
0
⇒ z 12w
tx f s
6t x 3f 2s
tx
f 3s
w 16 ⇒ 5.5w y 16
28w y 61
64w y 2 w 3w 61 ⇒
1
2
28w y 61
5.5w y 16
22.5w
657
45
w2
y5
x 4w 8
z 12w 1
Thus, 8 touchdowns, 5 extra-point kicks, 1 two-point
conversion, and 2 field goals were scored.
22
74
0
0
2t
f s 22
7t
3f 2s 74
tx
0
f 3s 0
Eq.1 Eq.3
Eq.2 Eq.3
2t
4s 22
7t
3f 2s 74
tx
0
f 3s 0
Eq.1 Eq.4
2t
7t
tx
4s 22
11s 74
0
f 3s 0
Eq.2 3Eq.4
t
7t
tx
2s 11
11s 74
0
f 3s 0
12 Eq.1
t
tx
2s 11
3s 3
0
f 3s 0
7Eq.1 Eq.2
3s 3 ⇒ s 1
t 21 11 ⇒ t 9
9x0 ⇒ x9
f 31 0 ⇒ f 3
There were 9 touchdowns, each with an extra point; and
there were 3 field goals and 1 safety.
75.
y0
x
0 ⇒ x y x y 10 0 ⇒ 2x 10 0
x 5
y 5
5
76.
2x 0
xy
2
2y 0
x y 4 0 ⇒ 2x 4 0
2x 4
x2
y2
4
658
77.
Chapter 7
Systems of Equations and Inequalities
2x 2x 0 ⇒ 2x1 0 ⇒ 1 or x 0
2y 0
y x2 0
If 1:
2y ⇒ y 1
2
x2 y ⇒ x ±
12 ± 22
If x 0:
x2 y ⇒ y 0
2y ⇒ 0
Solution: x ±
2
or x 0
1
2
y0
1
0
y
78.
2
2 2y 2 0
2x 1 0 ⇒ 2x 1
2x y 100 0
2 2y 22x 1 0 ⇒ 4x 2y 0 ⇒ 4x 2y 2x y 100 0 ⇒ 2x y 100 ⇒
0
4x 2y 200
4y 200
y 50
x 25
225 1 51
79. False. Equation 2 does not have a leading coefficient of 1.
80. True. If a system of three linear equations is inconsistent,
then it has no points in common to all three equations.
81. No, they are not equivalent. There are two arithmetic
errors. The constant in the second equation should be
11 and the coefficient of z in the third equation
should be 2.
82. When using Gaussian elimination to solve a system of
linear equations, a system has no solution when there
is a row representing a contradictory equation such as
0 N, where N is a nonzero real number.
For instance:
xy3
x y 3
Equation 1
Equation 2
xy0
06
Eq.1 Eq.2
No solution
83. There are an infinite number of linear systems that
have 4, 1, 2 as their solution. Two such systems
are as follows:
3x y z 9
x 2y z 0
x y 3z 1
x y z5
x
2z 0
2y z 0
84. There are an infinite number of linear systems that have
5, 2, 1 as their solution. Two systems are:
x y z 6
2x y 3z 15
x 4y z 14
2x y z 9
x 2y 2z 3
3x y 2z 11
Section 7.3
87. 0.07585 6.375
225 2x y 3z 28
6x 4y z 18
4x 2y 3z 19
x 2y 4z 9
y 2z 3
x
4z 4
88.
659
86. There are an infinite number of linear systems that have
32, 4, 7 as their solution. Two systems are:
85. There are an infinite number of linear systems that have
3, 12, 74 as their solution. Two such systems are
as follows:
x 2y 4z 5
x 4y 8z 13
x 6y 4z 7
Multivariable Linear Systems
x
150
100
4x y 2z 12
4y 2z 2
2x y z 0
89. 0.005n 400
90. 0.48n 132
n 80,000
225 1.5x
n 275
150% x
92. 6 3i 1 6i 6 1 3 6i
91. 7 i 4 2i 7 4 i 2i 11 i
7 3i
93. 4 i5 2i 20 8i 5i 2i2 20 3i 2
94. 1 2i3 4i 3 4i 6i 8i2
3 2i 81 11 2i
22 3i
95.
i
6
i1 i 61 i
1i 1i
1 i1 i
96.
i i2 6 6i
1 i2
1 4i 6 16i
17
73
7 7i
2
731 4i 176 16i
1773
7 7
i
2 2
73 292i 102 272i
1241
20
175
i
1241 1241
97. f x x 3 x2 12x
(a)
i
4i
8 3i
2i
i
2i
4 i 8 3i 4 i 4 i
8 3i 8 3i
98. f x 8x 4 32x2
x 3 x2 12x 0 (b)
(a) 8x 4 32x2 0
y
xx2 x 12 0
25
xx 4x 3 0
15
y
8x2x2 4 0
20
Zeros: x 4, 0, 3
(b)
36
30
Zeros: x 0, ± 2
24
18
−5
−3 −2 −1
x
1
2
4
−10
−15
−3
−20
99. f x 2x 3 5x2 21x 36
y
(b)
30
(a) 2x 3 5x2 21x 36 0
3
2
2
5
6
21
33
36
36
11
12
0
f x x 32x2 11x 12
x 3x 42x 3
Zeros: x 4, 32, 3
20
10
−5
x
−3 −2
1
− 30
− 40
− 50
− 60
2
4
−1
x
−6
1
3
660
Chapter 7
Systems of Equations and Inequalities
100. f x 6x3 29x2 6x 5
101. y 4x4 5
(a) 6x3 29x2 6x 5 0
5
29
30
6
6
5
5
5
6
1 1
0
f x x 56x2 x 1
x
2
0
2
4
5
y
4.9998
4.996
4.938
4
1
Horizontal asymptote: y 5
y
x 53x 12x 1
12
1
1
Zeros: x 5, 3, 2
(b)
10
8
6
y
4
2
20
x
−4 −2
2
4
6
x
−3 −2 −1
8
1
2
3
4
6
−4
−6
5
102. y 2 x1
103. y 1.90.8x 3
4
Horizontal asymptote: y 4
x
y
y
x
2
1
0
1
2
y
5.793
4.671
4
3.598
3.358
12
2
11.625
1
2.25
0
1.5
1
3
10
Horizontal asymptote: y 3
8
6
y
4
2
2
7
x
−4 −3 −2 −1
1
2
3
6
4
5
4
−6
3.6
2
2
1
−3 −2 −1
x
1
2
3
4
5
6
−2
104. y 3.5x2 6
105.
Horizontal asymptote: y 6
x
y
12
28.918
0
18.25
1
2
12.548
1
9.5
2
7
2xx 2yy 120
120
2x y 120
2x 4y 240
3y 120
y
40
y
18
Solution: 40, 40
x
−6
2Eq.2
x 240 120 ⇒ x 40
12
−1
Equation 1
Equation 2
1
2
3
4
Section 7.4
106.
10x6x 12y5y 35
72x 60y 36
50x
60y 25
x
22x
Partial Fractions
107. Answers will vary.
Equation 1
Equation 2
12Eq.1
5Eq.2
11
1
2
612 5y 3 ⇒ y 0
Solution:
12, 0
Section 7.4
■
Partial Fractions
You should know how to decompose a rational function
Nx
into partial fractions.
Dx
(a) If the fraction is improper, divide to obtain
Nx
N x
px 1
Dx
Dx
(a) where px is a polynomial.
(b) Factor the denominator completely into linear and irreducible quadratic factors.
(c) For each factor of the form px qm, the partial fraction decomposition includes the terms
A1
A2
Am
. . .
.
px q px q2
px qm
(d) For each factor of the form ax2 bx cn, the partial fraction decomposition includes the terms
B1x C1
B2x C2
Bnx Cn
. . .
.
ax2 bx c ax2 bx c2
ax2 bx cn
■
You should know how to determine the values of the constants in the numerators.
(a) Set
N1x
partial fraction decomposition.
Dx
(b) Multiply both sides by Dx to obtain the basic equation.
(c) For distinct linear factors, substitute the zeros of the distinct linear factors into the basic equation.
(d) For repeated linear factors, use the coefficients found in part (c) to rewrite the basic equation. Then use
other values of x to solve for the remaining coefficients.
(e) For quadratic factors, expand the basic equation, collect like terms, and then equate the coefficients of like terms.
Vocabulary Check
1. partial fraction decomposition
2. improper
3. m; n; irreducible
4. basic equation
1.
A
B
3x 1
xx 4
x
x4
Matches (b).
2.
3x 1
A
B
C
2
x2x 4
x
x
x4
Matches (c).
3.
3x 1
A Bx C
2
xx2 4
x
x 4
Matches (d).
661
662
4.
Chapter 7
Systems of Equations and Inequalities
B
C
3x 1
A
3x 1
xx2 4 xx 2x 2
x
x2 x2
5.
7
7
B
A
x2 14x xx 14
x
x 14
Matches (a).
6.
x2
B
A
x2
x2 4x 3 x 3x 1 x 3 x 1
7.
B
12
A
C
12
2
2
2
x 10x
x x 10
x
x
x 10
8.
B
x2 3x 2 A
C
x2 3x 2
2
2
3
2
4x 11x
x 4x 11
x
x
4x 11
9.
A
B
C
4x2 3
x 53 x 5 x 52 x 53
10.
6x 5
6x 5
x 24 x 2x 2x 2x 2
3
11.
2x 3
Bx C
2x 3
A
2
x3 10x xx2 10
x
x 10
A
C
D
B
x 2 x 22 x 23 x 24
12.
x6
Bx C
x6
A
2
2x3 8x
2xx2 4
2x
x 4
13.
x1
A Bx C
Dx E
2
2
xx2 12
x
x 1
x 12
14.
A
C
B
D
x4
2
x23x 12
x
x
3x 1 3x 12
15.
1
A
B
x2 1 x 1 x 1
1 Ax 1 Bx 1
Let x 1: 1 2A ⇒ A Let x 1: 1 2B ⇒ B 1
2
1
2
12
12
1 1
1
1
x2 1 x 1 x 1 2 x 1 x 1
16.
1
A
B
4x2 9 2x 3 2x 3
17.
1
A
B
x2 x
x
x1
1 Ax 1 Bx
1 A2x 3 B2x 3
3
1
Let x : 1 6A ⇒ A 2
6
Let x 0: 1 A
3
1
Let x : 1 6B ⇒ B 2
6
1
1
1
x x
x
x1
1
1
1
1
4x2 9 6 2x 3 2x 3
18.
2
3
A
B
x2 3x x 3
x
3 Ax Bx 3
Let x 3: 3 3A ⇒ A 1
Let x 0: 3 3B ⇒ B 1
1
1
3
x2 3x x 3
x
Let x 1: 1 B ⇒ B 1
19.
1
A
B
2x2 x 2x 1
x
1 Ax B2x 1
1
1
Let x : 1 A ⇒ A 2
2
2
Let x 0: 1 B
1
2
1
2x2 x
x
2x 1
Section 7.4
20.
5
A
B
x2 x 6 x 3 x 2
21.
Partial Fractions
3
A
B
x2 x 2 x 1 x 2
3 Ax 2 Bx 1
5 Ax 2 Bx 3
Let x 3: 5 5A ⇒ A 1
Let x 1: 3 3A ⇒ A 1
Let x 2: 5 5B ⇒ B 1
Let x 2: 3 3B ⇒ B 1
1
1
3
x2 x 2 x 1 x 2
x2
5
1
1
x6 x2 x3
22.
x1
x1
1
, x 1
x2 4x 3 x 3x 1 x 3
23.
x2 12x 12 A
B
C
x3 4x
x
x2 x2
x2 12x 12 Ax 2x 2 Bxx 2 Cxx 2
Let x 0: 12 4A ⇒ A 3
Let x 2: 8 8B ⇒ B 1
Let x 2: 40 8C ⇒ C 5
3
1
5
x2 12x 12
x3 4x
x
x2 x2
24.
x2
A
B
xx 4
x
x4
25.
x 2 Ax 4 Bx
Let x 0: 2 4A ⇒ A 4x2 2x 1 Axx 1 Bx 1 Cx2
Let x 0: 1 B
1
2
Let x 1: 1 C
3
Let x 4: 6 4B ⇒ B 2
1
1
3
x2
xx 4 2 x 4
x
4x2 2x 1 A
B
C
2
x2x 1
x
x
x1
Let x 1: 5 2A 2B C
5 2A 2 1
6 2A
3A
2x 1 3
1
1
2
x2x 1
x
x
x1
4x2
26.
2x 3
A
B
x 12 x 1 x 12
2x 3 Ax 1 B
27.
3x
A
B
x 32 x 3 x 32
3x Ax 3 B
Let x 1: 1 B
Let x 3: 9 B
Let x 0: 3 A B
Let x 0: 0 3A B
3 A 1
0 3A 9
2A
2x 3
2
1
x 12 x 1 x 12
3A
3
9
3x
x 32 x 3 x 32
663
664
28.
Chapter 7
Systems of Equations and Inequalities
6x2 1
A
B
C
D
2
x x 12
x
x
x 1 x 12
29.
2
6x2 1 Axx 12 Bx 12 Cx2x 1 Dx2
x2 1
A Bx C
2
xx2 1
x
x 1
x2 1 Ax2 1 Bx Cx
Let x 0 : 1 B
Ax2 A Bx2 Cx
Let x 1 : 7 D
A Bx2 Cx A
Substitute B and D into the equation, expand the binomials,
collect like terms, and equate the coefficients of like terms.
2x2 2x A Cx 3 2A Cx2 Ax
Equating coefficients of like terms gives
1 A B, 0 C, and 1 A.
Therefore, A 1, B 2, and C 0.
A2
1
2x
x2 1
2
xx2 1
x
x 1
2A C 2 ⇒ C 2 or
A C 0 ⇒ C 2
1
2
1
2
7
2
x2x 12
x
x
x 1 x 12
6x2
30.
x
A
Bx C
x 1x2 x 1 x 1 x2 x 1
x Ax2 x 1 Bx Cx 1
Ax2 Ax A Bx2 Bx Cx C
A Bx2 A B Cx A C
Equating coefficients of like powers gives 0 A B, 1 A B C, and 0 A C. Substituting A for
1
1
1
B and A for C in the second equation gives 1 3A, so A 3, B 3, and C 3.
1
1
x1
x
x 1x2 x 1 3 x 1 x2 x 1
31.
x
x
A
Bx C
2
x3 x2 2x 2 x 1x2 2 x 1
x 2
x Ax2 2 Bx Cx 1
Ax2 2A Bx2 Bx Cx C
A Bx2 C Bx 2A C
Equating coefficients of like terms gives 0 A B, 1 C B, and 0 2A C. Therefore, A 1, B 1, and C 2.
x
1
x2
x3 x2 2x 2 x 1 x2 2
32.
x6
x6
A
B
C
x3 3x2 4x 12
x 2x 2x 3 x 2 x 2 x 3
x 6 Ax 2x 3 Bx 2x 3 Cx 2x 2
⇒
9
5
C
Let x 2 : 4 20A ⇒
1
5
A
Let x 3 :
Let x 2 :
9 5C
8 4B ⇒ 2 B
1
9
2
1
1
10
9
x6
5 5 x3 3x2 4x 12
x2 x2 x3 5 x2 x2 x3
Section 7.4
33.
