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Precalculus

Chapter 9

PrecalculusChapter 9
  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Review Exercises
    15. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Equations
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  14. A | Basic Functions and Identities
  15. 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
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  16. Index

Try It

9.1 Systems of Linear Equations: Two Variables

1.

Not a solution.

2.

The solution to the system is the ordered pair ( −5,3 ). ( −5,3 ).

3.

( −2,−5 ) ( −2,−5 )

4.

( −6,−2 ) ( −6,−2 )

5.

( 10,−4 ) ( 10,−4 )

6.

No solution. It is an inconsistent system.

7.

The system is dependent so there are infinite solutions of the form (x,2x+5). (x,2x+5).

8.

700 children, 950 adults

9.2 Systems of Linear Equations: Three Variables

1.

( 1,−1,1 ) ( 1,−1,1 )

2.

No solution.

3.

Infinite number of solutions of the form ( x,4x−11,−5x+18 ). ( x,4x−11,−5x+18 ).

9.3 Systems of Nonlinear Equations and Inequalities: Two Variables

1.

( 1 2 , 1 2 ) ( 1 2 , 1 2 ) and ( 2,8 ) ( 2,8 )

2.

( −1,3 ) ( −1,3 )

3.

{ ( 1,3 ),( 1,−3 ),( −1,3 ),( −1,−3 ) } { ( 1,3 ),( 1,−3 ),( −1,3 ),( −1,−3 ) }

4.

9.4 Partial Fractions

1.

3 x−3 2 x−2 3 x−3 2 x−2

2.

6 x−1 5 ( x−1 ) 2 6 x−1 5 ( x−1 ) 2

3.

3 x−1 + 2x−4 x 2 +1 3 x−1 + 2x−4 x 2 +1

4.

x−2 x 2 −2x+3 + 2x+1 ( x 2 −2x+3 ) 2 x−2 x 2 −2x+3 + 2x+1 ( x 2 −2x+3 ) 2

9.5 Matrices and Matrix Operations

1.
A+B=[ 2 1 1 6 0 −3 ]+[ 3 1 −4 −2 5 3 ]=[ 2+3 1+1 1+(−4) 6+(−2) 0+5 −3+3 ]=[ 5 2 −3 4 5 0 ] A+B=[ 2 1 1 6 0 −3 ]+[ 3 1 −4 −2 5 3 ]=[ 2+3 1+1 1+(−4) 6+(−2) 0+5 −3+3 ]=[ 5 2 −3 4 5 0 ]
2.

−2B=[ −8 −2 −6 −4 ] −2B=[ −8 −2 −6 −4 ]

9.6 Solving Systems with Gaussian Elimination

1.

[ 4 −3 3 2 | 11 4 ] [ 4 −3 3 2 | 11 4 ]

2.

xy+z=5 2xy+3z=1 y+z=−9 xy+z=5 2xy+3z=1 y+z=−9

3.

( 2,1 ) ( 2,1 )

4.

[ 1 5 2 5 2 0 1 5 0 0 1 | 17 2 9 2 ] [ 1 5 2 5 2 0 1 5 0 0 1 | 17 2 9 2 ]

5.

( 1,1,1 ) ( 1,1,1 )

6.

$150,000 at 7%, $750,000 at 8%, $600,000 at 10%

9.7 Solving Systems with Inverses

1.
AB=[ 1 4 −1 −3 ] [ −3 −4 1 1 ]=[ 1(−3)+4(1) 1(−4)+4(1) −1(−3)+−3(1) −1(−4)+−3(1) ]=[ 1 0 0 1 ] BA=[ −3 −4 1 1 ] [ 1 4 −1 −3 ]=[ −3(1)+−4(−1) −3(4)+−4(−3) 1(1)+1(−1) 1(4)+1(−3) ]=[ 1 0 0 1 ] AB=[ 1 4 −1 −3 ] [ −3 −4 1 1 ]=[ 1(−3)+4(1) 1(−4)+4(1) −1(−3)+−3(1) −1(−4)+−3(1) ]=[ 1 0 0 1 ] BA=[ −3 −4 1 1 ] [ 1 4 −1 −3 ]=[ −3(1)+−4(−1) −3(4)+−4(−3) 1(1)+1(−1) 1(4)+1(−3) ]=[ 1 0 0 1 ]
2.

