SOLUTION MANUAL
Finite Mathematics & Its Applications
13th Edition by Larry J. Goldstein,
Chapters 1 - 12, Complete
, Contents
Chapter 1: Linear Equations and Straight Lines 1–1
Chapter 2: Matrices 2–1
Chapter 3: Linear Programming, A Geometric Approach 3–1
Chapter 4: The Simplex Method 4–1
Chapter 5: Sets and Counting 5–1
Chapter 6: Probability 6–1
Chapter 7: Probability and Statistics 7–1
Chapter 8: Markov Processes 8–1
Chapter 9: The Theory of Games 9–1
Chapter 10: The Mathematics of Finance 10–1
Chapter 11: Logic 11–1
Chapter 12: Difference Equations and Mathematical Models 12–1
, Chapter 1
Exercises 1.1 5
6. Left 1, down
2
1. Right 2, up 3 y
y
(2, 3)
x
x
( )
–1, – 52
7. Left 20, up 40
2. Left 1, up 4 y
y
(–20, 40)
(–1, 4)
x
x
8. Right 25, up 30
3. Down 2 y
y
(25, 30)
x
x
(0, –2)
9. Point Q is 2 units to the left and 2 units up or
4. Right 2
y (—2, 2).
10. Point P is 3 units to the right and 2 units down or
(3,—2).
x
(2, 0) 1
11. —2(1) + (3) = —2 +1 = —1so yes the point is
3
on the line.
5. Left 2, up 1 1
y 12. —2(2) + (6) = —1 is false, so no the point is not
3
on the line
(–2, 1)
x
Copyright © 2023 Pearson Education, Inc. 1-1
,Chapter 1: Linear Equations and Straight Lines ISM: Finite Math
1 24. 0 = 5
13 —2x + y = —1 Substitute the x and y no solution
3
. x-intercept: none
coordinates of the point into the equation:
f 1 hı f h When x = 0, y = 5
' , 3 → —2 ' 1 ı + 1 (3) = —1 → —1+1 = —1 is y-intercept: (0, 5)
y' ı 'y ıJ
2 J 2 3
a false statement. So no the point is not on 25. When y = 0, x = 7
theline. x-intercept: (7, 0)
f 1h f1 h 0=7
14 —2 ' ı + ' ı (—1) = —1 is true so yes the point is no solution
.
'y3 ıJ 'y3 ıJ y-intercept: none
on the line. 26. 0 = –8x
15. m = 5, b = 8 x=0
x-intercept: (0, 0)
16. m = –2 and b = –6 y = –8(0)
y=0
17. y = 0x + 3; m = 0, b = 3 y-intercept: (0, 0)
2 2 1
y = x + 0; m = , b = 0 27 0 = x –1
18 3
3 3 .
. x=3
19. 14x + 7 y = 21 x-intercept: (3, 0)
1
7 y = —14x + 21 y = (0) – 1
3
y = —2x + 3
y = –1
y-intercept: (0, –1)
20 x—y =3 y
. —y = —x + 3
y = x —3
(3, 0)
21. 3x = 5 x
5 (0, –1)
x=
3
1 2
28. When x = 0, y = 0.
22 – x+ y = 10
. 2 3 When x = 1, y = 2.
2 1 y
y= x +10
3 2
3
y = x +15 (1, 2)
4 x
(0, 0)
23. 0 = —4x + 8
4x = 8
x=2
x-intercept: (2, 0)
y = –4(0) + 8
y=8
y-intercept: (0, 8)
1-2 Copyright © 2023 Pearson Education, Inc.