Partial Fractions
x2
x2
x2
x4 2x2 8 x2 4x2 2 x 2x 2x2 2
A
B
Cx D
2
x2 x2
x 2
x2 Ax 2x2 2 Bx 2x2 2 Cx Dx 2x 2
Ax3 2x2 2x 4 Bx3 2x2 2x 4 Cx Dx2 4
Ax3 2Ax2 2Ax 4A Bx3 2Bx2 2Bx 4B Cx3 Dx2 4Cx 4D
A B Cx3 2A 2B Dx2 2A 2B 4Cx 4A 4B 4D
Equating coefficients of like terms gives
0 A B C, 1 2A 2B D, 0 2A 2B 4C, and 0 4A 4B 4D.
Using the first and third equation, we have A B C 0 and A B 2C 0;
by subtraction, C 0. Using the second and fourth equation, we have 2A 2B D 1
1
1
and 2A 2B 2D 0; by subtraction, 3D 1, so D 3. Substituting 0 for C and 3 for
D in the first and second equations, we have
A B 0 and 2A 2B 23, so A 16 and B 16.
x4
34.
1
1
x2
16
6 2 3
2
2x 8 x 2 x 2 x 2
1
1
1
3x2 2 6x 2 6x 2
1
2
1
1
6 x2 2 x 2 x 2
2x2 x 8 Ax B
Cx D
2
2
x2 42
x 4
x 42
2x2 x 8 Ax Bx2 4 Cx D
2x2 x 8 Ax3 Bx2 4A Cx 4B D
Equating coefficients of like powers:
0A
2B
1 4A C ⇒ C 1
8 4B D ⇒ D 0
2
x
2x2 x 8
2
x2 42
x 4 x2 42
35.
x
x
x
16x4 1 4x2 14x2 1 2x 12x 14x2 1
A
B
Cx D
2
2x 1 2x 1
4x 1
x A2x 14x2 1 B2x 14x2 1 Cx D2x 12x 1
A8x3 4x2 2x 1 B8x3 4x2 2x 1 Cx D4x2 1
8Ax3 4Ax2 2Ax A 8Bx3 4Bx2 2Bx B 4Cx3 4Dx2 Cx D
8A 8B 4Cx3 4A 4B 4Dx2 2A 2B Cx A B D
—CONTINUED—
665
666
Chapter 7
Systems of Equations and Inequalities
35. —CONTINUED—
Equating coefficients of like terms gives 0 8A 8B 4C, 0 4A 4B 4D, 1 2A 2B C,
and 0 A B D.
Using the first and third equations, we have 2A 2B C 0 and 2A 2B C 1;
1
by subtraction, 2C 1, so C 2.
Using the second and fourth equations, we have A B D 0 and A B D 0;
by subtraction 2D 0, so D 0.
1
Substituting 2 for C and 0 for D in the first and second equations, we have 8A 8B 2
and 4A 4B 0, so A 18 and B 18.
x
16x4
36.
1
1
8
2x 1
1
8
2x 1
12 x
4x2 1
1
1
x
82x 1 82x 1 24x2 1
1
1
1
4x
8 2x 1 2x 1 4x2 1
x1
A Bx C
2
x3 x
x
x 1
A Bx2 Cx A
Equating coefficients of like powers gives 0 A B, 1 C, and 1 A.
Therefore, A 1, B 1, and C 1.
1
x1
x1
2
3
x x
x
x 1
37.
x2 5
A
Bx C
x 1x2 2x 3 x 1 x2 2x 3
x2 5 Ax2 2x 3 Bx Cx 1
Ax2 2Ax 3A Bx2 Bx Cx C
A Bx2 2A B Cx 3A C
Equating coefficients of like terms gives 1 A B, 0 2A B C, and 5 3A C.
Subtracting both sides of the second equation from the first gives 1 3A C;
combining this with the third equation gives A 1 and C 2. Since A B 1,
we also have B 0.
1
2
x2 5
x 1x2 2x 3 x 1 x2 2x 3
38.
x2 4x 7
A
Bx C
x 1x2 2x 3 x 1 x2 2x 3
x2 4x 7 Ax2 2x 3 Bx Cx 1
Ax2 2Ax 3A Bx2 Bx Cx C
A Bx2 2A B Cx 3A C
Equating coefficients of like terms gives 1 A B, 4 2A B C, and 7 3A C.
Adding the second and third equations, and subtracting the first, gives 2 2C, so C 1. Therefore,
A 2, B 1, and C 1.
x2 4x 7
x1
2
x 1x2 2x 3 x 1 x2 2x 3
Section 7.4
39.
x2 x
2x 1
2x 1
1 2
1 2
x x1
x x1
x x1
40.
2
x2 4x
x2 x 6
Using long division gives
41.
2x3 x2 x 5
18x 19
2x 7 x2 3x 2
x 1x 2
18x 19
A
B
x 1x 2 x 1 x 2
18x 19 Ax 2 Bx 1
Let x 1: 1 A
Let x 2: 17 B ⇒ B 17
1
17
2x3 x2 x 5
2x 7 x2 3x 2
x1 x2
42.
x3 2x2 x 1
x2 3x 4
Using long division gives:
x3 2x2 x 1
6x 3
x1 2
x2 3x 4
x 3x 4
x3 2x2 x 1
6x 3
6x 3
A
B
x1 2
x2 3x 4
x 3x 4 x 4x 1
x4 x1
6x 3
A
B
x 4x 1
x4 x1
6x 3 Ax 1 Bx 4
6x 3 A Bx 4B A
AB6⇒ A6B
4B A 3 ⇒ 4B 6 B 3
5B 6 3
5B 3
B
3
5
A6
3 30 3 27
5
5
5
27
3
x3 2x2 x 1
5
5
x1
x2 3x 4
x4 x1
x 1 51 x 27 4 x 3 1
Partial Fractions
667
x2 4x
5x 6
1 2
.
x6
x x6
x2
668
Chapter 7
Systems of Equations and Inequalities
6x2 8x 3
x4
x4
3
x3
3
2
x 1
x 3x 3x 1
x 13
43.
6x2 8x 3
B
A
C
3
2
x 1
x 1 x 1
x 13
6x2 8x 3 Ax 12 Bx 1 C
Let x 1: 1 C
Let x 2: 11 A B 1
Let x 0: 3 A B 1
AB2
A B 10
So, A 6 and B 4.
4
6
1
x4
x3
x 13
x 1 x 12 x 13
16x4
16x4
24x2 16x 3
2x 3 3 3
2
2x 1
8x 12x 6x 1
2x 13
44.
24x2 16x 3
A
B
C
2x 12
2x 1
2x 12 2x 13
24x2 16x 3 A2x 12 B2x 1 C
Let x 1
:1C
2
24x2 16x 3 4Ax2 4Ax A 2Bx B 1
24x2 16x 3 4Ax2 4A 2Bx A B 1
Equating coefficients of like powers:
6 A, 3 A B 1
36B1
4B
6
4
1
16x4
2x 3 2x 13
2x 1
2x 12 2x 13
45.
5x
A
B
2x x 1 2x 1 x 1
2
2
x 5 Ax 1 B2x 1
−6
1 9 3
Let x : A ⇒ A 3
2 2 2
6
−6
Let x 1: 6 3B ⇒ B 2
5x
3
2
2x2 x 1 2x 1 x 1
46.
B
C
3x2 7x 2 A
x3 x
x
x1 x1
3x2 7x 2 Ax2 1 Bxx 1 Cxx 1
4
−6
6
Let x 0: 2 A ⇒ A 2
Let x 1: 8 2B ⇒ B 4
Let x 1: 6 2C ⇒ C 3
4
3
3x2 7x 2 2
x3 x
x
x1 x1
−4
Section 7.4
47.
x1
A
B
C
2
x3 x2
x
x
x1
48.
4x2 1
A
B
C
2xx 12 2x x 1 x 12
4x2 1 Ax 12 2Bxx 1 2Cx
x 1 Axx 1 Bx 1 Cx2
Let x 0: 1 A
Let x 1: 2 C
Let x 0: 1 B
Let x 1: 3 2C ⇒ C Let x 1: 0 2A 2B C
3 4 4B 3
2A
5
B
2
x1
2
1
2
2
x3 x2
x
x
x1
1
4x2 1
5
3
1
2xx 12 2
x
x 1 x 12
20
4
2
−6
−20
6
−4
49.
Cx D
x2 x 2 Ax B
2
2
x2 22
x 2
x 22
2
x2 x 2 Ax Bx2 2 Cx D
−3
3
x2 x 2 Ax3 Bx2 2A Cx 2B D
−2
Equating coefficients of like powers:
0A
1B
1 2A C ⇒ C 1
2 2B D ⇒ D 0
1
x
x x2
2
x2 22
x 2 x2 22
2
50.
B
D
x3
A
C
2
2
2
x 2 x 2
x 2 x 2
x 2 x 22
x3 Ax 2x 22 Bx 22 Cx 22x 2 Dx 22
Let x 2: 8 16B ⇒ B Let x 2: 8 16D ⇒ D 1
2
1
2
1
1
x3 Ax 2x 22 x 2 2 Cx 22x 2 x 22
2
2
x3 4x A Cx3 2A 2C x2 4A 4C x 8A 8C
—CONTINUED—
3
2
Let x 1: 3 4A 4B 2C
0 2A 2 2
−4
Partial Fractions
669
670
Chapter 7
Systems of Equations and Inequalities
50. —CONTINUED—
Equating coefficients of like powers:
0 2A 2C ⇒ A C
1AC
4
1 2A ⇒ A 1
1
⇒ C
2
2
−6
x3
1
1
1
1
1
x 22x 22 2 x 2 x 22 x 2 x 22
51.
6
2x3 4x2 15x 5
x5
2x 2
x 2x 8
x 2x 4
−4
x3 x 3
2x 1
x1
x2 x 2
x 2x 1
52.
2x 1
A
B
x 2x 1 x 2 x 1
B
x5
A
x 2x 4 x 2 x 4
2x 1 Ax 1 Bx 2
x 5 Ax 4 Bx 2
Let x 2: 3 6A ⇒ A Let x 2: 3 3A ⇒ A 1
1
2
Let x 1: 3 3B ⇒ B 1
3
Let x 4: 9 6B ⇒ B 2
2x3 4x2 15x 5
1 3
1
2x x2 2x 8
2 x4 x2
1
1
x3 x 3
x1
x2 x 2
x2 x1
5
20
−9
−6
9
6
−7
−20
53. (a)
x 12
A
B
xx 4
x
x4
x 12 A(x 4 Bx
Let x 0: 12 4A ⇒ A 3
Let x 4: 8 4B ⇒ B 2
3
2
x 12
xx 4
x
x4
(b) y x 12
xx 4
y
3
x
y
y
2
x4
y
y
8
8
8
6
6
6
4
4
4
2
2
2
x
−6 −4
2
8 10
x
–6
2
4
6
8 10
x
–6 –4 –2
2
8 10
–4
–6
−8
Vertical asymptotes: x 0
and x 4
–8
Vertical asymptote: x 0
(c) The combination of the vertical asymptotes of the terms of the decomposition are
the same as the vertical asymptotes of the rational function.
–8
Vertical asymptote: x 4
Section 7.4
54. (a) y 2x 12 A Bx C
2
xx2 1
x
x 1
2x 12
xx2 1
(b)
y
2x 12 Ax2 1 Bx2 Cx
Partial Fractions
2
4
and y 2
x
x 1
y
y
2x2 4x 2 A Bx2 Cx A
4
Equating coefficients of like powers gives
2 A B, 4 C, and 2 A.
2
y=
3
4
4
x2 + 1
2
1
1
x
–4 – 3 – 2 – 1
Therefore, A 2, B 0, and C 4.
1
2
3
x
−1
4
1
2
3
4
y= 2
x
2x 12 2
4
2
2
xx 1
x
x 1
–4
Vertical asymptote at x 0
y
2
is the same as the
x
vertical asymptote of the rational function.
2
has vertical asymptote x 0.
x
(c) The vertical asymptote of y 55. (a)
24x 3
A
B
x2 9
x3 x3
24x 3 A(x 3 Bx 3
Let x 3: 18 6A ⇒ A 3
Let x 3: 30 6B ⇒ B 5
24x 3
3
5
x2 9
x3 x3
(b) y 24x 3
x2 9
y
3
x3
y
5
x3
y
y
y
8
8
8
6
6
6
4
4
2
x
x
−4
4
6
–8 –6 –4
8
2
4
6
x
8
–4
2
−4
–4
–4
−6
–6
–6
−8
–8
–8
Vertical asymptotes: x ± 3
Vertical asymptote: x 3
B
24x2 15x 39 A
Cx D
2 2
x2x2 10x 26
x
x
x 10x 26
24x2 15x 39 Axx2 10x 26 Bx2 10x 26 Cx3 Dx2
8x2 30x 78 Ax 3 10Ax2 26Ax Bx2 10Bx 26B Cx 3 Dx2
A Cx 3 10A B Dx2 26A 10Bx 26B
Equating coefficients of like powers gives 0 A C, 8 10A B D, 30 26A 10B,
and 78 26B. Since 78 26B, B 3. Therefore, A 0, B 3, C 0, and D 5.
24x2 15x 39
3
5
2 2
x2x2 10x 26
x
x 10x 26
—CONTINUED—
4
6
8
Vertical asymptote: x 3
(c) The combination of the vertical asymptotes of the terms of the decomposition are
the same as the vertical asymptotes of the rational function.
56. (a) y 671
672
Chapter 7
Systems of Equations and Inequalities
56. —CONTINUED—
(b)
24x2 15x 39
x2x2 10x 26
3
5
and 2
2
x
x 10x 26
y
y
12
12
10
10
y = 32
x
y=
x
−4 − 2
2
4
6
2
4
6
8 10 12
−4
−4
Vertical asymptote is x 0.
(c) The vertical asymptote of y 57. (a)
x
−4 − 2
8 10 12
5
x 2 − 10x + 26
y
3
has vertical asymptote x 0.
x2
3
is the same as the vertical asymptote of the rational function.
x2
20004 3x
A
B
, 0 < x ≤ 1
11 7x7 4x 11 7x 7 4x
20004 3x A7 4x B11 7x
Let x 10,000 5
11
: A ⇒ A 2000
7
7
7
5
7
Let x : 2500 B ⇒ B 2000
4
4
2000
2000
20004 3x
2000
2000
,0 < x ≤ 1
11 7x7 4x 11 7x 7 4x
7 4x 11 7x
(b) Ymax Ymin 2000
7 4x
(c)
2000
11 7x
(d) Ymax0.5 400F
1000
Ymin0.5 266.7F
Ymax
Ymin
0
1
−100
58. One way to find the constants is to choose values of the variable that eliminate one or more of the constants in
the basic equation so that you can solve for another constant. If necessary, you can then use these constants
with other chosen values of the variable to solve for any remaining constants. Another way is to expand the basic
equation and collect like terms. Then you can equate coefficients of the like terms on each side of the equation
to obtain simple equations involving the constants. If necessary, you can solve these equations using substitution.
59. False. The partial fraction decomposition is
A
B
C
.
x 10 x 10 x 102
61.
1
A
B
, a is a constant.
2
a x
ax ax
2
1 Aa x Ba x
1
Let x a: 1 2aA ⇒ A 2a
Let x a: 1 2aB ⇒ B 1
1
1
1
a2 x2 2a a x a x
60. False. The expression is an improper rational expression,
so you must first divide before applying partial fraction
decomposition.
62.
1
A
B
, a is a constant.
xx a
x
xa
1 Ax a Bx
Let x 0: 1 aA ⇒ A 1
a
1
2a
Let x a: 1 aB ⇒ B 1
1 1
1
xx a a x
xa
1
a
Section 7.4
63.