A −1 =[ 3 5 1 5 2 5 1 5 ] A −1 =[ 3 5 1 5 2 5 1 5 ]

3.

A −1 =[ 1 1 2 2 4 −3 3 6 −5 ] A −1 =[ 1 1 2 2 4 −3 3 6 −5 ]

4.

X=[ 4 38 58 ] X=[ 4 38 58 ]

9.8 Solving Systems with Cramer's Rule

1.

( 3,7 ) ( 3,7 )

2.

10 10

3.

( 2, 3 5 , 12 5 ) ( 2, 3 5 , 12 5 )

9.1 Section Exercises

1.

No, you can either have zero, one, or infinitely many. Examine graphs.

3.

This means there is no realistic break-even point. By the time the company produces one unit they are already making profit.

5.

You can solve by substitution (isolating x x or y y ), graphically, or by addition.

7.

Yes

9.

Yes

11.

(−1,2) (−1,2)

13.

(−3,1) (−3,1)

15.

( 3 5 ,0 ) ( 3 5 ,0 )

17.

No solutions exist.

19.

( 72 5 , 132 5 ) ( 72 5 , 132 5 )

21.

( 6,−6 ) ( 6,−6 )

23.

( 1 2 , 1 10 ) ( 1 2 , 1 10 )

25.

No solutions exist.

27.

( 1 5 , 2 3 ) ( 1 5 , 2 3 )

29.

( x, x+3 2 ) ( x, x+3 2 )

31.

(−4,4) (−4,4)

33.

( 1 2 , 1 8 ) ( 1 2 , 1 8 )

35.

( 1 6 ,0 ) ( 1 6 ,0 )

37.

( x,2(7x−6) ) ( x,2(7x−6) )

39.

( 5 6 , 4 3 ) ( 5 6 , 4 3 )

41.

Consistent with one solution

43.

Consistent with one solution

45.

Dependent with infinitely many solutions

47.

( −3.08,4.91 ) ( −3.08,4.91 )

49.

( −1.52,2.29 ) ( −1.52,2.29 )

51.

( A+B 2 , AB 2 ) ( A+B 2 , AB 2 )

53.

( −1 AB , A AB ) ( −1 AB , A AB )

55.

( CEBF BDAE , AFCD BDAE ) ( CEBF BDAE , AFCD BDAE )

57.

They never turn a profit.

59.

(1,250,100,000) (1,250,100,000)

61.

The numbers are 7.5 and 20.5.

63.

24,000

65.

790 sophomores, 805 freshman

67.

56 men, 74 women

69.

10 gallons of 10% solution, 15 gallons of 60% solution

71.

Swan Peak: $750,000, Riverside: $350,000

73.

$12,500 in the first account, $10,500 in the second account.

75.

High-tops: 45, Low-tops: 15

77.

Infinitely many solutions. We need more information.

9.2 Section Exercises

1.

No, there can be only one, zero, or infinitely many solutions.

3.

Not necessarily. There could be zero, one, or infinitely many solutions. For example, ( 0,0,0 ) ( 0,0,0 ) is not a solution to the system below, but that does not mean that it has no solution.

    2x+3y−6z=1 −4x−6y+12z=−2       x+2y+5z=10     2x+3y−6z=1 −4x−6y+12z=−2       x+2y+5z=10

5.

Every system of equations can be solved graphically, by substitution, and by addition. However, systems of three equations become very complex to solve graphically so other methods are usually preferable.

7.

No

9.

Yes

11.

( −1,4,2 ) ( −1,4,2 )

13.

( 85 107 , 312 107 , 191 107 ) ( 85 107 , 312 107 , 191 107 )

15.

( 1, 1 2 ,0 ) ( 1, 1 2 ,0 )

17.

( 4,−6,1 ) ( 4,−6,1 )

19.

( x, 1 27 (65−16x), x+28 27 ) ( x, 1 27 (65−16x), x+28 27 )

21.

( 45 13 , 17 13 ,−2 ) ( 45 13 , 17 13 ,−2 )

23.

