Sumário

Preface
1Functions
2Linear Functions
3Polynomial and Rational Functions
4Exponential and Logarithmic Functions
5Trigonometric Functions
6Periodic Functions
7Trigonometric Identities and Equations
8Further Applications of Trigonometry
9Systems of Equations and Inequalities
10Analytic Geometry
11Sequences, Probability and Counting Theory
12Introduction to Calculus
ABasic Functions and Identities
Answer Key
Index

Review Exercises

Systems of Linear Equations: Two Variables

For the following exercises, determine whether the ordered pair is a solution to the system of equations.

$\begin{array}{l} 3 x - y = 4 \\ x + 4 y = - 3 \end{array} \begin{array}{l} 3 x - y = 4 \\ x + 4 y = - 3 \end{array}$ and $(- 1, 1) (- 1, 1)$

$\begin{array}{l} 6 x - 2 y = 24 \\ - 3 x + 3 y = 18 \end{array} \begin{array}{l} 6 x - 2 y = 24 \\ - 3 x + 3 y = 18 \end{array}$ and $(9, 15) (9, 15)$

For the following exercises, use substitution to solve the system of equations.

$\begin{array}{l} 10 x + 5 y = −5 \\ 3 x - 2 y = −12 \end{array} \begin{array}{l} 10 x + 5 y = −5 \\ 3 x - 2 y = −12 \end{array}$

$\begin{array}{l} \frac{4}{7} x + \frac{1}{5} y = \frac{43}{70} \\ \frac{5}{6} x - \frac{1}{3} y = - \frac{2}{3} \end{array} \begin{array}{l} \frac{4}{7} x + \frac{1}{5} y = \frac{43}{70} \\ \frac{5}{6} x - \frac{1}{3} y = - \frac{2}{3} \end{array}$

$\begin{array}{l} 5 x + 6 y = 14 \\ 4 x + 8 y = 8 \end{array} \begin{array}{l} 5 x + 6 y = 14 \\ 4 x + 8 y = 8 \end{array}$

For the following exercises, use addition to solve the system of equations.

$\begin{array}{l} 3 x + 2 y = −7 \\ 2 x + 4 y = 6 \end{array} \begin{array}{l} 3 x + 2 y = −7 \\ 2 x + 4 y = 6 \end{array}$

$\begin{array}{r} 3 x + 4 y = 2 \\ 9 x + 12 y = 3 \end{array} \begin{array}{r} 3 x + 4 y = 2 \\ 9 x + 12 y = 3 \end{array}$

$\begin{array}{l} 8 x + 4 y = 2 \\ 6 x - 5 y = 0.7 \end{array} \begin{array}{l} 8 x + 4 y = 2 \\ 6 x - 5 y = 0.7 \end{array}$

For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

A factory has a cost of production $C (x) = 150 x + 15, 000 C (x) = 150 x + 15, 000$ and a revenue function $R (x) = 200 x . R (x) = 200 x .$ What is the break-even point?

10.

A performer charges $C (x) = 50 x + 10, 000, C (x) = 50 x + 10, 000,$ where $x x$ is the total number of attendees at a show. The venue charges $75 per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?

Systems of Linear Equations: Three Variables

For the following exercises, solve the system of three equations using substitution or addition.

11.

$\begin{array}{l} 0.5 x - 0.5 y = 10 \\ - 0.2 y + 0.2 x = 4 \\ 0.1 x + 0.1 z = 2 \end{array} \begin{array}{l} 0.5 x - 0.5 y = 10 \\ - 0.2 y + 0.2 x = 4 \\ 0.1 x + 0.1 z = 2 \end{array}$

12.

$\begin{array}{r} 5 x + 3 y - z = 5 \\ 3 x - 2 y + 4 z = 13 \\ 4 x + 3 y + 5 z = 22 \end{array} \begin{array}{r} 5 x + 3 y - z = 5 \\ 3 x - 2 y + 4 z = 13 \\ 4 x + 3 y + 5 z = 22 \end{array}$

13.

$\begin{array}{r} x + y + z = 1 \\ 2 x + 2 y + 2 z = 1 \\ 3 x + 3 y = 2 \end{array} \begin{array}{r} x + y + z = 1 \\ 2 x + 2 y + 2 z = 1 \\ 3 x + 3 y = 2 \end{array}$

14.