B
A
1
ya y
y
ay
64.
1 Aa x Bx 1
1
Let y 0: 1 aA ⇒ A a
1
1
1 1
ya y a y
ay
Let x 1: 1 Aa 1 ⇒ A 1
a
Let x a: 1 Ba 1 ⇒ B Intercepts:
0, 18, 3, 0, 6, 0
2x 94 121
8
2
Vertex:
6
9
4,
121
8
y
2
4
8
2
8 10 12
−8
− 10
10
− 12
− 14
−4
− 16
67. f x x2x 3
68. f x 12x3 1
y
y
3 2, 0
Intercepts: 0, 1, 5
4
2
1
3
−3
−3 −2 −1
−1
x
1
2
4
x
−1
2
−3
−4
−3
x2 x 6
x5
−2
−2
5
−2
70. f x 3x 1
3x 1
x2 4x 12 x 6x 2
13, 0
x-intercepts: 3, 0, 2, 0
x-intercept:
6
y-intercept: 0, 5 Vertical asymptotes: x 6 and x 2
Vertical asymptote: x 5
Horizontal asymptote: y 0
Slant asymptote: y x 4
y
No horizontal asymptote.
8
y
6
4
2
5
x
10 15 20
6
−6
−2
5
4
−4
x
−2
x
−6 −4 −2
x-intercepts:
12, 0, 5, 0
4
2
− 20 − 15 − 10
66. f x 2x2 9x 5 2x 1x 5
8
Graph rises to the left and
rises to the right.
69. f x 1
a1
y
Graph rises to the left and
falls to the right.
1
a1
1
1
1
1
x 1a x a 1 x 1 a x
65. f x x2 9x 18 x 6x 3
Intercepts: 0, 0, 3, 0
673
1
A
B
, a is a positive integer.
x 1a x x 1 a x
1 Aa y By
Let y a: 1 aB ⇒ B Partial Fractions
− 4 −2
−4
−6
−8
− 10
x
4
6
3
674
Chapter 7
Systems of Equations and Inequalities
Systems of Inequalities
Section 7.5
■
You should be able to sketch the graph of an inequality in two variables.
(a) Replace the inequality with an equal sign and graph the equation. Use a dashed line for < or >,
a solid line for ≤ or ≥.
(b) Test a point in each region formed by the graph. If the point satisfies the inequality, shade the
whole region.
■
You should be able to sketch systems of inequalities.
Vocabulary Check
1. solution
2. graph
3. linear
4. solution
5. consumer surplus
1. y < 2 x 2
3. x ≥ 2
2. y2 x < 0
Using a dashed line, graph
y 2 x 2 and shade inside the
parabola.
y
Using a dashed line, graph the
parabola y2 x 0, and shade
the region inside this parabola.
(Use 1, 0 as a test point.)
Using a solid line, graph the
vertical line x 2 and shade
to the right of this line.
y
y
3
3
3
1
−3
−2
−1
2
2
x
1
2
1
1
3
−1
−1
−2
x
1
2
5
6. y ≤ 3
Using a solid line, graph the
horizontal line y 1 and
shade above this line.
Using a solid line, graph the
horizontal line y 3, and
shade below this line.
y
3
4
2
3
1
y
4
2
x
1
4
−3
y
−3
3
−2
5. y ≥ 1
Using a solid line, graph the
vertical line x 4, and shade
to the left of this line.
−2
5
x
1
−1
−3
4. x ≤ 4
−1
4
−1
−2
−3
−1
3
−1
2
3
2
1
1
5
−3
−2
x
−1
1
−2
2
3
−3
−2
−1
x
1
−1
−2
2
3
Section 7.5
7. y < 2 x
8. y > 2x 4
Using a dashed line, graph
y 2 x, and then shade below
the line. Use 0, 0 as a test point.
Systems of Inequalities
675
9. 2y x ≥ 4
Using a dashed line, graph
y 2x 4, and shade above
the line. (Use 0, 0 as a test point.)
Using a solid line, graph
2y x 4, and then shade above
the line. Use 0, 0 as a test point.
y
y
y
3
1
3
− 3 − 2 −1
2
x
1
3
4
5
1
−2
1
−2
4
2
4
−3
x
−1
1
2
3
−4
−4
4
−3
−2
1
−2
−2
12. x 12 y 42 > 9
11. x 12 y 22 < 9
10. 5x 3y ≥ 15
Using a solid line, graph
5x 3y 15, and shade above
the line. (Use 0, 0 as a test point.)
Using a dashed line, sketch the
circle x 12 y 22 9.
Using a dashed line, graph the
circle x 12 y 42 9
and shade the exterior. The circle
has center 1, 4 and radius 3, so
the origin could serve as a test
point.
Center: 1, 2
Radius: 3
y
Test point: 0, 0
2
−6
x
−1
x
−4
2
Shade the inside of the circle.
y
4
y
8
7
6
5
4
3
2
6
−6
4
−8
3
2
(1, 4)
(0, 0)
1
x
−5 −4
2
−3 − 2 − 1
3
x
1 2 3 4 5 6
−2
−2
1
1 x2
Using a solid line, graph y 13. y ≤
14. y >
1
, and then shade
1 x2
below the curve. Use 0, 0 as a test point.
15
x2 x 4
15
and then
x2 x 4
shade above the curve. (Use 0, 0 as a test point.)
Using a dashed line, graph y y
y
3
3
2
2
1
−3
−2
−3 −2 −1
x
−1
1
2
x
1
2
3
3
−2
−3
−2
−3
−5
16. y ≥ 6 lnx 5
15. y < ln x
2
0
10
4
6
−8
−9
−2
17. y < 3x4
1
9
−2
−2
676
Chapter 7
Systems of Equations and Inequalities
20. y ≤ 6 32x
19. y ≥ 23 x 1
18. y ≤ 22x0.5 7
9
4
4
−8
4
−6
6
−6
−12
−4
21. y < 3.8x 1.1
22. y ≥ 20.74 2.66x
2
23. x2 5y 10 ≤ 0
2
− 18
−3
9
−1
y ≤ 2
18
3
x2
5
3
−9
−2
9
− 22
−9
24. 2x2 y 3 > 0
25.
5
2y
3x2 6 ≥ 0
1
3
1
26. 10 x2 8 y < 4
y < 2x2 3
y ≥ 253x2 6
4
6
y >
4 2
15
x
2
−8
−6
2
3
8
6
−5
−4
4
−10
0
27. The line through 4, 0 and 0, 2 is y 2 x 2. For the
1
shaded region below the line, we have y ≤ 2x 2.
28. The parabola through 2, 0, 0, 4, 2, 0 is
y x2 4. For the shaded region inside the parabola,
we have y ≥ x2 4.
29. The line through 0, 2 and 3, 0 is y 23x 2. For the
shaded region above the line, we have
30. The circle shown is x2 y2 9. For the shaded region
inside the circle, we have x2 y2 ≤ 9.
1
y ≥ 23x 2.
31.
x ≥ 4
y > 3
y ≤ 8x 3
(a) 0 ≤ 80 3, False
0, 0 is not a solution.
(b) 3 > 3, False
1, 3 is not a solution.
(c) 4 ≥ 4, True
(d) 3 ≥ 4, True
0 > 3, True
11 > 3, True
0 ≤ 84 3, True
11 < 83 3, True
4, 0 is a solution.
3, 11 is a solution.
Section 7.5
32.
2x 5y ≥ 3
y < 4
4x 2y < 7
33.
(a) 20 52 ≥ 3, True
Systems of Inequalities
3x y > 1
y 12x2 ≤ 4
15x 4y > 0
30 10 >
(a)
10 2 < 4, True
1
2
2 0
1, True
≤ 4, True
40 22 < 7, True
150 410 >
0, 2 is a solution
0, 10 is a solution.
(b) 26 54 ≥ 3, True
677
0, True
(b) 30 1 > 1, False ⇒ 0, 1 is not a solution.
4 < 4, False
32 9 >
(c)
6, 4 is not a solution.
9 (c) 28 52 ≥ 3, True
1
2
2 2
1, True
≤ 4, True
152 49 >
2 < 4, True
0, True
2, 9 is a solution.
48 22 < 7, False
31 6 >
(d)
8, 2 is not a solution.
6 (d) 23 52 ≥ 3, True
1
2
2 1
1, True
≤ 4, True
151 46 >
0, True
1, 6 is a solution.
2 < 4, True
43 22 < 7, False
3, 2 is not a solution.
34.
x 2 y 2 ≥ 36
3x y ≤ 10
2
3x y ≥ 5
(a)
12 72 ≥ 36, True
(b) 52 12 ≥ 36, False
31 7 ≤ 10, True
5, 1 is not a solution.
7 ≥ 5, False
(d) 42 82 ≥ 36, True
1, 7 is not a solution.
34 8 ≤ 10, True
2
3 1
(c)
2
3 4
62 02 ≥ 36, True
4, 8 is a solution.
36 0 ≤ 0, True
2
3 6
8 ≥ 5, True
0 ≥ 5, False
6, 0 is not a solution.
35.
xy ≤ 1
x y ≤ 1
y ≥ 0
y
36.
3
2
First, find the points of
intersection of each pair
of equations.
(0, 1)
(− 1, 0)
(1, 0)
−2
1
x
3x 2y < 6
x
> 0
y > 0
y
3
2
First, find the points
of intersection of each
pair of equations.
2
1
(0, 0)
Vertex B
xy1
y0
Vertex C
x y 1
y0
0, 1
1, 0
1, 0
(2, 0)
x
−1
Vertex A
xy1
x y 1
(0, 3)
1
Vertex A
3x 2y 6
x0
0, 3
Vertex B
x0
y0
0, 0
3
Vertex C
3x 2y 6
y0
2, 0
678
37.
Chapter 7
Systems of Equations and Inequalities
x2 y ≤ 5
x
≥ 1
y ≥ 0
38.
y
6
(−1, 4) 4
First, find the points
of intersection of each
pair of equations.
2
1
(
5, 0(
x
−4 −3
1
2
3
y
(
(
(
2
− 2 ,1
−4
First, find the points
of intersection of each
pair of equations.
3
(−1, 0)
2x 2 y ≥ 2
x
≤ 2
y ≤ 1
(2, 1)
x
−2
4
−2
−4
4
(2, − 6)
−6
Vertex A
x2 y 5
x 1
Vertex B
x2 y 5
y0
Vertex C
x 1
y 0
1, 4
± 5, 0
1, 0
39. 2x y > 2
6x 3y < 2
40.
4
−1
x
2
3
4
Vertex C
2x2 y 2
y1
y
(6, 6)
6
2
(1, 0)
−1
−2
x
2
Vertex A
x 7y 36
5x 2y 5
1, 5
No solution
41.
x 7y > 36
5x 2y >
5
6x 5y >
6
First, find the points
of intersection of each
pair of equations.
1
−2
Vertex B
x2
y1
2, 1
Vertex A
2x y 2
x2
2, 6
y
The graphs of 2x y 2
and 6x 3y 2 are
parallel lines. The first
inequality has the region
above the line shaded. The
second inequality has the
region below the line shaded.
There are no points that
satisfy both inequalities.
3x 2y < 6
x 4y > 2
2x y < 3
42.
y
5
( 109 , 79 (
6
y
6
4
(0, 3)
1
(− 2, 0)
−3
5xx 2y3y <> 6
9
4
Vertex C
x 7y 36
6x 5y 6
6, 6
Vertex B
5x 2y 5
6x 5y 6
1, 0
Point of intersection:
0, 3
3
First, find the points
of intersection of each
pair of equations.
(
2
,1
2
x
−1
1
3
4
x
−2
2
−3
Vertex A
Point B
4
6
–2
Vertex C
3x 2y 6
x 4y 2
3x 2y 6
2x y 3
2, 0
0, 3
x 4y 2
2x y 3
109, 79 Note that B is not a vertex of the solution region.
x < y 2
43. x > y2
y
3
2
Points of intersection:
y2
y2
44.
y 1 y 2 0
y 1, 2
1, 1, 4, 2
y
Points of intersection:
(4, 2)
(4, 2)
2
1
1
y2
y20
x y2 > 0
xy > 2
x
−1
1
−2
−3
2
(1, − 1)
3
4
5
y2
y2
x
1
y2 y 2 0
−1
y 1 y 2 0
−2
y 1, 2
1, 1, 4, 2
(1, −1)
2
3
4
Section 7.5
45. x2 y2 ≤ 9
x y
2
2
y
≥ 1
4xx 3yy
2
46.
4
There are no points of
intersection. The region
common to both
inequalities is the region
between the circles.
2
Systems of Inequalities
≤ 25
≤ 0
y
6
2
x2 43x 25
2
−4
x
−2
2
4
25 2
9x
−2
−6
25
x
−2
(− 3, −4)
x ±3
−4
(3, 4)
4
Points of intersection:
2
679
2
4
6
−4
−6
3, 4, 3, 4
47. 3x 4 ≥ y2
xy < 0
4
(4, 4)
3
Points of intersection:
2
xy0⇒yx
1
x < 2y y2
0 < xy
48.
y
3y 4 y2
0 y2 3y 4
0 y 4 y 1
3
y 2y y2
1
2
3
4
5
1
y2 3y 0
−3
yy 3 0
−3
−4
−2
x
(0, 0)
−1
y 0, 3
0, 0, 3, 3
y 4 or y 1
x4
(− 3, 3)
Points of intersection:
x
(− 1, −1)
y
x 1
4, 4 and 1, 1
49. y ≤ 3x 1
x2 1
y ≥
50.
yy <> xx 2x4x 33
2
51.
2
5
7
y < x3 2x 1
y > 2x
x ≤ 1
5
−4
−5
8
7
−6
6
−3
−1
−3
52. y ≥ x4 2x2 1
y ≤ 1 x2
53.
x2y ≥ 1 ⇒ y ≥
1
x2
54.
0 < x ≤ 4
2
y ≤ 4
y ≤ ex 2
y ≥ 0
2 ≤ x ≤ 2
2
3
5
−3
3
−2
−3
−2
−1
−1
55.
y ≤ 4x
x ≥ 0
y ≥ 0
3
7
56. 0, 6, 3, 0, 0, 3
y < 6 2x
y ≥ x3
x ≥ 1
1
680
Chapter 7
Systems of Equations and Inequalities
57. Line through points 0, 4 and 4, 0: y 4 x
58. Circle: x2 y2 > 4
Line through points 0, 2 and 8, 0: y 2 14x
59.
y
y
x
y
≥
≥
≥
≥
4 x
2 14x
0
0
60. 0, 0, 0, 4, 8, 8
x2 y2 ≤ 16
x ≥ 0
y ≥ 0
x2 y2 ≤ 16
x ≤ y
x ≥ 0
62. Parallelogram with vertices at 0, 0, 4, 0, 1, 4, 5, 4
61. Rectangular region with vertices at
2, 1, 5, 1, 5, 7, and 2, 7
x
x
y
y
≥
≤
≥
≤
y
2
5
1
7
0, 0, 4, 0: y ≥ 0
8
(2, 7)
4, 0, 5, 4: 4x y ≤ 16
(5, 7)
6
4
2
This system may be written as:
(5, 1)
(2, 1)
2 ≤ x ≤ 5
1 ≤ y ≤ 7
−2
x
2
4
6
63. Triangle with vertices at 0, 0, 5, 0, 2, 3
0, 0, 1, 4: 4x y ≥ 0
2
x
(0, 0)
1, 0, 0, 1: y ≤ x 1
2, 3, 5, 0 Line: y x 5
0, 1, 1, 0:
5
(2, 3)
2
2
(5, 0)
(0, 0)
−1
Demand Supply
50 0.5x 0.125x
80 x
30
10 p
Point of equilibrium:
80, 10
4
5
(−1, 0)
x
−2
6
(1, 0)
x
−1
1
2
(b) The consumer surplus is the area of the triangular region
defined by
Consumer Surplus
Producer Surplus
50
40
(4, 0) 6
−1
p
50 0.625x
3
3
(0, 1)
3
2
y
3
1
65. (a)
1
y ≤ x 1
y ≤ x1
y ≤ x 1
y ≥ 0
y
2
(5, 4)
1
0, 0, 2, 3 Line: y 4
(1, 4)
3
4x y ≥ 0
4x y ≤ 16
0 ≤ y ≤ 4
1, 0, 1, 0: y ≥ 0
5
4
64. Triangle with vertices at 1, 0, 1, 0, 0, 1
3
2x
y ≤
y ≤ x 5
y ≥ 0
6
1, 4, 5, 4: y ≤ 4
0, 0, 5, 0 Line: y 0
3
2x
y
p = 50 − 0.5x
p = 0.125x
20
p ≤ 50 0.5x
p ≥ 10
x ≥ 0.