No solutions exist

25.

( 0,0,0 ) ( 0,0,0 )

27.

( 4 7 , 1 7 , 3 7 ) ( 4 7 , 1 7 , 3 7 )

29.

( 7,20,16 ) ( 7,20,16 )

31.

( −6,2,1 ) ( −6,2,1 )

33.

( 5,12,15 ) ( 5,12,15 )

35.

( −5,−5,−5 ) ( −5,−5,−5 )

37.

( 10,10,10 ) ( 10,10,10 )

39.

( 1 2 , 1 5 , 4 5 ) ( 1 2 , 1 5 , 4 5 )

41.

( 1 2 , 2 5 , 4 5 ) ( 1 2 , 2 5 , 4 5 )

43.

( 2,0,0 ) ( 2,0,0 )

45.

( 1,1,1 ) ( 1,1,1 )

47.

( 128 557 , 23 557 , 28 557 ) ( 128 557 , 23 557 , 28 557 )

49.

( 6,−1,0 ) ( 6,−1,0 )

51.

24, 36, 48

53.

70 grandparents, 140 parents, 190 children

55.

Your share was $19.95, Sarah’s share was $40, and your other roommate’s share was $22.05.

57.

There are infinitely many solutions; we need more information

59.

500 students, 225 children, and 450 adults

61.

The BMW was $49,636, the Jeep was $42,636, and the Toyota was $47,727.

63.

$400,000 in the account that pays 3% interest, $500,000 in the account that pays 4% interest, and $100,000 in the account that pays 2% interest.

65.

The United States consumed 26.3%, Japan 7.1%, and China 6.4% of the world’s oil.

67.

Saudi Arabia imported 16.8%, Canada imported 15.1%, and Mexico 15.0%

69.

Birds were 19.3%, fish were 18.6%, and mammals were 17.1% of endangered species

9.3 Section Exercises

1.

A nonlinear system could be representative of two circles that overlap and intersect in two locations, hence two solutions. A nonlinear system could be representative of a parabola and a circle, where the vertex of the parabola meets the circle and the branches also intersect the circle, hence three solutions.

3.

No. There does not need to be a feasible region. Consider a system that is bounded by two parallel lines. One inequality represents the region above the upper line; the other represents the region below the lower line. In this case, no points in the plane are located in both regions; hence there is no feasible region.

5.

Choose any number between each solution and plug into C(x) C(x) and R(x). R(x). If C(x)<R(x), C(x)<R(x), then there is profit.

7.

( 0,−3 ),( 3,0 ) ( 0,−3 ),( 3,0 )

9.

( 3 2 2 , 3 2 2 ),( 3 2 2 , 3 2 2 ) ( 3 2 2 , 3 2 2 ),( 3 2 2 , 3 2 2 )

11.

( −3,0 ),( 3,0 ) ( −3,0 ),( 3,0 )

13.

( 1 4 , 62 8 ),( 1 4 , 62 8 ) ( 1 4 , 62 8 ),( 1 4 , 62 8 )

15.

( 398 4 , 199 4 ),( 398 4 , 199 4 ) ( 398 4 , 199 4 ),( 398 4 , 199 4 )

17.

( 0,2 ),( 1,3 ) ( 0,2 ),( 1,3 )

19.

( 1 2 ( 5 −1 ) , 1 2 ( 1 5 ) ),( 1 2 ( 5 −1 ) , 1 2 ( 1 5 ) ) ( 1 2 ( 5 −1 ) , 1 2 ( 1 5 ) ),( 1 2 ( 5 −1 ) , 1 2 ( 1 5 ) )

21.

( 5,0 ) ( 5,0 )

23.

( 0,0 ) ( 0,0 )

25.

( 3,0 ) ( 3,0 )

27.

No Solutions Exist

29.

No Solutions Exist

31.

( 2 2 , 2 2 ),( 2 2 , 2 2 ),( 2 2 , 2 2 ),( 2 2 , 2 2 ) ( 2 2 , 2 2 ),( 2 2 , 2 2 ),( 2 2 , 2 2 ),( 2 2 , 2 2 )

33.

(2,0) (2,0)

35.

