Jay Abramson

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.

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

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

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

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

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

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

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

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

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

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

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

10.

A performer charges $C (x) = 50 x + 10, 000,$ where $x$ 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.

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

12.

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

13.

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

14.

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

15.

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

16.

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

17.

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

18.

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

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.

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

22.

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

23.

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

24.

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

25.

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

For the following exercises, graph the inequality.

26.

$y > x^{2} - 1$

27.

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

For the following exercises, graph the system of inequalities.

28.

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

29.

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

30.

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

Partial Fractions

For the following exercises, decompose into partial fractions.

31.

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

32.

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

33.

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

34.

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

35.

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

36.

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

37.

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

38.

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

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}]

39.

$- 4 A$

40.

$10 D - 6 E$

41.

$B + C$

42.

$A B$

43.

$B A$

44.

$B C$

45.

$C B$

46.

$D E$

47.

$E D$

48.

$E C$

49.

$C E$

50.

$A^{3}$

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.

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

52.

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

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

53.

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

54.

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

55.

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

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

56.

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

57.

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

58.

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

59.

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

60.

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

Solving Systems with Inverses

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

61.

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

62.

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

63.

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

64.

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

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

65.

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

66.

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

67.

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

68.

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

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 school club 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.

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

72.

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

73.

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

74.

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

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

75.

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

76.

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

77.

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

78.

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

79.

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

80.

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