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Algebra and Trigonometry

Review Exercises

Algebra and TrigonometryReview Exercises

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.

1.

3xy=4 x+4y=3 3xy=4 x+4y=3 and (1,1) (1,1)

2.

6x2y=24 3x+3y=18 6x2y=24 3x+3y=18 and (9,15) (9,15)

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

3.

10x+5y=−5 3x2y=−12 10x+5y=−5 3x2y=−12

4.

4 7 x+ 1 5 y= 43 70 5 6 x 1 3 y= 2 3 4 7 x+ 1 5 y= 43 70 5 6 x 1 3 y= 2 3

5.

5x+6y=14 4x+8y=8 5x+6y=14 4x+8y=8

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

6.

3x+2y=−7 2x+4y=6 3x+2y=−7 2x+4y=6

7.

3x+4y=2 9x+12y=3 3x+4y=2 9x+12y=3

8.

8x+4y=2 6x5y=0.7 8x+4y=2 6x5y=0.7

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

9.

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

10.

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

0.5x0.5y=10 0.2y+0.2x=4 0.1x+0.1z=2 0.5x0.5y=10 0.2y+0.2x=4 0.1x+0.1z=2

12.

5x+3yz=5 3x2y+4z=13 4x+3y+5z=22 5x+3yz=5 3x2y+4z=13 4x+3y+5z=22

13.

x+y+z=1 2x+2y+2z=1 3x+3y=2 x+y+z=1 2x+2y+2z=1 3x+3y=2

14.

2x3y+z=−1 x+y+z=−4 4x+2y3z=33 2x3y+z=−1 x+y+z=−4 4x+2y3z=33

15.

3x+2yz=−10 xy+2z=7 x+3y+z=−2 3x+2yz=−10 xy+2z=7 x+3y+z=−2

16.

3x+4z=−11 x2y=5 4yz=−10 3x+4z=−11 x2y=5 4yz=−10

17.

2x3y+z=0 2x+4y3z=0 6x2yz=0 2x3y+z=0 2x+4y3z=0 6x2yz=0

18.

6x4y2z=2 3x+2y5z=4 6y7z=5 6x4y2z=2 3x+2y5z=4 6y7z=5

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.

y= x 2 7 y=5x13 y= x 2 7 y=5x13

22.

y= x 2 4 y=5x+10 y= x 2 4 y=5x+10

23.

x 2 + y 2 =16 y=x8 x 2 + y 2 =16 y=x8

24.

x 2 + y 2 =25 y= x 2 +5 x 2 + y 2 =25 y= x 2 +5

25.

x 2 + y 2 =4 y x 2 =3 x 2 + y 2 =4 y x 2 =3

For the following exercises, graph the inequality.

26.

y> x 2 1 y> x 2 1

27.

1 4 x 2 + y 2 <4 1 4 x 2 + y 2 <4

For the following exercises, graph the system of inequalities.

28.

x 2 + y 2 +2x<3 y> x 2 3 x 2 + y 2 +2x<3 y> x 2 3

29.

x 2 2x+ y 2 4x<4 y<x+4 x 2 2x+ y 2 4x<4 y<x+4

30.

x 2 + y 2 <1 y 2 <x x 2 + y 2 <1 y 2 <x

Partial Fractions

For the following exercises, decompose into partial fractions.

31.

2x+6 x 2 +3x+2 2x+6 x 2 +3x+2

32.

10x+2 4 x 2 +4x+1 10x+2 4 x 2 +4x+1

33.

7x+20 x 2 +10x+25 7x+20 x 2 +10x+25

34.

x18 x 2 12x+36 x18 x 2 12x+36

35.

x 2 +36x+70 x 3 125 x 2 +36x+70 x 3 125

36.

5 x 2 +6x2 x 3 +27 5 x 2 +6x2 x 3 +27

37.

x 3 4 x 2 +3x+11 ( x 2 2) 2 x 3 4 x 2 +3x+11 ( x 2 2) 2

38.

4 x 4 2 x 3 +22 x 2 6x+48 x ( x 2 +4) 2 4 x 4 2 x 3 +22 x 2 6x+48 x ( x 2 +4) 2

Matrices and Matrix Operations

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

A=[ 4 2 1 3 ],B=[ 6 7 3 11 2 4 ],C=[ 6 7 11 2 14 0 ],D=[ 1 4 9 10 5 7 2 8 5 ],E=[ 7 14 3 2 1 3 0 1 9 ] A=[ 4 2 1 3 ],B=[ 6 7 3 11 2 4 ],C=[ 6 7 11 2 14 0 ],D=[ 1 4 9 10 5 7 2 8 5 ],E=[ 7 14 3 2 1 3 0 1 9 ]
39.