( 7 ,−3 ),( 7 ,3 ),( 7 ,−3 ),( 7 ,3 ) ( 7 ,−3 ),( 7 ,3 ),( 7 ,−3 ),( 7 ,3 )

37.

( 1 2 ( 73 −5 ) , 1 2 ( 7 73 ) ),( 1 2 ( 73 −5 ) , 1 2 ( 7 73 ) ) ( 1 2 ( 73 −5 ) , 1 2 ( 7 73 ) ),( 1 2 ( 73 −5 ) , 1 2 ( 7 73 ) )

39.
41.
43.
45.
47.
49.

( −2 70 383 ,−2 35 29 ),( −2 70 383 ,2 35 29 ),( 2 70 383 ,−2 35 29 ),( 2 70 383 ,2 35 29 ) ( −2 70 383 ,−2 35 29 ),( −2 70 383 ,2 35 29 ),( 2 70 383 ,−2 35 29 ),( 2 70 383 ,2 35 29 )

51.

No Solution Exists

53.

x=0,y>0 x=0,y>0 and 0<x<1, x <y< 1 x 0<x<1, x <y< 1 x

55.

12, 288

57.

2–20 computers

9.4 Section Exercises

1.

No, a quotient of polynomials can only be decomposed if the denominator can be factored. For example, 1 x 2 +1 1 x 2 +1 cannot be decomposed because the denominator cannot be factored.

3.

Graph both sides and ensure they are equal.

5.

If we choose x=−1, x=−1, then the B-term disappears, letting us immediately know that A=3. A=3. We could alternatively plug in x= 5 3 , x= 5 3 , giving us a B-value of −2. −2.

7.

8 x+3 5 x−8 8 x+3 5 x−8

9.

1 x+5 + 9 x+2 1 x+5 + 9 x+2

11.

3 5x−2 + 4 4x−1 3 5x−2 + 4 4x−1

13.

5 2( x+3 ) + 5 2( x−3 ) 5 2( x+3 ) + 5 2( x−3 )

15.

3 x+2 + 3 x−2 3 x+2 + 3 x−2

17.

9 5( x+2 ) + 11 5( x−3 ) 9 5( x+2 ) + 11 5( x−3 )

19.

8 x−3 5 x−2 8 x−3 5 x−2

21.

1 x−2 + 2 ( x−2 ) 2 1 x−2 + 2 ( x−2 ) 2

23.

6 4x+5 + 3 ( 4x+5 ) 2 6 4x+5 + 3 ( 4x+5 ) 2

25.

1 x−7 2 ( x−7 ) 2 1 x−7 2 ( x−7 ) 2

27.

4 x 3 2( x+1 ) + 7 2 ( x+1 ) 2 4 x 3 2( x+1 ) + 7 2 ( x+1 ) 2

29.

4 x + 2 x 2 3 3x+2 + 7 2 ( 3x+2 ) 2 4 x + 2 x 2 3 3x+2 + 7 2 ( 3x+2 ) 2

31.

x+1 x 2 +x+3 + 3 x+2 x+1 x 2 +x+3 + 3 x+2

33.

4−3x x 2 +3x+8 + 1 x−1 4−3x x 2 +3x+8 + 1 x−1

35.

2x−1 x 2 +6x+1 + 2 x+3 2x−1 x 2 +6x+1 + 2 x+3

37.

1 x 2 +x+1 + 4 x−1 1 x 2 +x+1 + 4 x−1

39.

2 x 2 −3x+9 + 3 x+3 2 x 2 −3x+9 + 3 x+3

41.

1 4 x 2 +6x+9 + 1 2x−3 1 4 x 2 +6x+9 + 1 2x−3

43.

1 x + 1 x+6 4x x 2 −6x+36 1 x + 1 x+6 4x x 2 −6x+36

45.

x+6 x 2 +1 + 4x+3 ( x 2 +1 ) 2 x+6 x 2 +1 + 4x+3 ( x 2 +1 ) 2

47.

x+1 x+2 + 2x+3 ( x+2 ) 2 x+1 x+2 + 2x+3 ( x+2 ) 2

49.