Consumer surplus 12baseheight 128040 $1600
10
(80, 10)
x
10 20 30 40 50 60 70 80
The producer surplus is the area of the triangular region
defined by
p ≥ 0.125x
p ≤ 10
x ≥ 0.
Producer surplus 12baseheight 128010 $400
Section 7.5
Demand Supply
66. (a)
p ≤ 100 0.05x
p ≥ 75
x ≥ 0.
75 0.15x
500 x
Consumer surplus 12baseheight 1250025 6250
75 p
Point of equilibrium: 500, 75
p
p = 100 − 0.05x
100
The producer surplus is the area of the triangular region
defined by
p ≤ 25 0.1x
p ≤ 75
x ≤ 0.
Consumer Surplus
Producer Surplus
150
681
(b) The consumer surplus is the area of the triangular region
defined by
100 0.05x 25 0.1x
200
Systems of Inequalities
Producer surplus 12baseheight 1250050 12,500
(500, 75)
50
p = 25 + 0.1x
x
200
400
600
Demand Supply
67. (a)
140 0.00002x 80 0.00001x
60 0.00003x
2,000,000 x
100 p
(b) The consumer surplus is the area of the triangular region
defined by
p ≤ 140 0.00002x
p ≥ 100
x ≥ 0.
Consumer surplus 12baseheight
Point of equilibrium: 2,000,000, 100
p
160
$40,000,000 or $40 million
Consumer Surplus
Producer Surplus
The producer surplus is the area of the triangular region
defined by
p = 140 − 0.00002x
140
122,000,00040
(2,000,000, 100)
120
100
80
p = 80 + 0.00001x
x
1,000,000
2,000,000
p ≥ 80 0.00001x
p ≤ 100
x ≥ 0.
Producer surplus 12baseheight
122,000,00020
$20,000,000 or $20 million
Demand Supply
68. (a)
400 0.0002x 225 0.0005x
175 0.0007x
250,000 x
350 p
Point of equilibrium: 250,000, 350
p
600
500
p = 400 − 0.0002x
300
p = 225 + 0.0005x
Producer surplus
1
1
2baseheight 2250,000125 15,625,000
100
x
400,000
Consumer surplus
12baseheight 12250,00050 6,250,000
p ≥ 225 0.0005x
p ≤ 350
x ≥ 0.
(250,000, 350)
200,000
p ≤ 400 0.0002x
p ≥ 350
x ≥ 0.
The producer surplus is the area of the triangular region
defined by
Consumer Surplus
Producer Surplus
400
200
(b) The consumer surplus is the area of the triangular region
defined by
682
Chapter 7
Systems of Equations and Inequalities
69. x number of tables
70. x number of model A
y number of chairs
y number of model B
x
4
3x
3
2y
3
2y
≤
≤
x ≥
y ≥
12
15
0
0
x
8x 12y
x
y
Assembly center
Finishing center
≥
≤
≥
≥
y
2y
200
4
2
24
20
16
12
y
8
12
4
10
x
8
12
16
20
24
6
4
2
x
2
4
6
8
10
72. x number of $30 tickets
71. x amount in smaller account
y number of $20 tickets
y amount in larger account
xy
y
x
y
≤ 20,000
2x
≥
≥ 5,000
≥ 5,000
x y
30x 20y
x
x
y
y
Account constraints:
15,000
10,000
≤ 3000
≥ 75,000
≤ 2000
≥
0
≥
0
3500
2500
1500
500
x
10,000
y
4500
x
15,000
500 1500 2500 3500 4500
74. Let x number of large trucks.
73. x number of packages of gravel
y number of bags of stone
Let y number of medium trucks.
The delivery requirements are:
55x 70y ≤ 7500
x ≥ 50
y ≥ 40
Weight
6x 4y
3x 6y
x
y
y
120
≥
≥
≥
≥
y
15
16
0
0
6
5
2
100
1
80
x
1
60
2
4
5
40
20
x
20
40
60
80 100 120
75. (a) x number of ounces of food X
y number of ounces of food Y
20x 10y
15x 10y
10x 20y
x
y
≥
≥
≥
≥
≥
300 calcium
150 iron
200 vitamin B
0
0
(b)
(c) Answers will vary. Some possible
solutions which would satisfy the
minimum daily requirements for
calcium, iron, and vitamin B:
y
30
0, 30 ⇒ 30 ounces of food Y
20, 0 ⇒ 20 ounces of food X
x
30
1313, 313 ⇒ 1313 ounces of food X and
313 ounces of food Y
Section 7.5
76. (a) Let y heart rate.
y
(b)
150
y ≤ 0.75220 x
125
100
x ≥ 20
683
(c) Answers will vary. For example, the
points 24, 98 and 24, 147 are on
the boundary of the solution set; a
person aged 24 should have a heart
rate between 98 and 147.
175
y ≥ 0.5220 x
Systems of Inequalities
75
50
x ≤ 70
25
x
25
77. (a) 9, 125.8, 10, 145.6, 11, 164.1,
12, 182.7, 13, 203.1
(b)
Linear model:
y 19.17t 46.61
50
75
100
h
(c) Area of a trapezoid: A a b
2
y ≤ 19.17t 46.61
t ≥ 8.5
t ≤ 13.5
y ≥ 0
h 13.5 8.5 5
a 19.178.5 46.61 116.335
225
b 19.1713.5 46.61 212.185
5
A 116.335 212.185
2
8
$821.3 billion
14
0
78. (a)
xy
2x y
x
y
≥ 500
≥ 125
0
≥
0
≥
y
(b)
Body-building space
Track Two semi–circles and two lengths
60
50
30
20
10
x
10
20
30
40
50
60
79. True. The figure is a rectangle with length of 9 units and
width of 11 units.
80. False. The graph shows the solution of the system
81. The graph is a half-line on the real number line;
on the rectangular coordinate system, the graph is
a half-plane.
82. Test a point on either side of the boundary.
y < 6
4x 9y < 6
3x y2 ≥ 2.
83. x radius of smaller circle
y radius of larger circle
(a) Constraints on circles:
y2
x2
(b)
4
≥ 10
y > x
−6
6
x > 0
−4
(c) The line is an asymptote to the
boundary. The larger the circles, the
closer the radii can be and the
constraint still be satisfied.
684
Chapter 7
Systems of Equations and Inequalities
84. (a) The boundary would be included in the solution.
(b) The solution would be the half-plane on the opposite
side of the boundary.
86. x2 y2 ≤ 16 ⇒ region inside the circle
x y ≤ 4 ⇒ region below the line
85. x2 y 2 ≤ 16 ⇒ region inside the circle
x y ≥ 4 ⇒ region above the line
Matches graph (d).
87. x2 y 2 ≥ 16 ⇒ region outside the circle
x y ≥ 4 ⇒ region above the line
Matches graph (c).
Matches graph (b).
88 . x2 y2 ≥ 16 ⇒ region outside the circle
x y ≤ 4 ⇒ region below the line
89. 2, 6, 4, 4
m
Matches graph (a).
4 6
10
5
4 2
6
3
5
y 4 x 4
3
3y 12 5x 20
5x 3y 8 0
90. 8, 0, 3, 1
91.
1 0
1
3 8
11
m
y0
y
34, 2, 27, 5
m
1
x 8
11
y 2 1
8
x
11
11
11y x 8
5 2
7
28
17 17
72 34
4
28
3
x
17
4
17y 34 28x 21
28x 17y 13 0
x 11y 8 0
92.
21, 0, 112, 12
m
93. 3.4, 5.2, 2.6, 0.8
12 0
12
2
11
1
6
2
2
21
y02 x y 2x 1
2x y 1 0
94. 4.1, 3.8, 2.9, 8.2
m
8.2 3.8 12
2.9 4.1
7
y 3.8 12
x 4.1
7
y 3.8 12
246
x
7
35
y
113
12
x
7
35
35y 60x 113
60x 35y 113 0
m
0.8 5.2
6
1
2.6 3.4
6
y 0.8 1x 2.6
y 0.8 x 2.6
x y 1.8 0
Section 7.6
95. (a) 8, 39.43, 9, 41.24, 10, 45.27, 11, 47.37 , 12, 48.40 , 13, 49.91
(b)
Linear Programming
685
60
y3
Linear model: y 2.17t 22.5
y1
y2
Quadratic model: y 0.241t 2 7.23t 3.4
Exponential model: y 271.05t
5
18
30
(c) The quadratic model is the best fit for the actual data.
(d) For 2008, use t 18: y 0.241182 7.2318 3.4 $48.66
96. A P 1 r
t
nt
A 4000 1 0.06 5 12
12
40001.00560
5395.40061
The amount after 5 years is $5395.40.
Section 7.6
■
Linear Programming
To solve a linear programming problem:
1. Sketch the solution set for the system of constraints.
2. Find the vertices of the region.
3. Test the objective function at each of the vertices.
Vocabulary Check
1. optimization
2. linear programming
3. objective
4. constraints; feasible solutions
5. vertex
1. z 4x 3y
2. z 2x 8y
3. z 3x 8y
At 0, 5: z 40 35 15
At 0, 4: z 20 84 32
At 0, 5: z 30 85 40
At 0, 0: z 40 30 0
At 0, 0: z 20 80 0
At 0, 0: z 30 80 0
At 5, 0: z 45 30 20
At 2, 0: z 22 80 4
At 5, 0: z 35 80 15
The minimum value is 0 at 0, 0.
The minimum value is 0 at 0, 0.
The minimum value is 0 at 0, 0.
The maximum value is 20 at 5, 0.
The maximum value is 32 at 0, 4.
The maximum value is 40 at 0, 5.
686
Chapter 7
Systems of Equations and Inequalities
4. z 7x 3y
5. z 3x 2y
6. z 4x 5y
At 0, 4: z 70 34 12
At 0, 5: z 30 25 10
At 0, 2: z 40 52 10
At 0, 0: z 70 30 0
At 4, 0: z 34 20 12
At 0, 4: z 40 54 20
At 2, 0: z 72 30 14
At 3, 4: z 33 24 17
At 3, 0: z 43 50 12
The minimum value is 0 at 0, 0.
At 0, 0: z 30 20 0
At 4, 3: z 44 53 31
The maximum value is 14 at 2, 0.
The minimum value is 0 at 0, 0.
The minimum value is 10 at 0, 2.
The maximum value is 17 at 3, 4.
The maximum value is 31 at 4, 3.
8. z 2x y
7. z 5x 0.5y
At 0, 5: z 50 52 52
At 0, 2: z 20 2 2
At 4, 0: z 54 20
At 0, 4: z 20 4 4
At 3, 4: z 53 17
At 3, 0: z 23 0 6
At 0, 0: z 50 0
At 4, 3: z 24 3 11
The minimum value is 0 at 0, 0.
The minimum value is 2 at 0, 2.
The maximum value is 20 at 4, 0.
The maximum value is 11 at 4, 3.
0
2
4
2
0
2
9. z 10x 7y
10. z 25x 35y
At 0, 45: z 100 745 315
At 0, 400: z 250 35400 14,000
At 30, 45: z 1030 745 615
At 0, 800: z 250 35800 28,000
At 60, 20: z 1060 720 740
At 450, 0: z 25450 350 11,250
At 60, 0: z 1060 70 600
At 900, 0: z 25900 350 22,500
At 0, 0: z 100 70 0
The minimum value is 11,250 at 450, 0.
The minimum value is 0 at 0, 0.
The maximum value is 28,000 at 0, 800.
The maximum value is 740 at 60, 20.
11. z 25x 30y
12. z 15x 20y
At 0, 45: z 250 3045 1350
At 0, 400: z 150 20400 8000
At 30, 45: z 2530 3045 2100
At 0, 800: z 150 20800 16,000
At 60, 20: z 2560 3020 2100
At 450, 0: z 15450 200 6750
At 60, 0: z 2560 300 1500
At 900, 0: z 15900 200 13,500
At 0, 0: z 250 300 0
The minimum value is 6750 at 450, 0.
The minimum value is 0 at 0, 0.
The maximum value is 16,000 at 0, 800.
The maximum value is 2100 at any point along the line
segment connecting 30, 45 and 60, 20.
13. z 6x 10y
y
At 0, 2: z 60 102 20
4
At 5, 0: z 65 100 30
3
At 0, 0: z 60 100 0
(0, 2)
1
The minimum value is 0 at 0, 0.
The maximum value is 30 at 5, 0.
(5, 0)
x
(0, 0)
−1
2
3
4
5
Section 7.6
14. z 7x 8y
Linear Programming
15. z 9x 24y
At 0, 8: z 70 88 64
At 0, 2: z 90 242 48
At 4, 0: z 74 80 28
At 5, 0: z 95 240 45
At 0, 0: z 70 80 0
At 0, 0: z 90 240 0
The minimum value is 0 at 0, 0.
The minimum value is 0 at 0, 0.
The maximum value is 64 at 0, 8.
The maximum value is 48 at 0, 2.
y
y
(0, 8)
8
4
3
6
(0, 2)
4
1
(5, 0)
2
x
(4, 0)
x
(0, 0)
2
6
8
(0, 0)
2
3
4
5
−1
17. z 4x 5y
16. z 7x 2y
At 0, 8: z 70 28 16
At 10, 0: z 410 50 40
At 4, 0: z 74 20 28
At 5, 3: z 45 53 35
At 0, 0: z 70 20 0
At 0, 8: z 40 58 40
The minimum value is 0 at 0, 0.
The minimum value is 35 at 5, 3.
The maximum value is 28 at 4, 0.
The region is unbounded. There is no maximum.
y
y
10
8
(0, 8)
(0, 8)
6
4
4
(5, 3)
2
2
x
(4, 0)
(0, 0)
2
x
2
6
4
6
8
8
(10, 0)
19. z 2x 7y
18. z 4x 5y
At 0, 0: z 40 50 0
At 10, 0: z 210 70 20
At 5, 0: z 45 50 20
At 5, 3: z 25 73 31
At 4, 1: z 44 51 21
At 0, 8: z 20 78 56
At 0, 3: z 40 53 15
The minimum value is 20 at 10, 0.
The minimum value is 0 at 0, 0.
The region is unbounded. There is no maximum.
