College Algebra, 2013a

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558 Systems of Equations and Matrices Now we enforce the conditions stated in Definition 8.3 for the variable y. To that end we need to get the coefficient of y in E2 equal to 1. We apply the second move listed in Theorem 8.1 and replace E2 with itself times − 1 2 . ⎧ ⎨ ⎩ (E1) x − y + z = 5 (E2) −2y + z = 6 (E3) 2y − 2z = −12 Replace E2 with− 1 2 E2 −−−−−−−−−−−−−−→ ⎧ ⎨ ⎩ (E1) x − y + z = 5 (E2) y − 1 2 z = −3 (E3) 2y − 2z = −12 To eliminate the y in E3, we add −2E2 toit. ⎧ ⎨ ⎩ (E1) x − y + z = 5 (E2) y − 1 2 z = −3 (E3) 2y − 2z = −12 Replace E3 with−2E2+E3 −−−−−−−−−−−−−−−−−→ ⎧ ⎨ ⎩ (E1) x − y + z = 5 (E2) y − 1 2 z = −3 (E3) −z = −6 Finally, we apply the second move from Theorem 8.1 one last time and multiply E3 by−1 to satisfy the conditions of Definition 8.3 for the variable z. ⎧ ⎨ ⎩ (E1) x − y + z = 5 (E2) y − 1 2 z = −3 (E3) −z = −6 Replace E3 with−1E3 −−−−−−−−−−−−−−→ ⎧ ⎨ ⎩ (E1) x − y + z = 5 (E2) y − 1 2 z = −3 (E3) z = 6 Now we proceed to substitute. Plugging in z =6intoE2 givesy − 3=−3 sothaty =0. With y = 0 and z =6,E1 becomes x − 0 + 6 = 5, or x = −1. Our solution is (−1, 0, 6). We leave it to the reader to check that substituting the respective values for x, y, andz into the original system results in three identities. Since we have found a solution, the system is consistent; since there are no free variables, it is independent. 2. Proceeding as we did in 1, our first step is to get an equation with x in the E1 position with 1 as its coefficient. Since there is no easy fix, we multiply E1 by 1 2 . ⎧ ⎨ ⎩ (E1) 2x +3y − z = 1 (E2) 10x − z = 2 (E3) 4x − 9y +2z = 5 Replace E1 with 1 2 E1 −−−−−−−−−−−−−→ ⎧ ⎨ ⎩ (E1) x + 3 2 y − 1 2 z = 1 2 (E2) 10x − z = 2 (E3) 4x − 9y +2z = 5 Nowit’stimetotakecareofthex’s in E2 andE3. ⎧ ⎨ ⎩ (E1) x + 3 2 y − 1 2 z = 1 2 (E2) 10x − z = 2 (E3) 4x − 9y +2z = 5 Replace E2 with−10E1+E2 −−−−−−−−−−−−−−−−−−→ Replace E3 with−4E1+E3 ⎧ ⎨ ⎩ (E1) x + 3 2 y − 1 2 z = 1 2 (E2) −15y +4z = −3 (E3) −15y +4z = 3
8.1 Systems of Linear Equations: Gaussian Elimination 559 Our next step is to get the coefficient of y in E2 equal to 1. To that end, we have ⎧ ⎪⎨ ⎪⎩ (E1) x + 3 2 y − 1 2 z = 1 2 (E2) −15y +4z = −3 (E3) −15y +4z = 3 Replace E2 with− 1 15 E2 −−−−−−−−−−−−−−−→ ⎧ ⎪⎨ ⎪⎩ (E1) x + 3 2 y − 1 2 z = 1 2 (E2) y − 4 15 z = 1 5 (E3) −15y +4z = 3 Finally, we rid E3 ofy. ⎧ ⎪⎨ ⎪⎩ (E1) x + 3 2 y − 1 2 z = 1 2 (E2) y − 4 15 z = 1 5 (E3) −15y +4z = 3 Replace E3 with15E2+E3 −−−−−−−−−−−−−−−−−→ ⎧ ⎪⎨ ⎪⎩ (E1) x − y + z = 5 (E2) y − 1 2 z = −3 (E3) 0 = 6 The last equation, 0 = 6, is a contradiction so the system has no solution. According to Theorem 8.1, since this system has no solutions, neither does the original, thus we have an inconsistent system. 