1 x 2 +3x+25 3x ( x 2 +3x+25 ) 2 1 x 2 +3x+25 3x ( x 2 +3x+25 ) 2

51.

1 8x x 8( x 2 +4 ) + 10x 2 ( x 2 +4 ) 2 1 8x x 8( x 2 +4 ) + 10x 2 ( x 2 +4 ) 2

53.

16 x 9 x 2 + 16 x−1 7 ( x−1 ) 2 16 x 9 x 2 + 16 x−1 7 ( x−1 ) 2

55.

1 x+1 2 ( x+1 ) 2 + 5 ( x+1 ) 3 1 x+1 2 ( x+1 ) 2 + 5 ( x+1 ) 3

57.

5 x−2 3 10( x+2 ) + 7 x+8 7 10( x−8 ) 5 x−2 3 10( x+2 ) + 7 x+8 7 10( x−8 )

59.

5 4x 5 2( x+2 ) + 11 2( x+4 ) + 5 4( x+4 ) 5 4x 5 2( x+2 ) + 11 2( x+4 ) + 5 4( x+4 )

9.5 Section Exercises

1.

No, they must have the same dimensions. An example would include two matrices of different dimensions. One cannot add the following two matrices because the first is a 2×2 2×2 matrix and the second is a 2×3 2×3 matrix. [ 1 2 3 4 ]+[ 6 5 4 3 2 1 ] [ 1 2 3 4 ]+[ 6 5 4 3 2 1 ] has no sum.

3.

Yes, if the dimensions of A A are m×n m×n and the dimensions of B B are n×m, n×m, both products will be defined.

5.

Not necessarily. To find AB, AB, we multiply the first row of A A by the first column of B B to get the first entry of AB. AB. To find BA, BA, we multiply the first row of B B by the first column of A A to get the first entry of BA. BA. Thus, if those are unequal, then the matrix multiplication does not commute.

7.

[ 11 19 15 94 17 67 ] [ 11 19 15 94 17 67 ]

9.

[ −4 2 8 1 ] [ −4 2 8 1 ]

11.

Undidentified; dimensions do not match

13.

[ 9 27 63 36 0 192 ] [ 9 27 63 36 0 192 ]

15.

[ −64 −12 −28 −72 −360 −20 −12 −116 ] [ −64 −12 −28 −72 −360 −20 −12 −116 ]

17.

[ 1,800 1,200 1,300 800 1,400 600 700 400 2,100 ] [ 1,800 1,200 1,300 800 1,400 600 700 400 2,100 ]

19.

[ 20 102 28 28 ] [ 20 102 28 28 ]

21.

[ 60 41 2 −16 120 −216 ] [ 60 41 2 −16 120 −216 ]

23.

[ −68 24 136 −54 −12 64 −57 30 128 ] [ −68 24 136 −54 −12 64 −57 30 128 ]

25.

Undefined; dimensions do not match.

27.

[ −8 41 −3 40 −15 −14 4 27 42 ] [ −8 41 −3 40 −15 −14 4 27 42 ]

29.

[ −840 650 −530 330 360 250 −10 900 110 ] [ −840 650 −530 330 360 250 −10 900 110 ]

31.

[ −350 1,050 350 350 ] [ −350 1,050 350 350 ]

33.

Undefined; inner dimensions do not match.

35.

[ 1,400 700 −1,400 700 ] [ 1,400 700 −1,400 700 ]

37.

[ 332,500 927,500 −227,500 87,500 ] [ 332,500 927,500 −227,500 87,500 ]

39.

[ 490,000 0 0 490,000 ] [ 490,000 0 0 490,000 ]

41.

[ −2 3 4 −7 9 −7 ] [ −2 3 4 −7 9 −7 ]

43.

[ −4 29 21 −27 −3 1 ] [ −4 29 21 −27 −3 1 ]

45.

[ −3 −2 −2 −28 59 46 −4 16 7 ] [ −3 −2 −2 −28 59 46 −4 16 7 ]

47.

[ 1 −18 −9 −198 505 369 −72 126 91 ] [ 1 −18 −9 −198 505 369 −72 126 91 ]

49.

[ 0 1.6 9 −1 ] [ 0 1.6 9 −1 ]

51.