The maximum value is 21 at 4, 1.
y
y
10
(0, 8)
4
(0, 3)
4
(5, 3)
2
2
(4, 1)
x
1
2
x
(0, 0)
2
3
(5, 0)
4
6
8
(10, 0)
687
688
Chapter 7
Systems of Equations and Inequalities
20. z 2x y
y
At 0, 0: z 20 0 0
4
At 5, 0: z 25 0 10
(0, 3)
At 4, 1: z 24 1 7
2
At 0, 3: z 20 3 3
(4, 1)
1
The minimum value is 3 at 0, 3.
x
(0, 0)
2
(5, 0)
3
The maximum value is 10 at 5, 0.
21. z 4x y
22. z x
15
At 36, 0: z 436 0 144
25
At 0, 0: z 0
At 40, 0: z 440 0 160
20
At 24, 8: z 424 8 104
At 12, 0: z 12
50
−5
At 10, 8: z 10
−5
The minimum value is 104 at 24, 8.
At 6, 16: z 6
The maximum value is 160 at 40, 0.
At 0, 20: z 0
40
−5
The minimum value is 0 at any point along the line
segment connecting 0, 0 and (0, 20. The maximum
value is 12 at 12, 0.
23. z x 4y
24. z y
15
At 0, 0: z 0
At 36, 0: z 36 40 36
At 40, 0: z 40 40 40
25
20
At 24, 8: z 24 48 56
At 12, 0: z 0
50
−5
At 10, 8: z 8
−5
The minimum value is 36 at 36, 0.
At 6, 16: z 16
The maximum value is 56 at 24, 8.
At 0, 20: z 20
40
−5
The minimum value is 0 at any point along the line
segment connecting 0, 0 and (12, 0.The maximum
value is 20 at 0, 20.
y
(0, 10)
8
6
(3, 6)
4
2
(0, 0)
−2
2 (5, 0)
x
6
Figure for Exercises 25–28
25. z 2x y
26. z 5x y
At 0, 10: z 20 10 10
At 0, 10: z 50 10 10
At 3, 6: z 23 6 12
At 3, 6: z 53 6 21
At 5, 0: z 25 0 10
At 5, 0: z 55 0 25
At 0, 0: z 20 0 0
At 0, 0: z 50 0 0
The maximum value is 12 at 3, 6.
The maximum value is 25 at 5, 0.
Section 7.6
27. z x y
Linear Programming
28. z 3x y
At 0, 10: z 0 10 10
At 0, 10: z 30 10 10
At 3, 6: z 3 6 9
At 3, 6: z 33 6 15
At 5, 0: z 5 0 5
At 5, 0: z 35 0 15
At 0, 0: z 0 0 0
At 0, 0: z 30 0 0
The maximum value is 10 at 0, 10.
The maximum value is 15 at any point along
the line segment connecting 3, 6 and 5, 0.
y
25
15
(223 , 196 )
10
(212 , 0)
(0, 5)
x
(0, 0)
10
15
Figure for Exercises 29–32
29. z x 5y
30. z 2x 4y
At 0, 5: z 0 55 25
At 0, 5: z 20 45 20
At : z 5 21
21
At 2 , 0: z 21
2 50 2
19
22
19
82
At 22
3 , 6 : z 2 3 4 6 3
At 0, 0: z 0 50 0
At 0, 0: z 20 40 0
The maximum value is 25 at 0, 5.
The maximum value is
22 19
3, 6
22
3
19
6
139
6
21
At 21
2 , 0: z 2 2 40 21
31. z 4x 5y
82
3
19
at 22
3 , 6 .
32. z 4x y
At 0, 5: z 40 55 25
At 0, 5: z 40 5 5
At : z 4 5 21
At 2 , 0: z 421
2 50 42
At 3 ,
At 0, 0: z 40 50 0
At 0, 0: z 40 0 0
22 19
3, 6
22
3
The maximum value is
19
6
271
6
at :
21
At 2 , 0:
271
6
22 19
3, 6
22 19
6
.
21
y
( 2019 , 4519 (
(0, 3)
At 0, 0: z 0
2
At 2, 0: z 5
1
45
95
At 20
19 , 19 : z 19 5
At 0, 3: z 3
The maximum value of 5 occurs at any point on the line segment
45
connecting 2, 0 and 20
19 , 19 .
z 421
2 0 42
The maximum value is 42 at 2 , 0.
33. Objective function: z 2.5x y
Constraints: x ≥ 0, y ≥ 0, 3x 5y ≤ 15, 5x 2y ≤ 10
19
65
z 422
3 6 2
(2, 0)
(0, 0)
1
x
3
689
690
Chapter 7
Systems of Equations and Inequalities
35. Objective function: z x 2y
34. Objective function: z x y
Constraints: x ≥ 0, y ≥ 0, x y ≤ 1, x 2y ≤ 4
Constraints: x ≥ 0, y ≥ 0, x ≤ 10, x y ≤ 7
At 0, 0: z 0 0 0
At 0, 0: z 0 20 0
At 0, 1: z 0 1 1
At 0, 7: z 0 27 14
At 2, 3: z 2 3 5
At 7, 0: z 7 20 7
The constraints do not form a closed set of points.
Therefore, z x y is unbounded.
The constraint x ≤ 10 is extraneous.
The maximum value of 14 occurs at 0, 7.
y
y
4
10
(2, 3)
3
(0, 7)
6
4
(0, 1)
2
(7, 0)
x
(0, 0)
1
2
3
x
4
(0, 0)
36. Objective function: z x y
2
4
6
y
Constraints: x ≥ 0, y ≥ 0, x y ≤ 0, 3x y ≥ 3
3
The feasible set is empty.
−3
x
−2
1
2
−1
−2
37. Objective function: z 3x 4y
38. Objective function: z x 2y
Constraints: x ≥ 0, y ≥ 0, x y ≤ 1, 2x y ≤ 4
Constraints: x ≥ 0, y ≥ 0, x 2y ≤ 4, 2x y ≤ 4
At 0, 0: z 30 40 0
At 0, 0: z 0 20 0
At 0, 1: z 30 41 4
At 0, 2: z 0 22 4
At 1, 0: z 31 40 3
The constraint 2x y ≤ 4 is extraneous.
The maximum value of 4 occurs at 0, 1.
At 43, 43 : z 43 243 4
At 2, 0: z 2 20 2
The maximum value is 4 at any point along the line
segment connecting 0, 2 and 43, 43 .
y
y
(0, 2)
( 43 , 43 (
3
2
1
(0, 1)
(2, 0)
(1, 0)
x
(0, 0)
3
4
x
(0, 0)
1
Section 7.6
39. x number of Model A
40. x number of Model A
y number of Model B
y number of Model B
Constraints: 2x 2.5y
4x y
x 0.75y
x
y
≤
≤
≤
≥
≥
4000
4800
1500
0
0
2.5x 2x Constraints:
Linear Programming
691
3y ≤ 4000
y ≤ 2500
0.75x 1.25y ≤ 1500
Objective function: P 45x 50y
x ≥
0
y ≥
0
Objective function: P 50x 52y
Vertices:
0, 0, 0, 1600, 750, 1000, 1050, 600, 1200, 0
Vertices:
6000
0, 0, 0, 1200, 4000
7 , 7 , 1000, 500, 1250, 0
At 0, 0: P 450 500 0
At 0, 0: P 500 520 0
At 0, 1600: P 450 501600 80,000
At 0, 1200: P 500 521200 62,400
At 750, 1000: P 45750 501000 83,750
At At 1050, 600: P 451050 50600 77,250
At 1000, 500: P 501000 52500 76,000
At 1200, 0: P 451200 500 54,000
At 1250, 0: P 501250 520 62,500
The optimal profit of $83,750 occurs when 750 units of
Model A and 1000 units of Model B are produced.
The optimal profit of $76,000 occurs when 1000 units of
Model A and 500 units of Model B are produced.
4000 6000
7 , 7
y
:
6000
P 504000
7 52 7 73,142.86
y
2000
(0, 1600)
,
( 4000
7
(750, 1000)
6000
7
(
(0, 1200)
(1050, 600)
1000
(1000, 500)
500
500
(0, 0)
x
x
500
(1250, 0)
500
(1200, 0)
41. x number of $250 models
y
y number of $300 models
Constraints: 250x 300y
x
y
x
y
≤
≤
≥
≥
65,000
250
0
0
Objective function: P 25x 40y
Vertices: 0, 0, 250, 0, 200, 50, 0, 21623 (0, 216 23 )
200
100
(200, 50)
(0, 0)
x
100
200
(250, 0)
At 0, 0: P 250 400 0
At 250, 0: P 25250 400 6250
At 200, 50: P 25200 4050 7000
At 0, 21623 : P 250 4021623 8666.67
An optimal profit of $8640 occurs when 0 units of the $250 model and 216 units of the $300 model are stocked in inventory.
A merchant cannot sell 23 of a unit.
Note:
692
Chapter 7
Systems of Equations and Inequalities
42. x number of acres for crop A
43. x number of bags of Brand X
y number of acres for crop B
y number of bags of Brand Y
y
Constraints:
x
x
Constraints:
y ≤ 150
2y ≤ 240
0.3x 0.1y ≤ 30
x ≥
0
y ≥
0
2x y
2x 9y
2x 3y
x
y
(60, 90)
100
(0, 120)
75
(75, 75)
50
25
(0, 0)
(100, 0)
x
25
50
75
≥
≥
≥
≥
≥
y
12
36
24
0
0
18
12
(0, 12)
6
(3, 6)
(9, 2)
(18, 0)
Objective function:
C 25x 20y
125
Objective function:
P 140x 235y
12
x
18
Vertices: 0, 12, 3, 6, 9, 2, 18, 0
Vertices: 0, 0, 100, 0, 0, 120, 60, 90, 75, 75
At 0, 12: C 250 2012 240
At 0, 0: P 1400 2350 0
At 3, 6: C 253 206 195
At 100, 0: P 140100 2350 14,000
At 9, 2: C 259 202 265
At 0, 120: P 1400 235120 28,200
At 18, 0: C 2518 200 450
At 60, 90 : P 14060 23590 29,550
To optimize cost, use three bags of Brand X and
six bags of Brand Y for an optimal cost of $195.
At 75, 75: P 14075 23575 28,125
To optimize the profit, the fruit grower should plant 60
acres of crop A and 90 acres of crop B. The optimal
profit is $29,550.
44. (a) x the proportion of regular gasoline
(c)
y
y the proportion of premium
C 1.84x 2.03y
3
4
(b) The constraints are:
1
2
x y
87x 93y
x
y
1
≥ 89
≥ 0
≥ 0
x
1
4
(d) Actually the only points of the plane which satisfy all
the constraints are the points of the line segment con2 1
necting (0, 1) and 3, 3 . Evaluate C 1.84x 2.03y
at the two endpoints to find that the lower cost occurs
at 23, 13 .
45. x number of audits
3
4
(e) The optimal cost is C 1.8423 2.0313 $1.90.
(f) This is lower than the national average of $1.96.
At 12, 0: R 250012 3500 30,000
At (5, 42: R 25005 35042 27,200
(0, 62)
≤ 900
≤ 155
0
≥
0
≥
1
2
At 0, 0: R 25000 3500 0
y
y number of tax returns
Constraints:
75x 12.5y
10x 2.5y
x
y
( 23 , 13 (
1
4
(5, 42)
40
At 0, 62: R 25000 35062 21,700
20
(0, 0)
−3
3
− 20
Objective function:
R 2500x 350y
Vertices: 0, 0, 12, 0, 5, 42, 0, 62
x
6
9
(12, 0)
15
The revenue will be optimal if 12 audits and 0 tax returns are
done each week. The optimal revenue is $30,000.
Section 7.6
Linear Programming
46. The modified objective function is R 2000x 350y.
The vertices of the region are at 0, 0, 0, 62, 5, 42, and 12, 0.
At 0, 0: R 20000 3500 0
At 0, 62: R 20000 35062 21,700
At 5, 42: R 20005 35042 24,700
At 12, 0: R 200012 3500 24,000
The optimal revenue of $24,700 occurs with 5 audits and 42 tax returns.
47. x amount of Type A
y amount of Type B
250,000
Constraints: x y ≤
1
≥ 4250,000
x
1
y ≥ 4250,000
y
250,000
187,500
(62,500, 187,500)
125,000
(187,500, 62,500)
Objective Function: P 0.08x 0.10y
Vertices: 62,500, 62,500, 62,500, 187,500, 187,500, 62,500
x
125,000 187,500
(62,500, 62,500)
At 62,500, 62,500: P 0.0862,500 0.1062,500 $11,250
At 62,500, 187,500: P 0.0862,500 0.10187,500 $23,750
At 187,500, 62,500: P 0.08187,500 0.1062,500 $21,250
To obtain an optimal return the investor should allocate $62,500 to Type A and $187,500 to Type B.
The optimal return is $23,750.
48. x amount in investment of Type A; y amount in investment of Type B
Constraints:
y
x y ≤ 450,000
400,000
x ≥ 225,000
300,000
y ≥ 112,500
200,000
(225,000, 225,000)
(337,500, 112,500)
Objective function: R 0.06x 0.1y
Vertices: 225,000, 112,500, 337,500, 112,500, 225,000, 225,000
At 225,000, 112,500: R 0.06225,000 0.1112,500 24,750
x
100,000
400,000
(225,000, 112,500)
At 337,500, 112,500: R 0.06337,500 0.1112,500 31,500
At 225,000, 225,000: R 0.06225,000 0.1225,000 36,000
The optimal return of $36,000 occurs for an investment of $225,000 to Type A and $225,000 to Type B.
49. True. The objective function has a maximum value at any point on the line segment connecting the two vertices.
Both of these points are on the line y x 11 and lie between 4, 7 and 8, 3.
50. True. If an objective function has a maximum value at more than one vertex, then any point on the line segment
connecting the points will produce the maximum value.
693
694
Chapter 7
Systems of Equations and Inequalities
51. Constraints: x ≥ 0, y ≥ 0, x 3y ≤ 15, 4x y ≤ 16
Vertex
y
(0, 5)
Value of z 3x ty
z0
0, 5
z 5t
3, 4
z 9 4t
4, 0
z 12
3
2
1
x
−1
5t ≥ 9 4t and 5t ≥ 12
t ≥
(4, 0)
(0, 0)
(a) For the maximum value to be at 0, 5, z 5t must be
greater than or equal to z 9 4t and z 12.
t ≥ 9
(3, 4)
4
0, 0
1
2
3
(b) For the maximum value to be at 3, 4, z 9 4t must
be greater than or equal to z 5t and z 12.
9 4t ≥ 5t and
12
5
9 4t ≥ 12
9 ≥ t
4t ≥ 3
Thus, t ≥ 9.
t ≥
Thus,
3
4
≤ t ≤ 9.
52. Constraints: x ≥ 0, y ≥ 0, x 2y ≤ 4, x y ≤ 1
y
z 3x ty
4
At 0, 0: z 30 t0 0
3
(0, 2)
At 1, 0: z 31 t0 3
x
(0, 0)
At 0, 2: z 30 t2 2t
(a) For the maximum value to be at 2, 1, z 6 t must
be greater than or equal to z 2t and z 3.
6t ≥ 3
6 ≥ t
(2, 1)
1
At 2, 1: z 32 t1 6 t
6 t ≥ 2t and
3
4
(1, 0)
4
(b) For maximum value to be at 0, 2, z 2t must be
greater than or equal to z 6 t and z 3.
2t ≥ 6 t and
t ≥ 3
t ≥ 6
Thus, 3 ≤ t ≤ 6.