4A 4A

40.

10D6E 10D6E

41.

B+C B+C

42.

AB AB

43.

BA BA

44.

BC BC

45.

CB CB

46.

DE DE

47.

ED ED

48.

EC EC

49.

CE CE

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.

[ 1 0 −3 0 1 2 0 0 0 | 7 −5 0 ] [ 1 0 −3 0 1 2 0 0 0 | 7 −5 0 ]

52.

[ 1 0 5 0 1 −2 0 0 0 | −9 4 3 ] [ 1 0 5 0 1 −2 0 0 0 | −9 4 3 ]

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

53.

2x+2y+z=7 2x8y+5z=0 19x10y+22z=3 2x+2y+z=7 2x8y+5z=0 19x10y+22z=3

54.

4x+2y3z=14 12x+3y+z=100 9x6y+2z=31 4x+2y3z=14 12x+3y+z=100 9x6y+2z=31

55.

x+3z=12 x+4y=0 y+2z=7 x+3z=12 x+4y=0 y+2z=7

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

56.

3x4y=7 6x+8y=14 3x4y=7 6x+8y=14

57.

3x4y=1 6x+8y=6 3x4y=1 6x+8y=6

58.

1.1x2.3y=6.2 5.2x4.1y=4.3 1.1x2.3y=6.2 5.2x4.1y=4.3

59.

2x+3y+2z=1 4x6y4z=2 10x+15y+10z=0 2x+3y+2z=1 4x6y4z=2 10x+15y+10z=0

60.

x+2y4z=8 3y+8z=4 7x+y+2z=1 x+2y4z=8 3y+8z=4 7x+y+2z=1

Solving Systems with Inverses

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

61.

[ 0.2 1.4 1.2 0.4 ] [ 0.2 1.4 1.2 0.4 ]

62.

[ 1 2 1 2 1 4 3 4 ] [ 1 2 1 2 1 4 3 4 ]

63.

[ 12 9 6 1 3 2 4 3 2 ] [ 12 9 6 1 3 2 4 3 2 ]

64.

[ 2 1 3 1 2 3 3 2 1 ] [ 2 1 3 1 2 3 3 2 1 ]

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

65.

0.3x0.1y=10 0.1x+0.3y=14 0.3x0.1y=10 0.1x+0.3y=14

66.

0.4x0.2y=0.6 0.1x+0.05y=0.3 0.4x0.2y=0.6 0.1x+0.05y=0.3

67.

4x+3y3z=4.3 5x4yz=6.1 x+z=0.7 4x+3y3z=4.3 5x4yz=6.1 x+z=0.7

68.

2x3y+2z=3 x+2y+4z=5 2y+5z=3 2x3y+2z=3 x+2y+4z=5 2y+5z=3

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.

| 100 0 0 0 | | 100 0 0 0 |

72.

| 0.2 0.6 0.7 1.1 | | 0.2 0.6 0.7 1.1 |

73.

| 1 4 3 0 2 3 0 0 3 | | 1 4 3 0 2 3 0 0 3 |

74.

| 2 0 0 0 2 0 0 0 2 | | 2 0 0 0 2 0 0 0 2 |

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

75.

4x2y=23 5x10y=35 4x2y=23 5x10y=35

76.

0.2x0.1y=0 0.3x+0.3y=2.5 0.2x0.1y=0 0.3x+0.3y=2.5

77.

0.5x+0.1y=0.3 0.25x+0.05y=0.15 0.5x+0.1y=0.3 0.25x+0.05y=0.15

78.

x+6y+3z=4 2x+y+2z=3 3x2y+z=0 x+6y+3z=4 2x+y+2z=3 3x2y+z=0

79.

4x3y+5z= 5 2 7x9y3z= 3 2 x5y5z= 5 2 4x3y+5z= 5 2 7x9y3z= 3 2 x5y5z= 5 2

80.

3 10 x 1 5 y 3 10 z= 1 50 1 10 x 1 10 y 1 2 z= 9 50 2 5 x 1 2 y 3 5 z= 1 5 3 10 x 1 5 y 3 10 z= 1 50 1 10 x 1 10 y 1 2 z= 9 50 2 5 x 1 2 y 3 5 z= 1 5

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