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College Algebra

Review Exercises

College AlgebraReview Exercises
  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Key Terms
    6. Key Concepts
    7. Review Exercises
    8. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  8. 7 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 7.1 Systems of Linear Equations: Two Variables
    3. 7.2 Systems of Linear Equations: Three Variables
    4. 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 7.4 Partial Fractions
    6. 7.5 Matrices and Matrix Operations
    7. 7.6 Solving Systems with Gaussian Elimination
    8. 7.7 Solving Systems with Inverses
    9. 7.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  9. 8 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 8.1 The Ellipse
    3. 8.2 The Hyperbola
    4. 8.3 The Parabola
    5. 8.4 Rotation of Axes
    6. 8.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  10. 9 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 9.1 Sequences and Their Notations
    3. 9.2 Arithmetic Sequences
    4. 9.3 Geometric Sequences
    5. 9.4 Series and Their Notations
    6. 9.5 Counting Principles
    7. 9.6 Binomial Theorem
    8. 9.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
  12. Index

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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