3
2t ≥ 3
t ≥
3
2
Thus, t ≥ 6.
53. There are an infinite number of objective functions that
would have a maximum at 0, 4. One such objective
function is z x 5y.
54. There are an infinite number of objective functions that
would have a maximum at 4, 3. One such objective
function is z x y.
55. There are an infinite number of objective functions that
would have a maximum at 5, 0. One such objective
function is z 4x y.
56. There are an infinite number of objective functions that
would have a minimum at 5, 0. One such objective
function is z 10x y.
57.
9
x
9
x
9
6
6 2x
x
2
x
x
x2
x
x2
2
4
x 4
x
x
x
x
1 x x
9
9
23 x 23 x 2x 3,
x0
2
58.
x
x2 4 x2
x
x
1
x 2x 2 x 2, x 0, 2
Section 7.6
x 9 x 2
59.
1
1
x 3 x 3
4
2
2
4x 2 2x2 9
x 2x2 9
x 3 x 3
x2 9
2x2 4x 26
x 2x2 9
2x2 2x 13
x 22x
x2 2x 13
, x ±3
xx 2
1
x2 9
2x
61. e2x 2ex 15 0
62. e2x 10e x 24 0
ex 5ex 3 0
e x 4e x 6 0
ex 4
ex 5 or ex 3
x ln 3
x 1.099
No real
solution.
x3
2x 1
3
3
2
2x 4x 2
2x 12
x 1 2
1
60.
Linear Programming
2x 12
3
x3
2x 1
x 3x 1
, x 1
3
63. 862 ex4 192
62 ex4 24
or e x 6
ex4 38
x ln 4
x ln 6
x 1.386
x 1.792
ex4 38
x
ln 38
4
x 4 ln 38
x 14.550
64.
150
75
ex 4
150 75ex 300
65. 7 ln 3x 12
66. lnx 92 2
12
7
2 lnx 9 2
ln 3x 75ex 450
lnx 9 1
3x e127
ex 6
x9e
e127
x
3
x ln 6
x ln 6
xe9
x 6.282
x 1.851
x 1.792
67.
x 2 y 3 z 23
2x 6y z 17
5y z 8
68.
x 2y 3z 23
2y 5z 29
5y z 8
x 2y 2y 23
2z
161
2
2Eq.1 Eq.2
3z 23
5z 29
161
23
2z 2
5
2Eq.2 Eq.3
⇒ z 7
2y 57 29 ⇒ y 3
x 23 37 23 ⇒ x 27 23
⇒ x 4
Solution: 4, 3, 7
7x 3y 5z 28
4x
4z 16
7x 2y z 0
7x 3y 5z 28
12y 8z 0
5y 6z 28
7x 3y 12y 4Eq.1 7Eq.2
1Eq.1 Eq.3
5z 28
8z 0
112z 336
7x 3y 5z 28
3y 2z 0
z 3
3y 23 5Eq.2 12Eq.3
14 Eq.2
1121 Eq.3
0 ⇒ y
2
7x 32 53 28 ⇒ x 1
Solution: 1, 2, 3
695
696
Chapter 7
Systems of Equations and Inequalities
Review Exercises for Chapter 7
1. x y 2
xy0 ⇒ xy
2. 2x 3y 3
x y0 ⇒ xy
xx2
2y 3y 3
2x 2
y 3
x1
y 3
y1
x 3
Solution: 1, 1
3.
Solution: 3, 3
0.5x y 0.75 ⇒ y 0.75 0.5x
1.25x
4.5y 2.5
2
3
4. x 5 y 5
x 15 y 45
1.25x 4.50.75 0.5x 2.5
Multiply both equations by 5 to clear the denominators.
1.25x 3.375 2.25x 2.5
2y 3
5x
5x y 4 ⇒ 5x 4 y
3.50x 0.875
4 y 2y 3
x 0.25
4 y 3
y 0.625
y7
Solution: 0.25, 0.625
5x 4 7
5x 11
x 11
5
Solution:
5. x2 y2 9
x y 1 ⇒ xy1
6.
115, 7
x2 y2 169
1
3x 2y 39 ⇒ x 3 39 2y
1339 2y 2 y2 169
y 12 y2 9
1
9 1521
2y 1 9
156y 4y2 y2 169
1521 156y 4y 2 9y 2 1521
y4
13y 2 156y 0
x5
13yy 12 0 ⇒ y 0, 12
Solution: 5, 4
x
1
3 39
20 13
y 12: x 1
3 39
212 5
y 0:
Solution: 13, 0, 5, 12
7. y 2x2
y x4 2x2 ⇒ 2x2 x4 2x2
0
0
0
x
y
x4 4x2
x2x2 4
x2x 2x 2
0, x 2, x 2
0, y 8, y 8
Solutions: 0, 0, 2, 8, 2, 8
8.
xx yy
2
3
1
y 3 y2 1
0 y2 y 2
0 y 2 y 1 ⇒ y 2, 1
y 2: x 2 3 5
y 1: x 1 3 2
Solution: 5, 2, 2, 1
Review Exercises for Chapter 7
697
10. 8x 3y 3
2x 5y 28
9. 2x y 10
x 5y 6
The point of intersection appears to be at 1.5, 5.
Point of intersection: 4, 2
y
y
6
8
4
7
2
x
−6
−2
6
−4
(4, −2)
(1.5, 5)
5
8 10
4
3
−6
2
−8
1
−10
x
−1
11. y 2x2 4x 1
y x2 4x 3
1
2
3
4
5
3
4
5
6
7
8
y
21
18
Point of intersection: 1.41, 0.66, 1.41, 10.66
15
12
(−1.41, 10.66)
9
x
−3 −2 −1
−3
(1.41, − 0.66)
12. y2 2y x 0 ⇒ y 12 1 x ⇒ y 1 ± 1 x
4
x y 0 ⇒ y x
Points of intersection: 0, 0 and 3, 3
−5
2
−2
13.
2e yy 02e⇒ y 2e
x
x
x
Point of intersection: 0, 2
2
14. y lnx 1 3
y4
4
1
2x
0
12
Point of intersection:
9.68, 0.84
−4
−6
6
−6
15. Let x number of kits.
C 12x 50,000
R 25x
Break-even:
RC
25x 12x 50,000
13x 50,000
x 3846.15
You would need to sell 3847 kits to cover your costs.
16. y 35,000 0.015x
y 32,500 0.02x
35,000 0.015x 32,500 0.02x
2500 0.005x
$500,000 x
For the second offer to be better, you would have to sell
more than $500,000 per year.
698
Chapter 7
Systems of Equations and Inequalities
2l 2w 480
17.
18. 2l 2w 68
w 89l
l 1.50w
2l 289 l 68
21.50w 2w 480
34
9l
5w 480
l 18
w 96
w
l 144
19. 2x y 2 ⇒ 16x 8y 16
6x 8y 39 ⇒ 6x 8y 39
8
9l
16
The width of the rectangle is 16 feet, and the
length is 18 feet.
The dimensions are 96 144 meters.
22x
20.
30y 24 ⇒ 40x 30y 24
40x
20x 50y 14 ⇒ 40x 100y 28
130y 52
y 25
55
5
x 55
22 2
Back-substitute x 52 into Equation 1.
Back-substitute y 25 in Equation 1.
252 y 2
40x 3025 24
y3
Solution:
5
2,
68
40x 12
3
3
x 10
Solution:
103 , 25 21. 0.2x 0.3y 0.14 ⇒ 20x 30y 14 ⇒ 20x 30y 14
0.4x 0.5y 0.20 ⇒ 4x 5y 2 ⇒ 20x 25y 10
5y 4
y 45
4
5
Back-substitute y into Equation 2.
4x 545 2
4x 2
x 12
1 4
Solution: 2, 5 0.5, 0.8
22. 12x 42y 17 ⇒ 36x 126y 51
30x 18y 19 ⇒ 210x 126y 133
246x
82
1
x
3
23.
6x
x
Back-substitute x 13 in Equation 1.
12
1
3
3x 2y 0 ⇒ 3x 2y 0
3x 2 y 5 10 ⇒ 3x 2y 0
Back-substitute x 0 into Equation 1.
42y 17
30 2y 0
42y 21
2y 0
y
Solution:
12
y0
13, 12 Solution: 0, 0
7x 12y 63
7x2x 12y3y 6315 ⇒
⇒ 8x 12y 60
24. 7x 12y 63
2x 3y 2 21
x 3
0
0
Back-substitute x 3 in Equation 1.
73 12y 63
12y 84
y7
Solution: 3, 7
Review Exercises for Chapter 7
25.
⇒ 5x 8y 14
1.25x5x 2y8y 3.5
14 ⇒ 5x 8y 14
0
26. 1.5x 2.5y 8.5 ⇒ 3x 5y 17
6x 10y 24 ⇒ 3x 5y 12
0
5
0
The system is inconsistent. There is no solution.
There are infinitely many solutions.
Let y a, then 5x 8a 14 ⇒ x Solution:
8
5a
14
5,
699
8
5a
14
5.
a where a is any real number.
28. 3x y 7
9x 3y 21
27. x 5y 4 ⇒
x 5y 4
x 3y 6 ⇒ x 3y 6
8y 2 ⇒ y 41
3x y 7 ⇒ y 3x 7;
The graph contains 0, 7 and 2, 1.
Matches graph (d). The system has one solution and is
consistent.
9x 3y 21 ⇒ 3y 9x 21 ⇒ y 3x 7;
The graph is the same as the previous graph.
The graph of the system matches (c). The system has
infinitely many solutions and is consistent.
29.
⇒
6x 2y 14
6x3x 2yy 87 ⇒
6x 2y 8
30. 2x y 3
x 5y 4
0 22
2x y 3 ⇒ y 2x 3 ⇒ y 2x 3;
The graph contains 0, 3 and 2, 1.
Matches graph (b). The system has no solution and is
inconsistent.
x 5y 4 ⇒ 5y x 4 ⇒ y 15x 45;
The graph contains 0, 45 and 4, 0.
The graph of the system matches (a). The system has one
solution and is consistent.
32. 45 0.0002x 120 0.0001x
31. 37 0.0002x 22 0.00001x
0.0003x 75
15 0.00021x
x
500,000 159
Point of equilibrium:
,
7
7
33.
x 250,000 units
159
500,000
,p
7
7
p $95.00
Point of equilibrium: 250,000, 95
x 4y 3z 3
y z 1
z 5
34.
y 5 1 ⇒ y 4
x 44 35 Solution: 2, 4, 5
3 ⇒ x
2
x 7y 8z 85
y 9z 35
z
3
y 93 35 ⇒ y 8
x 78 83 85 ⇒ x 5
Solution: 5, 8, 3
700
35.
Chapter 7
x
3x 4x
x
x
x
Systems of Equations and Inequalities
2y 2y 6z 4
z 4
2z 16
2y 6z 8y 17z 8y 22z 4
8
0
2y 6z 8y 17z 5z 4
8
8
2y 6z 4
8y 17z 8
z 85
8y 17
8
x 2 6 4
24 22
8
Solution: 5 , 5 , 5 22
5
37.
85
85
36.
3Eq.1 Eq.2
4Eq.1 Eq.3
Eq.2 Eq.3
⇒ y
⇒ x
24
5
x 2y z 6
2x 3y
7
x 3y 3z 11
x 2y z 6
y 2z 5
y 2z 5
x 2y z 6
y 2z 5
0 0
15Eq.3
22
5
Eq.2 Eq.3
3y z 5z y 2z 2Eq.1 Eq.2
4Eq.1 Eq.3
x
3y z 13
6y 3z 3
17
63
2z 2
136 Eq.2 Eq.3
x
y 2a 5
x 22a 5 a 6
y
x 3a 4
Solution: 3a 4, 2a 5, a where a is any real number.
z
1
2z
13
1
2
63
z 17
1
2
16 Eq.2
172 Eq.3
⇒ y
40
17
⇒ x
38
17
2x
3x 3x 6z 9
2y 11z 16
y 7z 11
Equation 1
Equation 2
Equation 3
x 3x 3x 2y 5z 7
2y 11z 16
y 7z 11
1Eq.2 Eq.1
x 2y 5z 4y 4z 5y 8z 7
5
10
2y 5z 4y 4z 3y
7
5
0
2y 5z y z
y
7
x x 3a 10 6
3y 6317 12
63
x 340
17 17 13
40
63
Solution: 38
17 , 17 , 17 Equation 1
Equation 2
Equation 3
3y z 13
6y 3z 3
13y 2z 38
x Let z a, then:
13
23
14
x
y
38.
2Eq.1 Eq.2
Eq.1 Eq.3
x
2x
4x 5
4
0
3Eq.1 Eq.2
3Eq.1 Eq.3
2Eq.2 Eq.3
14 Eq.2
13 Eq.3
0 z ⇒ z 54
x 20 5 54 7 ⇒ x 34
5
4
3
5
Solution: 4, 0, 4 39. 5x 12y 7z 16 ⇒ 15x 36y 21z 48
3x 7y 4z 9 ⇒ 15x 35y 20z 45
y z 3
Let z a. Then y a 3 and
5x 12a 3 7a 16 ⇒ x a 4.
Solution: a 4, a 3, a where a is any real number.
40. 2x 5y 19z 34 ⇒ 6x 15y 57z 102
3x 8y 31z 54 ⇒ 6x 16y 62z 108
y 5z 6
Let z a. Then:
y 5a 6 ⇒ y 5a 6
2x 55a 6 19a 34 ⇒ x 3a 2
Solution: 3a 2, 5a 6, a where a is any
real number.
Review Exercises for Chapter 7
41. y ax2 bx c through 0, 5, 1, 2, and 2, 5.
0, 5: 5 c ⇒
1, 2: 2 a b c ⇒
2, 5: 5 4a 2b c ⇒
The equation of the parabola is y a
2a
2a
a
a
2x2
42. y ax2 bx c through 5, 6, 1, 0,2, 20.
c 5
b 3
b 5
b 5
b 3
2
b 1
5, 6: 6 25a 5b c
1, 0:
0
a b c ⇒ c a b
2, 20: 20 4a 2b c
24a 6b 6
24a3a 6bb 620 ⇒
⇒ 24a 8b 160
14b 154
x 5.
b 11
3a 11 20 ⇒ a 3
10
−5
701
c 3 11 ⇒ c 14
5
The equation of the parabola
is y 3x2 11x 14.
−10
24
−12
12
−24
43. x2 y2 Dx Ey F 0 through 1, 2, 5, 2 and 2, 1.
1,2: 5 D 2E F 0 ⇒
D 2E F 5
5,2: 29 5D 2E F 0 ⇒ 5D 2E F 29
2, 1: 5 2D E F 0 ⇒ 2D E F 5
From the first two equations we have
6D 24
D 4.
Substituting D 4 into the second and third equations yields:
20 2E F 29 ⇒ 2E F 9
8 E F 5 ⇒ E F 3
3E
12
E
4
F 1
The equation of the circle is x2 y2 4x 4y 1 0.
To verify the result using a graphing utility, solve the equation for y.
4
−6
9
x 2 4x 4 y 2 4y 4 1 4 4
x 22 y 22 9
−6
y 2 9 x 2
2
2
y 2 ± 9 x 22
Let y1 2 9 x 22 and y2 2 9 x 22.
702
Chapter 7
Systems of Equations and Inequalities
44. x2 y2 Dx Ey F 0 through 1, 4, 4, 3, 2, 5.