[ 2 24 −4.5 12 32 −9 −8 64 61 ] [ 2 24 −4.5 12 32 −9 −8 64 61 ]

53.

[ 0.5 3 0.5 2 1 2 10 7 10 ] [ 0.5 3 0.5 2 1 2 10 7 10 ]

55.

[ 1 0 0 0 1 0 0 0 1 ] [ 1 0 0 0 1 0 0 0 1 ]

57.

[ 1 0 0 0 1 0 0 0 1 ] [ 1 0 0 0 1 0 0 0 1 ]

59.

B n ={ [ 1 0 0 0 1 0 0 0 1 ],neven, [ 1 0 0 0 0 1 0 1 0 ],nodd. B n ={ [ 1 0 0 0 1 0 0 0 1 ],neven, [ 1 0 0 0 0 1 0 1 0 ],nodd.

9.6 Section Exercises

1.

Yes. For each row, the coefficients of the variables are written across the corresponding row, and a vertical bar is placed; then the constants are placed to the right of the vertical bar.

3.

No, there are numerous correct methods of using row operations on a matrix. Two possible ways are the following: (1) Interchange rows 1 and 2. Then R 2 = R 2 −9 R 1 . R 2 = R 2 −9 R 1 . (2) R 2 = R 1 −9 R 2 . R 2 = R 1 −9 R 2 . Then divide row 1 by 9.

5.

No. A matrix with 0 entries for an entire row would have either zero or infinitely many solutions.

7.

[ 0 16 9 −1 | 4 2 ] [ 0 16 9 −1 | 4 2 ]

9.

[ 1 5 8 12 3 0 3 4 9 | 16 4 −7 ] [ 1 5 8 12 3 0 3 4 9 | 16 4 −7 ]

11.

−2x+5y=5 6x−18y=26 −2x+5y=5 6x−18y=26

13.

3x+2y=3 x−9y+4z=−1 8x+5y+7z=8 3x+2y=3 x−9y+4z=−1 8x+5y+7z=8

15.

4x+5y−2z=12         y+58z=2 8x+7y−3z=−5 4x+5y−2z=12         y+58z=2 8x+7y−3z=−5

17.

No solutions

19.

(−1,−2) (−1,−2)

21.

( 6,7 ) ( 6,7 )

23.

( 3,2 ) ( 3,2 )

25.

( 1 5 , 1 2 ) ( 1 5 , 1 2 )

27.

( x, 4 15 (5x+1) ) ( x, 4 15 (5x+1) )

29.

( 3,4 ) ( 3,4 )

31.

( 196 39 , 5 13 ) ( 196 39 , 5 13 )

33.

( 31,−42,87 ) ( 31,−42,87 )

35.

( 21 40 , 1 20 , 9 8 ) ( 21 40 , 1 20 , 9 8 )

37.

( 18 13 , 15 13 , 15 13 ) ( 18 13 , 15 13 , 15 13 )

39.

( x,y, 1 2 (1−2x−3y) ) ( x,y, 1 2 (1−2x−3y) )

41.

( x, x 2 ,−1 ) ( x, x 2 ,−1 )

43.

( 125,−25,0 ) ( 125,−25,0 )

45.

( 8,1,−2 ) ( 8,1,−2 )

47.

( 1,2,3 ) ( 1,2,3 )

49.

( x, 31 28 3x 4 , 1 28 (−7x−3) ) ( x, 31 28 3x 4 , 1 28 (−7x−3) )

51.

No solutions exist.

53.

860 red velvet, 1,340 chocolate

55.

4% for account 1, 6% for account 2

57.

$126

59.

Banana was 3%, pumpkin was 7%, and rocky road was 2%

61.

100 almonds, 200 cashews, 600 pistachios

9.7 Section Exercises

1.

If A −1 A −1 is the inverse of A, A, then A A −1 =I, A A −1 =I, the identity matrix. Since A A is also the inverse of A −1 , A −1 A=I. A −1 , A −1 A=I. You can also check by proving this for a 2×2 2×2 matrix.

3.

No, because ad ad and bc bc are both 0, so adbc=0, adbc=0, which requires us to divide by 0 in the formula.

5.