$\begin{array}{l} 2 x - 3 y + z = −1 \\ x + y + z = −4 \\ 4 x + 2 y - 3 z = 33 \end{array} \begin{array}{l} 2 x - 3 y + z = −1 \\ x + y + z = −4 \\ 4 x + 2 y - 3 z = 33 \end{array}$

15.

$\begin{array}{l} 3 x + 2 y - z = −10 \\ x - y + 2 z = 7 \\ - x + 3 y + z = −2 \end{array} \begin{array}{l} 3 x + 2 y - z = −10 \\ x - y + 2 z = 7 \\ - x + 3 y + z = −2 \end{array}$

16.

$\begin{array}{r} 3 x + 4 z = −11 \\ x - 2 y = 5 \\ 4 y - z = −10 \end{array} \begin{array}{r} 3 x + 4 z = −11 \\ x - 2 y = 5 \\ 4 y - z = −10 \end{array}$

17.

$\begin{array}{r} 2 x - 3 y + z = 0 \\ 2 x + 4 y - 3 z = 0 \\ 6 x - 2 y - z = 0 \end{array} \begin{array}{r} 2 x - 3 y + z = 0 \\ 2 x + 4 y - 3 z = 0 \\ 6 x - 2 y - z = 0 \end{array}$

18.

$\begin{array}{r} 6 x - 4 y - 2 z = 2 \\ 3 x + 2 y - 5 z = 4 \\ 6 y - 7 z = 5 \end{array} \begin{array}{r} 6 x - 4 y - 2 z = 2 \\ 3 x + 2 y - 5 z = 4 \\ 6 y - 7 z = 5 \end{array}$

For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

19.

Three odd numbers sum up to 61. The smaller is one-third the larger and the middle number is 16 less than the larger. What are the three numbers?

20.

A local theatre sells out for their show. They sell all 500 tickets for a total purse of $8,070.00. The tickets were priced at $15 for students, $12 for children, and $18 for adults. If the band sold three times as many adult tickets as children’s tickets, how many of each type was sold?

Systems of Nonlinear Equations and Inequalities: Two Variables

For the following exercises, solve the system of nonlinear equations.

21.

$\begin{array}{l} y = x^{2} - 7 \\ y = 5 x - 13 \end{array} \begin{array}{l} y = x^{2} - 7 \\ y = 5 x - 13 \end{array}$

22.

$\begin{array}{l} y = x^{2} - 4 \\ y = 5 x + 10 \end{array} \begin{array}{l} y = x^{2} - 4 \\ y = 5 x + 10 \end{array}$

23.

$\begin{array}{l} x^{2} + y^{2} = 16 \\ y = x - 8 \end{array} \begin{array}{l} x^{2} + y^{2} = 16 \\ y = x - 8 \end{array}$

24.

$\begin{array}{l} x^{2} + y^{2} = 25 \\ y = x^{2} + 5 \end{array} \begin{array}{l} x^{2} + y^{2} = 25 \\ y = x^{2} + 5 \end{array}$

25.

$\begin{array}{r} x^{2} + y^{2} = 4 \\ y - x^{2} = 3 \end{array} \begin{array}{r} x^{2} + y^{2} = 4 \\ y - x^{2} = 3 \end{array}$

For the following exercises, graph the inequality.

26.

$y > x^{2} - 1 y > x^{2} - 1$

27.

$\frac{1}{4} x^{2} + y^{2} < 4 \frac{1}{4} x^{2} + y^{2} < 4$

For the following exercises, graph the system of inequalities.

28.

$\begin{array}{l} x^{2} + y^{2} + 2 x < 3 \\ y > - x^{2} - 3 \end{array} \begin{array}{l} x^{2} + y^{2} + 2 x < 3 \\ y > - x^{2} - 3 \end{array}$

29.

$\begin{array}{l} x^{2} - 2 x + y^{2} - 4 x < 4 \\ y < - x + 4 \end{array} \begin{array}{l} x^{2} - 2 x + y^{2} - 4 x < 4 \\ y < - x + 4 \end{array}$

30.

$\begin{array}{l} x^{2} + y^{2} < 1 \\ y^{2} < x \end{array} \begin{array}{l} x^{2} + y^{2} < 1 \\ y^{2} < x \end{array}$

Partial Fractions

For the following exercises, decompose into partial fractions.

31.

$\frac{- 2 x + 6}{x^{2} + 3 x + 2} \frac{- 2 x + 6}{x^{2} + 3 x + 2}$

32.