1, 4:
17 D 4E F 0
4, 3:
25 4D 3E F 0
To verify the result using a graphing utility, solve the
equation for y.
x 2 2x 1 y 2 2y 1 23 1 1
x 12 y 12 25
2, 5: 29 2D 5E F 0
y 12 25 x 12
D 4E F 17
4D 3E F 25
2D 5E F 29
Equation 1
Equation 2
Equation 3
D 4E F 17
13E 3F 43
3E 3F 63
4Eq.1 Eq.2
2Eq.1 Eq.3
D 4E F 17
3E 3F 63
13E 3F 43
y 1 ± 25 x 12
Let y1 1 25 x 12 and
y2 1 25 x 12.
7
−10
11
Interchange equations.
−7
D 4E F 17
3E 3F 63
10F 230
133 Eq.2 Eq.3
F 23, E 2, D 2
The equation of the circle is x 2 y 2 2x 2y 23 0.
45. 3, 101.7, 4, 108.4, 5, 121.1
(a) n 3,
3
3
x 12, x
i
i1
2
i 3
50,
i1
x
3
i 3
216,
i1
x
4
i 3
962,
i1
3
3
y 331.2, x y 1344.2, x
i
2
i yi
i i
i1
i1
5677.2
i1
3c 12b 50a 331.2
12c 50b 216a 1344.2
50c 216b 962a 5677.2
Solving this system yields c 117.6, b 14.3, a 3.
Quadratic model: y 3x2 14.3x 117.6
(b)
The model is a good fit
to the data. The actual
points lie on the parabola.
130
0
(c) For 2008, use x 8:
y 382 14.38 117.6
195.2 million online shoppers
This answer seems reasonable.
6
80
46. From the following chart we obtain our system of equations.
A
B
C
Mixture X
1
5
2
5
2
5
Mixture Y
0
0
1
Mixture Z
1
3
1
3
1
3
Desired Mixture
6
27
8
27
13
27
1
5x
2
5x
2
5x
13z 13z x 6
27
8
27
10
27 ,
z 12
27
5
y 13z 13
27 ⇒ y 27
To obtain the desired mixture, use 10 gallons of spray X, 5
gallons of spray Y, and 12 gallons of spray Z.
47. Let x amount invested at 7%
y amount invested at 9%
z amount invested at 11%.
y x 3000 and
z x 5000 ⇒ y z 2x 8000
x
y
z
40,000
0.07x 0.09y 0.11z 3500
y
z 2x 8000
40,000 ⇒ x 16,000
x 2x 8000 y 16,000 3000 ⇒ y 13,000
z 16,000 5000 ⇒ z 11,000
Thus, $16,000 was invested at 7%, $13,000 at 9% and
$11,000 at 11%.
Review Exercises for Chapter 7
48. s 12at2 v0 t s0
1
(a) When t 1: s 134: 2a12 v01 s0 134 ⇒ a 2v0 2s0 268
1
When t 2: s 86: 2a22 v02 s0 86 ⇒ 2a 2v0 s0 86
1
When t 3: s 6: 2a32 v03 s0 6 ⇒ 9a 6v0 2s0 12
a 2v0 2s0 268
2a 2v0 s0 86
9a 6v0 2s0 12
a 2v0 2s0 268
2v0 3s0 450
12v0 16s0 2400
2Eq.1 Eq.2
9Eq.1 Eq.3
a 2v0 2s0 268
2v0 3s0 450
3v0 4s0 600
14 Eq.3
a 2v0 2s0 268
2v0 3s0 450
s0 150
3Eq.2 2Eq.3
s0 150 ⇒ s0 150
2v0 3150 450 ⇒ v0 0
a 20 2150 268 ⇒ a 32
1
The position equation is s 232t2 0t 150, or s 16t2 150.
1
(b) When t 1: s 184: 2a12 v01 s0 184 ⇒ a 2v0 2s0 368
1
When t 2: s 116: 2a22 v02 s0 116 ⇒ 2a 2v0 s0 116
1
When t 3: s 16: 2a32 v03 s0 16 ⇒ 9a 6v0 2s0 32
a 2v0 2s0 368
2a 2v0 s0 116
9a 6v0 2s0 32
a 2v0 2s0 368
2v0 3s0 620
12v0 16s0 3280
2Eq.1 Eq.2
9Eq.1 Eq.3
a 2v0 2s0 368
2v0 3s0 620
3v0 4s0 820
14 Eq.3
a 2v0 2s0 368
2v0 3s0 620
s0 220
3Eq.2 2Eq.3
s0 220 ⇒ s0 220
2v0 3220 620 ⇒ v0 20
a 220 2220 368 ⇒ a 32
1
The position equation is s 232t2 20t 220, or s 16t2 20t 220.
49.
3
3
A
B
x2 20x xx 20
x
x 20
50.
x8
x8
A
B
x2 3x 28 x 7x 4 x 7 x 4
51.
3x 4
3x 4
A
B
C
2
x3 5x2 x2x 5
x
x
x5
52.
Dx E
A Bx C
x2
2
2
xx2 22
x
x 2
x 22
703
704
53.
Chapter 7
Systems of Equations and Inequalities
A
B
4x
x2 6x 8 x 2 x 4
54.
A
B
x
x2 3x 2 x 1 x 2
x Ax 2 Bx 1
4 x Ax 4 Bx 2
Let x 2: 6 2A ⇒ A 3
Let x 1: 1 A
Let x 4: 8 2B ⇒ B 4
Let x 2: 2 B ⇒ B 2
x2
55.
3
4
4x
6x 8 x 2 x 4
x2
x2
2x 15
1 2
x 2x 15
x 2x 15
56.
2
Let x 3: 9 6A ⇒ A 2x 15 Ax 3 Bx 5
Let x 5: 25 8A ⇒ A x2
57.
25
8
3
2
Let x 3: 9 6B ⇒ B 1
3
3
9
x2 9 2 x 3 x 3
9
8
3
2
25
9
x2
1
2x 15
8x 5 8x 3
x2 2x
A
x2 2x
Bx C
2
x3 x2 x 1
x 1x2 1 x 1
x 1
58.
x2 2x Ax2 1 Bx Cx 1
Ax2
A
A B
x2
Bx2
3
Bx Cx C
B Cx A C
1
1
3
x3
2 x 1 x2 1
3x3 4x Ax Bx2 1 Cx D
Ax3 Bx2 A Cx B D
3A
0B
43C ⇒ C1
0BD ⇒ D0
3x
x
3x3 4x
x2 12 x2 1 x2 12
4
B
3
Let x 2:
4
8
4
A ⇒ A
3
3
3
3x3 4x
Ax B
Cx D
2
2
x2 12
x 1
x 12
Equating coefficients of like powers:
Let x 1:
4
4
4x
3x 12 3x 1 3x 12
3
2 x 2
x2 2x
2
2
3
x x2 x 1
x1
x 1
4x
A
B
3x 12 x 1 x 12
4
x Ax 1 B
3
Equating coefficients of like terms gives 1 A B,
2 B C, and 0 A C. Adding both sides of all
three equations gives 3 2A. Therefore,
A 32, B 12, and C 32.
59.
A
B
9
x2 9 x 3 x 3
9 Ax 3 Bx 3
2x 15
A
B
x 5x 3 x 5 x 3
Let x 3: 9 8B ⇒ B 1
2
x
3x 2 x 1 x 2
60.
Bx C
4x2
A
2
x 1x2 1 x 1
x 1
4x2 Ax2 1 Bx Cx 1
Ax2 A Bx2 Bx Cx C
A Bx2 B Cx A C
Equating coefficients of like terms gives 4 A B,
0 B C, and 0 A C. Adding both sides of all
three equations gives 4 2A, so A 2. Then B 2 and
C 2.
4x2
2
2x 2
x 1x2 1 x 1 x2 1
2
x 1 1 xx 11
2
Review Exercises for Chapter 7
61. y ≤ 5 12 x
y
62. 3y x ≥ 7
705
y
10
8
8
6
6
4
4
2
x
−2
−2
2
4
6
8
10
−6
−4
−4
3
x2 2
64. y ≤
y
4
Using a solid line, graph
3
y 2
, and shade
x 2
below the curve. Use 0, 0
as a test point.
2
1
−2
x
−1
1
2
3
−2
x 2y
3x y
x
y
≤
≤
≥
≥
y
4
3
3
65.
2
− 4 − 3 −2 −1
−1
−2
−3
−4
y
160
180
0
0
100
(0, 80)
(40, 60)
60
40
20
(0, 0)
(60, 0)
x
20
Vertex A
x 2y 160
3x y 180
40, 60
66.
2x 3y
2x y
x
y
≤
≤
≥
≥
Vertex B
x 2y 160
x0
0, 80
40
80
Vertex C
3x y 180
y0
60, 0
100
Vertex D
x0
y0
0, 0
y
24
16
0
0
16
12
Vertices: 0, 0, 0, 8, 6, 4, 8, 0
(0, 8)
(6, 4)
4
(0, 0)
x
4 (8, 0)
67.
3x 2y
x 2y
2 ≤ x
y
≥
≥
≤
≤
12
16
y
24
12
15
15
16
(2, 15)
(15, 15)
12
(2, 9)
8
(15, − 32 (
4
(6, 3)
x
4
Vertex A
3x 2y 24
x 2y 12
6, 3
2
−2
63. y 4x2 > 1
−3
x
−2
Vertex B
3x 2y 24
x2
2, 9
Vertex C
x2
y 15
2, 15
12
Vertex D
x 15
y 15
15, 15
Vertex E
x 2y 12
x 15
15, 32 x
1
2
3
4
706
68.
Chapter 7
2x
x
0
0
y
3y
≤ x
≤ y
≥
≥
≤
≤
Systems of Equations and Inequalities
16
18
25
25
69.
yy <> xx 11
2
Vertices:
x 1 x2 1
Vertices: 6, 4, 0, 16, 0, 25, 25, 25, 25, 0, 18, 0
0 x2 x 2 x 1x 2
y
x 1 or x 2
(25, 25)
(0, 25)
y
y 0 or y 3
15
(0, 16)
1, 0
6
5
2, 3
4
3
10
(2, 3)
2
(6, 4)
5
(18, 0)
(−1, 0)
(25, 0)
x
−4 − 3
x
5
1
2
3
4
15
−2
70. y ≤ 6 2x x2
y ≥ x6
71.
x 6 6 2x x2
Vertices:
2x 3y ≥ 0
2x y ≤ 8
y ≥ 0
y
8
6
x2 3x 0
4
xx 3 0 ⇒ x 0, 3
0, 6, 3, 3
(6, 4)
2
(0, 0)
(4, 0)
2
y
4
6
x
8
−2
8
(0, 6)
4
(− 3, 3)
2
−6
x
−2
Vertex B
2x 3y 0
y0
0, 0
Vertex A
2x 3y 0
2x y 8
6, 4
4
−2
72.
x2
y2 ≤ 9 ⇒ y2 ≤ 9 x2
x 32 y2 ≤ 9 ⇒ y2 ≤ 9 (x 32
9
x2
9 x 3
x 3 x2
0
6x 9 x2
0
Vertices:
2
x2
2
x
32,
±
33
2
( 32 , 3 2 3 (
2
x
2
−4
−6
20x 30y
12x 8y
x
y
4
( 32 , − 3 2 3 (
8
≤
≤
≥
≥
24,000
12,400
0
0
1600
1600
−400
6
−4
y number of units of Product II
− 400
y
4
3
2
73. x number of units of Product I
Vertex C
2x y 8
y0
4, 0
Review Exercises for Chapter 7
74. (a) Let x amount of Food X.,
y
(b)
(c) Answers may vary. For example,
15, 8 or 16, 9 represent acceptable quantities x, y for Foods X
and Y.
Let y amount of Food Y.
12x 15y
10x 20y
20x 12y
x
y
25
≥
≥
≥
≥
≥
300
280
300
0
0
20
10
5
x
5
75. (a)
175
10
20
25
76. (a)
Consumer Surplus
Producer Surplus
p
p
Consumer Surplus
Producer Surplus
200
p = 160 − 0.0001x
p = 130 − 0.0002x
150
150
125
(200,000, 90)
100
100
(300,000, 130)
75
50
50
p = 70 + 0.0002x
p = 30 + 0.0003x
x
x
100,000
100,000 200,000 300,000
160 0.0001x 70 0.0002x
300,000
130 0.0002x 30 0.0003x
90 0.0003x
100 0.0005x
x 300,000 units
x 200,000 units
p $130
p $90
Point of equilibrium: 300,000, 130
Point of equilibrium: 200,000, 90
(b) Consumer surplus:
1
2 300,00030
$4,500,000
1
(b) Consumer surplus: 2200,00040 $4,000,000
Producer surplus: 2300,00060 $9,000,000
1
77. Objective function: z 3x 4y
Constraints:
x
y
2x 5y
4x y
≥ 0
≥ 0
≤ 50
≤ 28
1
Producer surplus: 2200,00060 $6,000,000
78. z 10x 7y
At 0, 100: z 100 7100 700
At 25, 50: z 1025 750 600
At 75, 0: z 1075 70 750
At 0, 0: z 0
The minimum value is 600 at 25, 50.
At 0, 10: z 40
There is no maximum value.
y
At 5, 8: z 47
100
At 7, 0: z 21
The minimum value is 0 at 0, 0.
75
The maximum value is 47 at 5, 8.
50
(0, 100)
(25, 50)
25
y
(75, 0)
25
15
12
(0, 10)
(5, 8)
9
6
3
(0, 0)
3
(7, 0)
6
9
x
12
15
707
75
100
x
708
Chapter 7
Systems of Equations and Inequalities
80. z 50x 70y
79. Objective function: z 1.75x 2.25y
Constraints:
x
y
2x y
3x 2y
≥ 0
≥ 0
≥ 25
≥ 45
At 0, 25: z 56.25
At 5, 15: z 42.5
At 0, 0: z 500 700 0
y
27
24
21
18
15
12
9
6
3
At 0, 750: z 500 70750 52,500
(0, 25)
At 500, 500: z 50500 70500 60,000
(5, 15)
At 700, 0: z 50700 700 35,000
The minimum value is 0 at 0, 0.
(15, 0)
3 6
At 15, 0: z 26.25
The maximum value is 60,000 at 500, 500.
x
9 12 15 18 21 24 27
y
The minimum value is 26.25 at 15, 0.
(0, 750)
Since the region in unbounded, there is no maximum value.
600
(500, 500)
400
200
(700, 0)
(0, 0)
200
x
y
x 3y
3x 2y
≥ 0
≥ 0
≤ 12
≤ 15
At 0, 0: z 0
At 5, 0: z 25
At 2, 5: z 22 5 1
6
At 7, 0: z 27 0 14
5
(0, 4)
The minimum value is 14 at 7, 0.
(3, 3)
3
There is no maximum value.
2
1
At 3, 3: z 48
x
800
At 0, 10: z 20 10 10
y
4
600
82. z 2x y
81. Objective function: z 5x 11y
Constraints:
400
(0, 0)
y
(5, 0)
x
1
2
3
4
5
6
(0, 10)
10
At 0, 4: z 44
The minimum value is 0 at 0, 0.
6
The maximum value is 48 at 3, 3.