,ISM: Finite Math Chapter 1: Linear Equations and Straight Lines
5 32 x +0 = 3
29 0=
2 . x =3
.
no solution x-intercept: (3, 0)
x-intercept: 0+ y =3
none
5 y =3
When x = 0, y =
y-intercept: (0, 3)
f 5 h2 y
y-intercept: '0, ı
'y 2 ıJ
y
(0, 3)
(3, 0) x
( 0, 52)
x
5
33 x =—
. 2
30. The line coincides with the y-axis.
y
x=0
x
1 1
34. x — (0) = —1
2 3
31. 3x + 4(0) = 24 x = —2
x =8 x intercept (–2, 0)
x-intercept: (8, 0) 1 1
(0)— y = —1
3(0) + 4 y = 24 2 3
y =6 y =3
y-intercept: (0, 6) y intercept (0, 3)
y
(0, 6)
(8, 0)
x
,Chapter 1: Linear Equations and Straight Lines ISM: Finite Math
35. 2x + 3y = 6 1
36. x –5 y =1
3y = —2x + 6 2 1
2 –5 y = +1
y =— x + 2 – x
3 2
1 1
y= x–
a. 4x + 6 y = 12 10 5
6 y = —4x +12 1
2 a. =1
y =— x +2 2x – y
3 5
1
Yes — y = —2x +1
5
b. Yes y = 10x — 5
No
3
c. x = 3— y
2 b. x = 5 y + 25
3 y = x —2
y = —x + 3 1 2
2 y = x—
2
y =— x +2 5 5
3 No
2
y=– x +2
3 c. 2 —5x +10 y = 0
Yes —10 y = —5x + 2
1 1
d. 6 — 2x — y = 0 y = x—
2 5
y = 6 — 2x = —2x + 6 No
No
d. y = .1(x — 2)
2 2
e. y = 2– x = – x +2 y = .1x —.2
3 3 1 1
Yes y = x—
10 5
f. x + y =1 Yes
y = —x +1 e. 10 y — x = —2
No
10 y = x — 2
1 1
y = x—
10 5
Yes
f. 1+.5x = 2 + 5 y
5 y = .5x —1
1 1
y = x—
10 5
Yes
1-4 Copyright © 2023 Pearson Education, Inc.
,ISM: Finite Math Chapter 1: Linear Equations and Straight Lines
37. a. x+ y =3 c. 2004f 1—1960
h = 44
y = —x + 3 y = ' ı (44) + 70
' ı
m = –1, b = 3 y6 J
L3 y = 7.33 + 70
y = 77.33
b. 2x — y = —2 A person born in 2004 has a life expectancy
—y = —2x — 2 of 77.3 years.
y = 2x + 2
m = 2, b = 2 f' 1 hı
41. a. x-intercept: '161 , 0ı
L1 y 9 J
y-intercept: (0, 5.8)
c. x = 3y + 33y
= x —3
1
y = x —1
3
1
m = , b = –1
3
L2
b. In 2004, 5.8 trillion cigarettes were sold.
38. a. No; 5 + 4 ≠ 3
c. 5.5 = –.036x + 5.8
b. No; 2 ≠ 1 – 1
x = 8.33
c. Yes; 2(2) = 1 + 3 and 2(4) = 5 + 3 2004 + 8 = 2012
y = 30x + 72 d. 2028 – 2004 = 24
39
y = –.036(24) + 5.8
.
a. When x = 0, y = 72. This is the temperature y = 4.936
of the water at time = 0 before the kettle 4.9 trillion
is turned on.
42. a. x-intercept: (–12.17, 0)
y-intercept: (0, 14)
b. y = 30(3) + 72
y =162∘ F
c. Water boils when y = 212 so we have
212 = 30x + 72. Solving for x gives
x =4 32 minutes or 4 minutes 40 seconds.
40. a. A person born in 1960 has a life b. In 2010 the income from ecotourism was
expectancyof 70 years. $14,000.
f 1h
b. 75 = ' ı x + 70 c. 20 = 1.15x + 14
' ı x ≈ 5.22
y6 J
f 1 ıh 2010 + 5.22 = 2015.22
5= x'
The year 2015 should have had $20,000 in
'y6 Jı ecotourism income.
x = 30
1960 + 30 = 1990
,Chapter 1: Linear Equations and Straight Lines ISM: Finite Math
d. 2032 – 2010 = 22 c. 1180 = 30x +1000
y = 1.15(22) + 14 180 = 30x
y = 39.3
x =6
$39,300
The balance will be $1180 after 6 years.