$\frac{10 x + 2}{4 x^{2} + 4 x + 1} \frac{10 x + 2}{4 x^{2} + 4 x + 1}$

33.

$\frac{7 x + 20}{x^{2} + 10 x + 25} \frac{7 x + 20}{x^{2} + 10 x + 25}$

34.

$\frac{x - 18}{x^{2} - 12 x + 36} \frac{x - 18}{x^{2} - 12 x + 36}$

35.

$\frac{- x^{2} + 36 x + 70}{x^{3} - 125} \frac{- x^{2} + 36 x + 70}{x^{3} - 125}$

36.

$\frac{- 5 x^{2} + 6 x - 2}{x^{3} + 27} \frac{- 5 x^{2} + 6 x - 2}{x^{3} + 27}$

37.

$\frac{x^{3} - 4 x^{2} + 3 x + 11}{{(x^{2} - 2)}^{2}} \frac{x^{3} - 4 x^{2} + 3 x + 11}{{(x^{2} - 2)}^{2}}$

38.

$\frac{4 x^{4} - 2 x^{3} + 22 x^{2} - 6 x + 48}{x {(x^{2} + 4)}^{2}} \frac{4 x^{4} - 2 x^{3} + 22 x^{2} - 6 x + 48}{x {(x^{2} + 4)}^{2}}$

Matrices and Matrix Operations

For the following exercises, perform the requested operations on the given matrices.

A = [\begin{array}{r} 4 & - 2 \\ 1 & 3 \end{array}], B = [\begin{array}{r} 6 & 7 & - 3 \\ 11 & - 2 & 4 \end{array}], C = [\begin{array}{r} \begin{matrix} 6 & 7 \\ 11 & - 2 \end{matrix} \\ \begin{matrix} 14 & 0 \end{matrix} \end{array}], D = [\begin{array}{r} 1 & - 4 & 9 \\ 10 & 5 & - 7 \\ 2 & 8 & 5 \end{array}], E = [\begin{array}{r} 7 & - 14 & 3 \\ 2 & - 1 & 3 \\ 0 & 1 & 9 \end{array}] A = [\begin{array}{r} 4 & - 2 \\ 1 & 3 \end{array}], B = [\begin{array}{r} 6 & 7 & - 3 \\ 11 & - 2 & 4 \end{array}], C = [\begin{array}{r} \begin{matrix} 6 & 7 \\ 11 & - 2 \end{matrix} \\ \begin{matrix} 14 & 0 \end{matrix} \end{array}], D = [\begin{array}{r} 1 & - 4 & 9 \\ 10 & 5 & - 7 \\ 2 & 8 & 5 \end{array}], E = [\begin{array}{r} 7 & - 14 & 3 \\ 2 & - 1 & 3 \\ 0 & 1 & 9 \end{array}]

39.

$- 4 A - 4 A$

40.

$10 D - 6 E 10 D - 6 E$

41.

$B + C B + C$

42.

$A B A B$

43.

$B A B A$

44.

$B C B C$

45.

$C B C B$

46.

$D E D E$

47.

$E D E D$

48.

$E C E C$

49.

$C E C E$

50.

$A^{3} A^{3}$

Solving Systems with Gaussian Elimination

For the following exercises, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.

51.

$[\begin{array}{r} 1 & 0 & −3 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{array} | \begin{array}{r} 7 \\ −5 \\ 0 \end{array}] [\begin{array}{r} 1 & 0 & −3 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{array} | \begin{array}{r} 7 \\ −5 \\ 0 \end{array}]$

52.

$[\begin{array}{r} 1 & 0 & 5 \\ 0 & 1 & −2 \\ 0 & 0 & 0 \end{array} | \begin{array}{r} −9 \\ 4 \\ 3 \end{array}] [\begin{array}{r} 1 & 0 & 5 \\ 0 & 1 & −2 \\ 0 & 0 & 0 \end{array} | \begin{array}{r} −9 \\ 4 \\ 3 \end{array}]$

For the following exercises, write the augmented matrix from the system of linear equations.

53.

$\begin{array}{r} - 2 x + 2 y + z = 7 \\ 2 x - 8 y + 5 z = 0 \\ 19 x - 10 y + 22 z = 3 \end{array} \begin{array}{r} - 2 x + 2 y + z = 7 \\ 2 x - 8 y + 5 z = 0 \\ 19 x - 10 y + 22 z = 3 \end{array}$

54.