4
(2, 5)
2
(7, 0)
6
2
83. Let x number of haircuts
y number of permanents.
y
25
Objective function: Optimize R 25x 70y subject
to the following constraints:
20
10
x ≥ 0
y ≥ 0
70
20
60 x 60 y ≤ 24 ⇒ 2x 7y ≤ 144
At 0, 0: R 0
At 72, 0: R 1800
At 0, 144
7 : R 1440
The revenue is optimal if the student does 72 haircuts
and no permanents. The maximum revenue is $1800.
x
10
(0, 1447)
15
5
(72, 0)
(0, 0)
20
x
40
60
Review Exercises for Chapter 7
84. x number of walking shoes
y number of running shoes
Objective function: Optimize P 18x 24y subject to the following constraints:
4x 2y ≤ 24
y
x 2y ≤ 9
12
x y ≤ 8
10
x ≥ 0
6
y ≥ 0
At 0, 0: P 180 240 0
At 6, 0: P 186 240 108
( )
9
0, 2
(5, 2)
x
(0, 0)
2
(6, 0)
8
12
At 5, 2: P 185 242 138
9
9
At 0, 2 : P 190 242 108
The optimal profit of $138 occurs when 5 walking shoes and 2 running shoes are produced.
85. Let x the number of bags of Brand X, and
y the number of bags of Brand Y.
Objective function: Optimize C 15x 30y.
8x 2y
x y
Constraints: 2x 7y
x
y
≥
≥
≥
≥
≥
16
5
20
0
0
y
(0, 8)
8
(1, 4)
4
(3, 2)
2
(10, 0)
x
4
8
10
At 0, 8: C 150 308 240
At 1, 4: C 151 304 135
At 3, 2: C 153 302 105
At 10, 0: C 1510 300 150
To optimize cost, use three bags of Brand X and two bags of Brand Y. The minimum cost is $105.
86. x fraction of regular
y
y fraction of premium
Constraints:
87x 93y
x y
x
y
(0, 1)
≥ 89
1
≥ 0
≥ 0
Objective function: Minimize C 1.63x 1.83y.
1
2
( 23 , 13(
x
1
2
1
Note that the “region” defined by the constraints is actually the
line segment connecting 0, 1 and 23, 13 .
At 0, 1: C 1.630 1.831 1.83
At 23, 13 : C 1.6323 1.8313 1.70
The minimum cost is $1.70 and occurs with a mixture of 32 gallon of regular and 13 gallon of premium.
87. False. The system y ≤ 5, y ≥ 2, y ≤ 72 x 9, and y ≤ 72 x 26 represents the
region covered by an isosceles trapezoid.
709
710
Chapter 7
Systems of Equations and Inequalities
88. False. A linear programming problem either has one
optional solution or infinitely many optimal solutions.
(However, in real-life situations where the variables must
have integer values, it is possible to have exactly ten
integer-valued solutions.)
90. There are an infinite number of linear systems with the
solution 5, 4. One possible system is:
3xx yy 119
89. There are an infinite number of linear systems with the
solution 6, 8. One possible solution is:
xx yy 142
91. There are infinite linear systems with the solution 43, 3.
One possible solution is:
6x3x 3yy 71
92. There are an infinite number of linear systems with the
solution 1, 94 . One possible system is:
x3x 4y8y 2110
93. There are an infinite number of linear systems with the
solution 4, 1, 3. One possible system is as follows:
x yz6
xyz0
xyz2
94. There are an infinite number of linear systems with the
solution 3, 5, 6. One possible system is:
95. There are an infinite number of linear systems with the
solution 5, 32, 2. One possible solution is:
x 2y z 7
2x y 4z 25
x 3y z 12
96. There are an infinite number of linear systems with the
3
solution 4, 2, 8. One possible system is:
2x 2y 3z 7
x 2y z 4
x 4y z 1
97. A system of linear equations is inconsistent if it has
no solution.
4x y z 7
8x 3y 2z 16
4x 2y 3z 31
98. The lines are distinct and parallel.
99. If the solution to a system of equations is at fractional or
irrational values, then the substitution method may yield
an exact answer. The graphical method works well when
the solution is at integer values, otherwise we can usually
only approximate the solution.
2xx 2y4y 39
Problem Solving for Chapter 7
y
1. The longest side of the triangle is a diameter of the circle and has a length of 20.
The lines y 2x 5 and y 2x 20 intersect at the point 6, 8.
1
8
The distance between 10, 0 and 6, 8 is:
d1 6 10 8 0 320 85
2
2
a
b
(10, 0)
c
−8
−4
−8
− 12
Since 3202 802 202
400 400
the sides of the triangle satisfy the Pythagorean Theorem. Thus, the triangle is a right triangle.
x
4
−4
The distance between 6, 8 and 10, 0 is:
d2 10 62 0 82 80 45
(6, 8)
12
(−10, 0)
8
Problem Solving for Chapter 7
2. The system will have infinite solutions when the lines
coincide, or are identical.
2x3x k5yy 8k
1
2
711
3. The system will have exactly one solution when the slopes
of the line are not equal.
⇒ 6x 10y 16
⇒ 6x 3k1y 3k2
3k1 10 ⇒ k1 10
3
e
a
ax by e ⇒ y x b
b
c
f
cx dy f ⇒ y x d
d
a
c
b
d
3k2 16 ⇒ k2 16
3
a
c
b d
ad bc
4. (a)
5xx 4y6y 313
Eq. 1
Eq. 2
(b)
4x2x 3y6y 147
Eq.1
Eq.2
y
y
4
4
3
3
2
2
1
1
x
−2 −1
−1
1
3
4
5
6
−2 −1
−1
−2
−2
−4
−4
x 14y4y 328
5Eq.1 Eq.2
y2
x
1
2
2x 3y0 70
4
5
6
2Eq.1 Eq.2
The lines coincide. Infinite solutions.
x 42 3 ⇒ x 5
Solution: 5, 2
Let y a, then 2x 3a 7 ⇒ x y
Solution:
4
3a 7
2
3a 2 7, a
3
The solution(s) remain the same at each step of the process.
1
− 2 −1
−1
x
1
2
3
4
5
6
−2
−3
−4
5. There are a finite number of solutions.
(a) If both equations are linear,
then the maximum number of
solutions to a finite system is
one.
(b) If one equation is linear and
the other is quadratic, then the
maximum number of solutions
is two.
(c) If both equations are quadratic,
then the maximum number of
solutions to a finite system is
four.
712
Chapter 7
Systems of Equations and Inequalities
6. B total votes cast for Bush
BB KK 354,912 118,304,000
3,320,000
BB KK 117,949,088
3,320,000
K total votes cast for Kerry
N total votes cast for Nader
B K N 118,304,000
BK
3,320,000
N 0.003118,304,000
2B 121,269,088
B 60,634,544
N 354,912
K 57,314,544
Bush: 60,634,544 votes
Kerry: 57,314,544 votes
Nader: 354,912 votes
7. The point where the two sections meet is at a depth
of 10.1 feet. The distance between 0, 10.1 and
252.5, 0 is:
d 252.5 02 0 10.12 63858.26
d 252.7
8. Let C weight of a carbon atom.
Let H weight of a hydrogen atom.
6H 30.07 ⇒ 8C 24H 120.28
2C
3C 8H 44.097 ⇒ 9C 24H 132.291
C
Each section is approximately 252.7 feet long.
y
Each carbon atom weighs 12.011 u.
10
(−252.5, 0)
− 250
(252.5, 0)
− 50
50
(0, −10.1)
2x + 50y = − 505 − 20
Each hydrogen atom weighs 1.008 u.
x
250
2x − 50y = 505
9. Let x cost of the cable, per foot.
Let y cost of a connector.
15.50 ⇒ 6x 2y 15.50
6x3x 2y2y 10.25
⇒ 3x 2y 10.25
3x
5.25
x
1.75
y
2.50
For a four-foot cable with a connector on each end the cost should be 41.75 22.50 $12.00
10. Let t time that the 9:00 A.M. bus is on the road.
d
1
4
Then t time that the 9:15 A.M. bus is on the road.
40
30
d 30t
d 40t 14 20
40t 14 30t
40t 10 30t
10
t
1
−2
1
2
1
3
2
2
10t 10
t1
The 9:15 A.M. bus will catch up with the 9:00 A.M. bus in one hour. At that point both buses have traveled
30 miles and are 5 miles from the airport.
12.011
C
12.011
H
1.008
Problem Solving for Chapter 7
1
1
1
11. Let X , Y , and Z .
x
y
z
(a)
12
x
3
x
12
7 ⇒ 12X 12Y 7 ⇒ 12X 12Y 7
y
4
0 ⇒ 3X 4Y 0 ⇒ 9X 12Y 0
y
7
21X
X
1
3
Y
1
4
1 1
1
1
⇒ x 3 and ⇒ y 4.
x
3
y
4
Thus,
Solution: 3, 4
2 1 3
x
y
z
(b)
4 ⇒
2X Y 3Z 4
Eq.1
2Z 10
Eq.2
2 3 13
8 ⇒ 2X 3Y 13Z 8
x
y
z
Eq.3
4
x
2X 2X 2
10 ⇒
z
4X
Y 3Z 4
2Y 8Z 2
4Y 16Z 4
2Eq.1 Eq.2
Eq.1 Eq.3
Y 3Z 4
2Y 8Z 2
00
2Eq.2 Eq.3
The system has infinite solutions.
Let Z a, then Y 4a 1 and X Then
x
a 5
.
2
1 1
1
1
a ⇒ z
4a 1 ⇒ y z
a, y
4a 1
a 5
2
⇒ x
.
2
a 5
Solution:
a2 5, 4a 1 1, a1, a 5, 41, 0
12. Solution: 1, 2, 3
x 2y 3z a ⇒ 1 22 33 12 a
x y z b ⇒ 1 2 3 4 b
2x 3y 2z c ⇒ 21 32 23 10 c
Thus, a 12, b 4, and c 10.
713
714
Chapter 7
Systems of Equations and Inequalities
13. Solution: 1, 1, 2
4x 2 y 5 z 16
x y
0
x 3y 2z 6
Equation 1
Equation 2
Equation 3
(a) 4x 2y 5z 16
x y
0
(b)
4xx 2yy 5z 160
Interchange the equations.
x y
0
6y 5z 16
4Eq.1 Eq.2
5a 16
5a 16
Let z a, then y and x .
6
6
Solution:
5a6 16, 5a 6 16, a
x y
0
x 3y 2z 6
x y
0
2y 2z 6
Eq.1 Eq.2
Solution: c 3, c 3, c
When c 2 we have the original solution.
x1 x2 2x3
3x1 2x2 4x3
x2 x3
2x1 2x2 4x3
2x1 2x2 4x3
2x4
4x4
x4
5x4
4x4
x1 x2 2x3
x1
x2 x3
2x1 2x2 4x3
2x1 2x2 4x3
2x4 6x5 6
2
x4 3x5 3
5x4 15x5 10
4x4 13x5 13
6x5
12x5
3x5
15x5
13x5
6
14
3
10
13
2Eq.1 Eq.2
x1 x2
0
x1
2
x2 x3 x4 3x5 3
2x1 2x2 4x3 5x4 15x5 10
2x1 2x2 4x3 4x4 13x5 13
Eq.1 2Eq.3
2
2
x2 x3 x4 3x5 3
2x1 2x2 4x3 5x4 15x5 10
2x1 2x2 4x3 4x4 13x5 13
Eq.1 Eq.2
x2
x1
—CONTINUED—
x 14y3y 13z2z 406
Let z b, then y Interchange the equations.
4Eq.1 Eq.2
11b 36
13b 40
and x 14
14
11b14 36, 13b14 40, b
When b 2 we have the original solution.
(d) Each of these systems has infinite solutions.
Let z c, then y c 3 and x c 3
14.
x4x 2y3y 5z2z 166
Solution:
When a 2 we have the original solution.
(c)
x4x 2y3y 5z2z 166
Problem Solving for Chapter 7
715
14. —CONTINUED—
Substitute into the subsequent equations and simplify:
x1
x2
2 x3 x4 3x5
22 22 4x3 5x4 15x5
22 22 4x3 4x4 13x5
x1
x2
2
2
3
10
13
2
2
x3 x4 3x5 1
4x3 5x4 15x5 2
4x3 4x4 13x5 5
2
2
x3 x4 3x5 1
x4 3x5 2
x5 1
x1
x2
x1
x2
x3
x4
x5
2
2
3
5
1
Eq.3
Eq.4 4Eq.3
Eq.5 4Eq.3
Eq.3 Eq.4
Eq.4 3Eq.5
16. x number of inches by which a person’s height exceeds
4 feet 10 inches
15. t amount of terrestrial vegetation in kilograms
a amount of aquatic vegetation in kilograms
a t ≤ 32
y person’s weight in pounds
t
(a)
0.15a ≥ 1.9
30
193a 4193t ≥ 11,000
25
20
(c) For someone 6 feet tall,
x 14 inches.
10
5
−5
−5
17. (a) x HDL cholesterol (good)
y 91 3.7x
y ≤ 119 4.8x
x ≥ 0, y ≥ 0
(b)
300
0
45
0
Minimum weight: 91 3.714 142.8 pounds
a
5 10 15 20 25 30
Maximum weight: 119 4.814 186.2 pounds
(c) y 120 is in the region since 0 < y < 130.
y LDL cholesterol (bad)
x 90 is in the region since 35 < x < 200.
x y 210 is not in the region since x y < 200.
0 < y < 130
x ≥ 35
x y ≤ 200
(b)
(d) If the LDL reading is 150 and the HDL reading is 40, then
x ≥ 35 and x y ≤ 200 but y < 130.
y
(e)
250
x y < 4x
200
(70, 130)
150
y < 3x
100
50
xy
< 4
x
The point 50, 120 is in the region and 120 < 350.
(35,130)
x
50
100
150
250
716
Chapter 7
Systems of Equations and Inequalities
Chapter 7
Practice Test
For Exercises 1–3, solve the given system by the method of substitution.
1. 3x y 1
3x y 15
2.
x 3y 3
x 6y 2
3.
5
x y z
6
2x y 3z 0
5x 2y z 3
4. Find the two numbers whose sum is 110 and product is 2800.
5. Find the dimensions of a rectangle if its perimeter is 170 feet and its area is
1500 square feet.
For Exercises 6–8, solve the linear system by elimination.
6. 2x 15y 4
x
7.
3y 23
38x 19y 7
x
y2
8. 0.4x 0.5y 0.112
0.3x 0.7y 0.131
9. Herbert invests $17,000 in two funds that pay 11% and 13% simple interest, respectively.
If he receives $2080 in yearly interest, how much is invested in each fund?
10. Find the least squares regression line for the points 4, 3, 1, 1, 1, 2,
and 2, 1.
For Exercises 11–12, solve the system of equations.
11.
xy
2
2x y z 11
12. 3x 2y z 5
6x y 5z 2
4y 3z 20
13. Find the equation of the parabola y ax2 bx c passing through the points
0, 1, 1, 4 and 2, 13.
Practice Test for Chapter 7
For Exercises 14–15, write the partial fraction decomposition of the rational functions.
14.
10x 17
x2 7x 8
15.
x2 4
x 4 x2
16. Graph x2 y2 ≥ 9.
17. Graph the solution of the system.
xy ≤ 6
x ≥ 2
y ≥ 0
18. Derive a set of inequalities to describe the triangle with vertices 0, 0, 0, 7,
and 2, 3.
19. Find the maximum value of the objective function, z 30x 26y, subject to the following constraints.
x ≥ 0
y ≥ 0
2x 3y ≤ 21
5x 3y ≤ 30
20. Graph the system of inequalities.
x2 y2 ≤ 4
x 2 y
2
2
≥ 4
For Exercises 21–22, write the partial fraction decomposition for the rational expression.
21.
1 2x
x2 x
22.
6x 17
x 32
717