Yes. Consider the matrix [ 0 1 1 0 ]. [ 0 1 1 0 ]. The inverse is found with the following calculation: A −1 = 1 0(0)−1(1) [ 0 −1 −1 0 ]=[ 0 1 1 0 ]. A −1 = 1 0(0)−1(1) [ 0 −1 −1 0 ]=[ 0 1 1 0 ].

7.

AB=BA=[ 1 0 0 1 ]=I AB=BA=[ 1 0 0 1 ]=I

9.

AB=BA=[ 1 0 0 1 ]=I AB=BA=[ 1 0 0 1 ]=I

11.

AB=BA=[ 1 0 0 0 1 0 0 0 1 ]=I AB=BA=[ 1 0 0 0 1 0 0 0 1 ]=I

13.

1 29 [ 9 2 −1 3 ] 1 29 [ 9 2 −1 3 ]

15.

1 69 [ −2 7 9 3 ] 1 69 [ −2 7 9 3 ]

17.

There is no inverse

19.

4 7 [ 0.5 1.5 1 −0.5 ] 4 7 [ 0.5 1.5 1 −0.5 ]

21.

1 17 [ −5 5 −3 20 −3 12 1 −1 4 ] 1 17 [ −5 5 −3 20 −3 12 1 −1 4 ]

23.

1 209 [ 47 −57 69 10 19 −12 −24 38 −13 ] 1 209 [ 47 −57 69 10 19 −12 −24 38 −13 ]

25.

[ 18 60 −168 −56 −140 448 40 80 −280 ] [ 18 60 −168 −56 −140 448 40 80 −280 ]

27.

( −5,6 ) ( −5,6 )

29.

( 2,0 ) ( 2,0 )

31.

( 1 3 , 5 2 ) ( 1 3 , 5 2 )

33.

( 2 3 , 11 6 ) ( 2 3 , 11 6 )

35.

( 7, 1 2 , 1 5 ) ( 7, 1 2 , 1 5 )

37.

( 5,0,−1 ) ( 5,0,−1 )

39.

1 34 ( −35,−97,−154 ) 1 34 ( −35,−97,−154 )

41.

1 690 ( 65,−1136,−229 ) 1 690 ( 65,−1136,−229 )

43.

( 37 30 , 8 15 ) ( 37 30 , 8 15 )

45.

( 10 123 ,−1, 2 5 ) ( 10 123 ,−1, 2 5 )

47.

1 2 [ 2 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 ] 1 2 [ 2 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 ]

49.

1 39 [ 3 2 1 7 18 53 32 10 24 36 21 9 9 46 16 5 ] 1 39 [ 3 2 1 7 18 53 32 10 24 36 21 9 9 46 16 5 ]

51.

[ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 1 1 1 1 ] [ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 1 1 1 1 ]

53.

Infinite solutions.

55.

50% oranges, 25% bananas, 20% apples

57.

10 straw hats, 50 beanies, 40 cowboy hats

59.

Tom ate 6, Joe ate 3, and Albert ate 3.

61.

124 oranges, 10 lemons, 8 pomegranates

9.8 Section Exercises

1.

A determinant is the sum and products of the entries in the matrix, so you can always evaluate that product—even if it does end up being 0.

3.

The inverse does not exist.

5.

2 2

7.

7 7

9.

4 4

11.

0 0

13.

7,990.7 7,990.7

15.

3 3

17.

1 1

19.

224 224

21.

15 15

23.

17.03 17.03

25.

( 1,1 ) ( 1,1 )

27.

( 1 2 , 1 3 ) ( 1 2 , 1 3 )

29.

( 2,5 ) ( 2,5 )

31.

( 1, 1 3 ) ( 1, 1 3 )

33.

( 15,12 ) ( 15,12 )

35.

( 1,3,2 ) ( 1,3,2 )

37.

( 1,0,3 ) ( 1,0,3 )

39.

( 1 2 ,1,2 ) ( 1 2 ,1,2 )

41.

( 2,1,4 ) ( 2,1,4 )

43.

Infinite solutions

45.

24 24

47.

1 1

49.

Yes; 18, 38

51.

Yes; 33, 36, 37

53.