43. a. x-intercept: (–18.5, 0)
45. a. In 2010, 6.1% of entering college freshmen
y-intercept: (0, 869)
intended to major in biology.
b. 2019 — 2010 = 9
y = 1 6(9) + 6.1
y = 7.6
7.6% of college freshmen in 2019
intendedto major in biology.
c. 6.8 = 1 x + 6.1
6
b. In 2014 the car insurance rate for a small car 0.7 = 16xx
was $869. = 4.2
2010 + 4 = 2014
c. 2017 – 2014 = 3 In 2014, the percent of college freshmen
y = 47(3) + 869 who intended to major in biology was 6.8%.
y = 1010
$1010 46. a. In 2010, 46.4% of college freshmen
considered themselves middle-of-the-
d. 1480 = 47x + 869 roadpolitically.
611 = 47x
x = 13 b. 2016 — 2010 = 6
2014 +13 = 2027 y = 46.4 — 0.31(6)
The yearly rate will be $1480 in 2027. y = 44.54
f 100 , 0hı
44. a. x-intercept: ''— 44.5% of college freshmen considered
y 3 ıJ themselves middle-of-the-road politically in
y-intercept: (0, 1000) 2016.
c. 43.9 = 46.4 — 0.31x
—2.5 = —0.31x
x ≈8
2010 + 8 = 2018
In 2018, the percent of college freshmen that
considered themselves middle-of-the-
roadwas 43.9%.
47. a. 2018— 2012 = 6
b. y = 30(2) +1000 y = 433(6) + 21, 593
y = 60 +1000 y = 24,191
y = 1060 $24,191 was the approximate average
$1060 will be in the account after 2 years. tuitionin 2018.
1-6 Copyright © 2023 Pearson Education, Inc.
,ISM: Finite Math Chapter 1: Linear Equations and Straight Lines
b. 28, 600 = 433x + 21, 593 52. Since the equation is parallel to the y axis, it will
7007 = 433x be in the form x = a. Therefore, the equation
willbe x = 5.
x ≈ 16.2
2012 +16 = 2028 53. On the x-axis, y = 0.
In 2028, the approximate average cost of
tuition will be more than $28,600. 54. No, because two straight lines (the graphed
line and the x-axis) cannot intersect more than
48. a. 2015— 2011 = 4 once.
y = 1121(4) +17,182
y = 21, 666 55. The equation of a line parallel to the y axis will
Approximately 21,666 bachelor’s degrees in be in the form x = a.
mathematics and statistics were awarded in
2015. 56. y = b is an equation of a line parallel to the x-
axis.
b. 34, 000 = 1121x +17,182 57. 2x — y = —3
16,818 = 1121x
x ≈15.002 58. —3x + y = —4
2011+15 = 2026 2
In 2026, there will be more than 34,000 59. x + y = —5
bachelor’s degrees in mathematics 3
andstatistics awarded. 2x + 3y = —15
5
60 4x — y =
49 y = mx + b 6
.
. 8 = m(0) + b 24x — 6 y = 5
b =8
0 = m(16) + 8 0) and (0,b) are points on the line the slope of
61. the line is (b-0)/(0-a) = -b/a. Since the y
e
(
a
,
1 intercept is (0,b), the equation of the line is
m =—
2 y = —(b / a)x + b or ay = —bx + ab. In general
1 form, the equation is bx + ay = ab.
y =— x + 8
2
62. If (5, 0) and (0, 6) are on the line, then a = 5 and
50 y = mx + b b = 6. Substituting these values into the
. equationbx + ay = ab gives 6x + 5y = 30.
0.9 = m(0) + b
b = 0.9 66. One possible equation
63. One possible equation
0 = m(0.6) + 0.9 is
m = —1.5 67. One possible equation
y = —1.5x + 0.9 64. One possible equation
is
51. y = mx + b
5 = m(0) + b
b=5 65. One possible equation
0 = m(4) + 5 is
5
, Chapter
y 1: Linear Equations and Straight Lines 9. y = x yISM:
= x + 2.Finite Math
= +10.
x y = x + 7.
— y = x — 6.
m=–
4
5 68. One possible equation y = x.
y =– x +5 is
4 y = x +9.
69. One possible equation
is
1-8 Copyright © 2023 Pearson Education, Inc.