$\begin{array}{l} 4 x + 2 y - 3 z = 14 \\ - 12 x + 3 y + z = 100 \\ 9 x - 6 y + 2 z = 31 \end{array} \begin{array}{l} 4 x + 2 y - 3 z = 14 \\ - 12 x + 3 y + z = 100 \\ 9 x - 6 y + 2 z = 31 \end{array}$

55.

$\begin{array}{r} x + 3 z = 12 \\ - x + 4 y = 0 \\ y + 2 z = - 7 \end{array} \begin{array}{r} x + 3 z = 12 \\ - x + 4 y = 0 \\ y + 2 z = - 7 \end{array}$

For the following exercises, solve the system of linear equations using Gaussian elimination.

56.

$\begin{array}{r} 3 x - 4 y = - 7 \\ - 6 x + 8 y = 14 \end{array} \begin{array}{r} 3 x - 4 y = - 7 \\ - 6 x + 8 y = 14 \end{array}$

57.

$\begin{array}{r} 3 x - 4 y = 1 \\ - 6 x + 8 y = 6 \end{array} \begin{array}{r} 3 x - 4 y = 1 \\ - 6 x + 8 y = 6 \end{array}$

58.

$\begin{array}{l} - 1.1 x - 2.3 y = 6.2 \\ - 5.2 x - 4.1 y = 4.3 \end{array} \begin{array}{l} - 1.1 x - 2.3 y = 6.2 \\ - 5.2 x - 4.1 y = 4.3 \end{array}$

59.

$\begin{array}{r} 2 x + 3 y + 2 z = 1 \\ - 4 x - 6 y - 4 z = - 2 \\ 10 x + 15 y + 10 z = 0 \end{array} \begin{array}{r} 2 x + 3 y + 2 z = 1 \\ - 4 x - 6 y - 4 z = - 2 \\ 10 x + 15 y + 10 z = 0 \end{array}$

60.

$\begin{array}{r} - x + 2 y - 4 z = 8 \\ 3 y + 8 z = - 4 \\ - 7 x + y + 2 z = 1 \end{array} \begin{array}{r} - x + 2 y - 4 z = 8 \\ 3 y + 8 z = - 4 \\ - 7 x + y + 2 z = 1 \end{array}$

Solving Systems with Inverses

For the following exercises, find the inverse of the matrix.

61.

$[\begin{array}{r} - 0.2 & 1.4 \\ 1.2 & - 0.4 \end{array}] [\begin{array}{r} - 0.2 & 1.4 \\ 1.2 & - 0.4 \end{array}]$

62.

$[\begin{array}{r} \frac{1}{2} & - \frac{1}{2} \\ - \frac{1}{4} & \frac{3}{4} \end{array}] [\begin{array}{r} \frac{1}{2} & - \frac{1}{2} \\ - \frac{1}{4} & \frac{3}{4} \end{array}]$

63.

$[\begin{matrix} 12 & 9 & - 6 \\ - 1 & 3 & 2 \\ - 4 & - 3 & 2 \end{matrix}] [\begin{matrix} 12 & 9 & - 6 \\ - 1 & 3 & 2 \\ - 4 & - 3 & 2 \end{matrix}]$

64.

$[\begin{matrix} 2 & 1 & 3 \\ 1 & 2 & 3 \\ 3 & 2 & 1 \end{matrix}] [\begin{matrix} 2 & 1 & 3 \\ 1 & 2 & 3 \\ 3 & 2 & 1 \end{matrix}]$

For the following exercises, find the solutions by computing the inverse of the matrix.

65.

$\begin{array}{l} 0.3 x - 0.1 y = - 10 \\ - 0.1 x + 0.3 y = 14 \end{array} \begin{array}{l} 0.3 x - 0.1 y = - 10 \\ - 0.1 x + 0.3 y = 14 \end{array}$

66.

$\begin{array}{l} 0.4 x - 0.2 y = - 0.6 \\ - 0.1 x + 0.05 y = 0.3 \end{array} \begin{array}{l} 0.4 x - 0.2 y = - 0.6 \\ - 0.1 x + 0.05 y = 0.3 \end{array}$

67.

$\begin{array}{r} 4 x + 3 y - 3 z = - 4.3 \\ 5 x - 4 y - z = - 6.1 \\ x + z = - 0.7 \end{array} \begin{array}{r} 4 x + 3 y - 3 z = - 4.3 \\ 5 x - 4 y - z = - 6.1 \\ x + z = - 0.7 \end{array}$

68.