$7,000 in first account, $3,000 in second account.

55.

120 children, 1,080 adult

57.

4 gal yellow, 6 gal blue

59.

13 green tomatoes, 17 red tomatoes

61.

Strawberries 18%, oranges 9%, kiwi 10%

63.

100 for movie 1, 230 for movie 2, 312 for movie 3

65.

20–29: 2,100, 30–39: 2,600, 40–49: 825

67.

300 almonds, 400 cranberries, 300 cashews

Review Exercises

1.

No

3.

( 2,3 ) ( 2,3 )

5.

( 4,1 ) ( 4,1 )

7.

No solutions exist.

9.

(300,60,000) (300,60,000)

10.

(400,30,000) (400,30,000)

11.

( 10,10,10 ) ( 10,10,10 )

13.

No solutions exist.

15.

( 1,2,3 ) ( 1,2,3 )

17.

( x, 8x 5 , 14x 5 ) ( x, 8x 5 , 14x 5 )

19.

11, 17, 33

21.

( 2,3 ),( 3,2 ) ( 2,3 ),( 3,2 )

23.

No solution

25.

No solution

27.
29.
31.

2 x+2 , 4 x+1 2 x+2 , 4 x+1

33.

7 x+5 , 15 (x+5) 2 7 x+5 , 15 (x+5) 2

35.

3 x5 , 4x+1 x 2 +5x+25 3 x5 , 4x+1 x 2 +5x+25

37.

x4 ( x 2 2) , 5x+3 ( x 2 2) 2 x4 ( x 2 2) , 5x+3 ( x 2 2) 2

39.

[ 16 8 4 12 ] [ 16 8 4 12 ]

41.

undefined; dimensions do not match

43.

undefined; inner dimensions do not match

45.

[ 113 28 10 44 81 41 84 98 42 ] [ 113 28 10 44 81 41 84 98 42 ]

47.

[ 127 74 176 2 11 40 28 77 38 ] [ 127 74 176 2 11 40 28 77 38 ]

49.

undefined; inner dimensions do not match

51.

x3z=7 y+2z=5 x3z=7 y+2z=5 with infinite solutions

53.

[ 2 2 1 2 8 5 19 10 22   |    7 0 3 ] [ 2 2 1 2 8 5 19 10 22   |    7 0 3 ]

55.

[ 1 0 3 −1 4 0 0 1 2   |    12 0 −7 ] [ 1 0 3 −1 4 0 0 1 2   |    12 0 −7 ]

57.

No solutions exist.

59.

No solutions exist.

61.

1 8 [ 2 7 6 1 ] 1 8 [ 2 7 6 1 ]

63.

No inverse exists.

65.

( 20,40 ) ( 20,40 )

67.

( 1,0.2,0.3 ) ( 1,0.2,0.3 )

69.

17% oranges, 34% bananas, 39% apples

71.

0

73.

6

75.

( 6, 1 2 ) ( 6, 1 2 )

77.

(x, 5x + 3)

79.

( 0,0, 1 2 ) ( 0,0, 1 2 )

Practice Test

1.

Yes

3.

No solutions exist.

5.

1 20 ( 10,5,4 ) 1 20 ( 10,5,4 )

7.

( x, 16x 5 13x 5 ) ( x, 16x 5 13x 5 )

9.

(2 2 , 17 ),( 2 2 , 17 ),( 2 2 , 17 ),( 2 2 , 17 ) (2 2 , 17 ),( 2 2 , 17 ),( 2 2 , 17 ),( 2 2 , 17 )

11.
13.

5 3x+1 2x+3 (3x+1) 2 5 3x+1 2x+3 (3x+1) 2

15.

[ 17 51 8 11 ] [ 17 51 8 11 ]

17.

[ 12 20 15 30 ] [ 12 20 15 30 ]

19.

1 8 1 8

21.

[ 14 2 13 2 3 6 1 5 12   |    140 1 11 ] [ 14 2 13 2 3 6 1 5 12   |    140 1 11 ]

23.

No solutions exist.

25.

( 100,90 ) ( 100,90 )

27.

( 1 100 ,0 ) ( 1 100 ,0 )

29.

32 or more cell phones per day

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