Finite Mathematics & Its Applications
13th Edition by Larry J. Goldstein,
Chapters 1 - 12, Complete
, Contents
Chapter 1: Linear Equations and Straight Lines 1–1
Chapter 2: Matrices 2–1
Chapter 3: Linear Programming, A Geometric Approach 3–1
Chapter 4: The Simplex Method 4–1
Chapter 5: Sets and Counting 5–1
Chapter 6: Probability 6–1
Chapter 7: Probability and Statistics 7–1
Chapter 8: Markov Processes 8–1
Chapter 9: The Theory of Games 9–1
Chapter 10: The Mathematics of Finance 10–1
Chapter 11: Logic 11–1
Chapter 12: Difference Equations and Mathematical Models 12–1
, Chapter 1
Exercises 1.1 5
6. Left 1, down
2
1. Right 2, up 3 y
y
(2, 3)
x
x
( )
–1, – 52
7. Left 20, up 40
2. Left 1, up 4 y
y
(–20, 40)
(–1, 4)
x
x
8. Right 25, up 30
3. Down 2 y
y
(25, 30)
x
x
(0, –2)
9. Point Q is 2 units to the left and 2 units up or
4. Right 2
y (—2, 2).
10. Point P is 3 units to the right and 2 units down or
(3,—2).
x
(2, 0) 1
11. —2(1) + (3) = —2 +1 = —1so yes the point is
3
on the line.
5. Left 2, up 1 1
y 12. —2(2) + (6) = —1 is false, so no the point is not
3
on the line
(–2, 1)
x
Copyright © 2023 Pearson Education, Inc. 1-1
,Chapter 1: Linear Equations and Straight Lines ISM: Finite Math
1 24. 0 = 5
13 —2x + y = —1 Substitute the x and y no solution
3
. x-intercept: none
coordinates of the point into the equation:
f 1 hı f h When x = 0, y = 5
' , 3 → —2 ' 1 ı + 1 (3) = —1 → —1+1 = —1 is y-intercept: (0, 5)
y' ı 'y ıJ
2 J 2 3
a false statement. So no the point is not on 25. When y = 0, x = 7
theline. x-intercept: (7, 0)
f 1h f1 h 0=7
14 —2 ' ı + ' ı (—1) = —1 is true so yes the point is no solution
.
'y3 ıJ 'y3 ıJ y-intercept: none
on the line. 26. 0 = –8x
15. m = 5, b = 8 x=0
x-intercept: (0, 0)
16. m = –2 and b = –6 y = –8(0)
y=0
17. y = 0x + 3; m = 0, b = 3 y-intercept: (0, 0)
2 2 1
y = x + 0; m = , b = 0 27 0 = x –1
18 3
3 3 .
. x=3
19. 14x + 7 y = 21 x-intercept: (3, 0)
1
7 y = —14x + 21 y = (0) – 1
3
y = —2x + 3
y = –1
y-intercept: (0, –1)
20 x—y =3 y
. —y = —x + 3
y = x —3
(3, 0)
21. 3x = 5 x
5 (0, –1)
x=
3
1 2
28. When x = 0, y = 0.
22 – x+ y = 10
. 2 3 When x = 1, y = 2.
2 1 y
y= x +10
3 2
3
y = x +15 (1, 2)
4 x
(0, 0)
23. 0 = —4x + 8
4x = 8
x=2
x-intercept: (2, 0)
y = –4(0) + 8
y=8
y-intercept: (0, 8)
1-2 Copyright © 2023 Pearson Education, Inc.