$\begin{array}{r} \begin{array}{l} - 2 x - 3 y + 2 z = 3 \end{array} \\ - x + 2 y + 4 z = - 5 \\ - 2 y + 5 z = - 3 \end{array} \begin{array}{r} \begin{array}{l} - 2 x - 3 y + 2 z = 3 \end{array} \\ - x + 2 y + 4 z = - 5 \\ - 2 y + 5 z = - 3 \end{array}$

For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

69.

Students were asked to bring their favorite fruit to class. 90% of the fruits consisted of banana, apple, and oranges. If oranges were half as popular as bananas and apples were 5% more popular than bananas, what are the percentages of each individual fruit?

70.

A sorority held a bake sale to raise money and sold brownies and chocolate chip cookies. They priced the brownies at $2 and the chocolate chip cookies at $1. They raised $250 and sold 175 items. How many brownies and how many cookies were sold?

Solving Systems with Cramer's Rule

For the following exercises, find the determinant.

71.

$| \begin{matrix} 100 & 0 \\ 0 & 0 \end{matrix} | | \begin{matrix} 100 & 0 \\ 0 & 0 \end{matrix} |$

72.

$| \begin{matrix} 0.2 & - 0.6 \\ 0.7 & - 1.1 \end{matrix} | | \begin{matrix} 0.2 & - 0.6 \\ 0.7 & - 1.1 \end{matrix} |$

73.

$| \begin{matrix} - 1 & 4 & 3 \\ 0 & 2 & 3 \\ 0 & 0 & - 3 \end{matrix} | | \begin{matrix} - 1 & 4 & 3 \\ 0 & 2 & 3 \\ 0 & 0 & - 3 \end{matrix} |$

74.

$| \begin{matrix} \sqrt{2} & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \end{matrix} | | \begin{matrix} \sqrt{2} & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \end{matrix} |$

For the following exercises, use Cramer’s Rule to solve the linear systems of equations.

75.

$\begin{array}{r} 4 x - 2 y = 23 \\ - 5 x - 10 y = - 35 \end{array} \begin{array}{r} 4 x - 2 y = 23 \\ - 5 x - 10 y = - 35 \end{array}$

76.

$\begin{array}{l} 0.2 x - 0.1 y = 0 \\ - 0.3 x + 0.3 y = 2.5 \end{array} \begin{array}{l} 0.2 x - 0.1 y = 0 \\ - 0.3 x + 0.3 y = 2.5 \end{array}$

77.

$\begin{array}{r} - 0.5 x + 0.1 y = 0.3 \\ - 0.25 x + 0.05 y = 0.15 \end{array} \begin{array}{r} - 0.5 x + 0.1 y = 0.3 \\ - 0.25 x + 0.05 y = 0.15 \end{array}$

78.

$\begin{array}{l} x + 6 y + 3 z = 4 \\ 2 x + y + 2 z = 3 \\ 3 x - 2 y + z = 0 \end{array} \begin{array}{l} x + 6 y + 3 z = 4 \\ 2 x + y + 2 z = 3 \\ 3 x - 2 y + z = 0 \end{array}$

79.

$\begin{array}{r} 4 x - 3 y + 5 z = - \frac{5}{2} \\ 7 x - 9 y - 3 z = \frac{3}{2} \\ x - 5 y - 5 z = \frac{5}{2} \end{array} \begin{array}{r} 4 x - 3 y + 5 z = - \frac{5}{2} \\ 7 x - 9 y - 3 z = \frac{3}{2} \\ x - 5 y - 5 z = \frac{5}{2} \end{array}$

80.

$\begin{array}{r} \frac{3}{10} x - \frac{1}{5} y - \frac{3}{10} z = - \frac{1}{50} \\ \frac{1}{10} x - \frac{1}{10} y - \frac{1}{2} z = - \frac{9}{50} \\ \frac{2}{5} x - \frac{1}{2} y - \frac{3}{5} z = - \frac{1}{5} \end{array} \begin{array}{r} \frac{3}{10} x - \frac{1}{5} y - \frac{3}{10} z = - \frac{1}{50} \\ \frac{1}{10} x - \frac{1}{10} y - \frac{1}{2} z = - \frac{9}{50} \\ \frac{2}{5} x - \frac{1}{2} y - \frac{3}{5} z = - \frac{1}{5} \end{array}$