,ISM: Finite Math Chapter 1: Linear Equations and Straight Lines
5 32 x +0 = 3
29 0=
2 . x =3
.
no solution x-intercept: (3, 0)
x-intercept: 0+ y =3
none
5 y =3
When x = 0, y =
y-intercept: (0, 3)
f 5 h2 y
y-intercept: '0, ı
'y 2 ıJ
y
(0, 3)
(3, 0) x
( 0, 52)
x
5
33 x =—
. 2
30. The line coincides with the y-axis.
y
x=0
x
1 1
34. x — (0) = —1
2 3
31. 3x + 4(0) = 24 x = —2
x =8 x intercept (–2, 0)
x-intercept: (8, 0) 1 1
(0)— y = —1
3(0) + 4 y = 24 2 3
y =6 y =3
y-intercept: (0, 6) y intercept (0, 3)
y
(0, 6)
(8, 0)
x
,Chapter 1: Linear Equations and Straight Lines ISM: Finite Math
35. 2x + 3y = 6 1
36. x –5 y =1
3y = —2x + 6 2 1
2 –5 y = +1
y =— x + 2 – x
3 2
1 1
y= x–
a. 4x + 6 y = 12 10 5
6 y = —4x +12 1
2 a. =1
y =— x +2 2x – y
3 5
1
Yes — y = —2x +1
5
b. Yes y = 10x — 5
No
3
c. x = 3— y
2 b. x = 5 y + 25
3 y = x —2
y = —x + 3 1 2
2 y = x—
2
y =— x +2 5 5
3 No
2
y=– x +2
3 c. 2 —5x +10 y = 0
Yes —10 y = —5x + 2
1 1
d. 6 — 2x — y = 0 y = x—
2 5
y = 6 — 2x = —2x + 6 No
No
d. y = .1(x — 2)
2 2
e. y = 2– x = – x +2 y = .1x —.2
3 3 1 1
Yes y = x—
10 5
f. x + y =1 Yes
y = —x +1 e. 10 y — x = —2
No
10 y = x — 2
1 1
y = x—
10 5
Yes
f. 1+.5x = 2 + 5 y
5 y = .5x —1
1 1
y = x—
10 5
Yes
1-4 Copyright © 2023 Pearson Education, Inc.
,ISM: Finite Math Chapter 1: Linear Equations and Straight Lines
37. a. x+ y =3 c. 2004f 1—1960
h = 44
y = —x + 3 y = ' ı (44) + 70
' ı
m = –1, b = 3 y6 J
L3 y = 7.33 + 70
y = 77.33
b. 2x — y = —2 A person born in 2004 has a life expectancy
—y = —2x — 2 of 77.3 years.
y = 2x + 2
m = 2, b = 2 f' 1 hı
41. a. x-intercept: '161 , 0ı
L1 y 9 J
y-intercept: (0, 5.8)
c. x = 3y + 33y
= x —3
1
y = x —1
3
1
m = , b = –1
3
L2
b. In 2004, 5.8 trillion cigarettes were sold.
38. a. No; 5 + 4 ≠ 3
c. 5.5 = –.036x + 5.8
b. No; 2 ≠ 1 – 1
x = 8.33
c. Yes; 2(2) = 1 + 3 and 2(4) = 5 + 3 2004 + 8 = 2012
y = 30x + 72 d. 2028 – 2004 = 24
39
y = –.036(24) + 5.8
.
a. When x = 0, y = 72. This is the temperature y = 4.936
of the water at time = 0 before the kettle 4.9 trillion
is turned on.
42. a. x-intercept: (–12.17, 0)
y-intercept: (0, 14)
b. y = 30(3) + 72
y =162∘ F
c. Water boils when y = 212 so we have
212 = 30x + 72. Solving for x gives
x =4 32 minutes or 4 minutes 40 seconds.
40. a. A person born in 1960 has a life b. In 2010 the income from ecotourism was
expectancyof 70 years. $14,000.
f 1h
b. 75 = ' ı x + 70 c. 20 = 1.15x + 14
' ı x ≈ 5.22
y6 J
f 1 ıh 2010 + 5.22 = 2015.22
5= x'
The year 2015 should have had $20,000 in
'y6 Jı ecotourism income.
x = 30
1960 + 30 = 1990
,Chapter 1: Linear Equations and Straight Lines ISM: Finite Math
d. 2032 – 2010 = 22 c. 1180 = 30x +1000
y = 1.15(22) + 14 180 = 30x
y = 39.3
x =6
$39,300
The balance will be $1180 after 6 years.
43. a. x-intercept: (–18.5, 0)
45. a. In 2010, 6.1% of entering college freshmen
y-intercept: (0, 869)
intended to major in biology.
b. 2019 — 2010 = 9
y = 1 6(9) + 6.1
y = 7.6
7.6% of college freshmen in 2019
intendedto major in biology.
c. 6.8 = 1 x + 6.1
6
b. In 2014 the car insurance rate for a small car 0.7 = 16xx
was $869. = 4.2
2010 + 4 = 2014
c. 2017 – 2014 = 3 In 2014, the percent of college freshmen
y = 47(3) + 869 who intended to major in biology was 6.8%.
y = 1010
$1010 46. a. In 2010, 46.4% of college freshmen
considered themselves middle-of-the-
d. 1480 = 47x + 869 roadpolitically.
611 = 47x
x = 13 b. 2016 — 2010 = 6
2014 +13 = 2027 y = 46.4 — 0.31(6)
The yearly rate will be $1480 in 2027. y = 44.54
f 100 , 0hı
44. a. x-intercept: ''— 44.5% of college freshmen considered
y 3 ıJ themselves middle-of-the-road politically in
y-intercept: (0, 1000) 2016.
c. 43.9 = 46.4 — 0.31x
—2.5 = —0.31x
x ≈8
2010 + 8 = 2018
In 2018, the percent of college freshmen that
considered themselves middle-of-the-
roadwas 43.9%.
47. a. 2018— 2012 = 6
b. y = 30(2) +1000 y = 433(6) + 21, 593
y = 60 +1000 y = 24,191
y = 1060 $24,191 was the approximate average
$1060 will be in the account after 2 years. tuitionin 2018.
1-6 Copyright © 2023 Pearson Education, Inc.
,ISM: Finite Math Chapter 1: Linear Equations and Straight Lines
b. 28, 600 = 433x + 21, 593 52. Since the equation is parallel to the y axis, it will
7007 = 433x be in the form x = a. Therefore, the equation
willbe x = 5.
x ≈ 16.2
2012 +16 = 2028 53. On the x-axis, y = 0.
In 2028, the approximate average cost of
tuition will be more than $28,600. 54. No, because two straight lines (the graphed
line and the x-axis) cannot intersect more than
48. a. 2015— 2011 = 4 once.
y = 1121(4) +17,182
y = 21, 666 55. The equation of a line parallel to the y axis will
Approximately 21,666 bachelor’s degrees in be in the form x = a.
mathematics and statistics were awarded in
2015. 56. y = b is an equation of a line parallel to the x-
axis.
b. 34, 000 = 1121x +17,182 57. 2x — y = —3
16,818 = 1121x
x ≈15.002 58. —3x + y = —4
2011+15 = 2026 2
In 2026, there will be more than 34,000 59. x + y = —5
bachelor’s degrees in mathematics 3
andstatistics awarded. 2x + 3y = —15
5
60 4x — y =
49 y = mx + b 6
.
. 8 = m(0) + b 24x — 6 y = 5
b =8
0 = m(16) + 8 0) and (0,b) are points on the line the slope of
61. the line is (b-0)/(0-a) = -b/a. Since the y
e
(
a
,
1 intercept is (0,b), the equation of the line is
m =—
2 y = —(b / a)x + b or ay = —bx + ab. In general
1 form, the equation is bx + ay = ab.
y =— x + 8
2
62. If (5, 0) and (0, 6) are on the line, then a = 5 and
50 y = mx + b b = 6. Substituting these values into the
. equationbx + ay = ab gives 6x + 5y = 30.
0.9 = m(0) + b
b = 0.9 66. One possible equation
63. One possible equation
0 = m(0.6) + 0.9 is
m = —1.5 67. One possible equation
y = —1.5x + 0.9 64. One possible equation
is
51. y = mx + b
5 = m(0) + b
b=5 65. One possible equation
0 = m(4) + 5 is
5
, Chapter
y 1: Linear Equations and Straight Lines 9. y = x yISM:
= x + 2.Finite Math
= +10.
x y = x + 7.
— y = x — 6.
m=–
4
5 68. One possible equation y = x.
y =– x +5 is
4 y = x +9.
69. One possible equation
is
1-8 Copyright © 2023 Pearson Education, Inc.