3. For our last system, we begin by multiplying E1 by 1 3 to get a coefficient of 1 on x 1. ⎧ ⎨ ⎩ (E1) 3x 1 + x 2 + x 4 = 6 (E2) 2x 1 + x 2 − x 3 = 4 (E3) x 2 − 3x 3 − 2x 4 = 0 Replace E1 with 1 3 E1 −−−−−−−−−−−−−→ ⎧ ⎨ ⎩ (E1) x 1 + 1 3 x 2 + 1 3 x 4 = 2 (E2) 2x 1 + x 2 − x 3 = 4 (E3) x 2 − 3x 3 − 2x 4 = 0 Next we eliminate x 1 from E2 ⎧ ⎪⎨ ⎪⎩ (E1) x 1 + 1 3 x 2 + 1 3 x 4 = 2 (E2) 2x 1 + x 2 − x 3 = 4 (E3) x 2 − 3x 3 − 2x 4 = 0 Replace E2 −−−−−−−−−−→ with −2E1+E2 ⎧ ⎪⎨ ⎪⎩ (E1) x 1 + 1 3 x 2 + 1 3 x 4 = 2 (E2) 1 3 x 2 − x 3 − 2 3 x 4 = 0 (E3) x 2 − 3x 3 − 2x 4 = 0 We switch E2 andE3 togetacoefficientof1forx 2 . ⎧ ⎪⎨ ⎪⎩ (E1) x 1 + 1 3 x 2 + 1 3 x 4 = 2 (E2) 1 3 x 2 − x 3 − 2 3 x 4 = 0 (E3) x 2 − 3x 3 − 2x 4 = 0 Finally, we eliminate x 2 in E3. Switch E2 and E3 −−−−−−−−−−−→ ⎧ ⎪⎨ ⎪⎩ (E1) x 1 + 1 3 x 2 + 1 3 x 4 = 2 (E2) x 2 − 3x 3 − 2x 4 = 0 (E3) 1 3 x 2 − x 3 − 2 3 x 4 = 0
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College Algebra Version ⌊π⌋ Co
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Table of Contents Preface ix 1 Rela
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Table of Contents v 4.3.3 Answers .
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Preface Thank you for your interest
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xi text and we have gone to great l
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Chapter 1 Relations and Functions 1
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1.1 Sets of Real Numbers and the Ca
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1.2 Relations 21 Solution. 1. To gr
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1.2 Relations 23 lines y = 1 and y
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1.2 Relations 25 ‘connecting the
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1.2 Relations 27 (x − 2) 2 + y 2
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1.2 Relations 29 1.2.2 Exercises In
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1.2 Relations 31 29. 5 4 3 y 30. 2
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1.2 Relations 33 1.2.3 Answers 1. y
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1.2 Relations 35 13. y 14. y 3 2
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1.2 Relations 37 41. y = x 2 +1 The
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1.2 Relations 39 45. y = √ x −
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1.2 Relations 41 49. (x +2) 2 + y 2
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1.3 Introduction to Functions 43 1.
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1.3 Introduction to Functions 45 So
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1.3 Introduction to Functions 47 Al
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1.3 Introduction to Functions 51 23
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1.4 Function Notation 55 1.4 Functi
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1.4 Function Notation 57 (c) To fin
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1.4 Function Notation 59 1 − 4x x
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1.4 Function Notation 61 these limi
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1.4 Function Notation 63 1.4.2 Exer
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1.4 Function Notation 65 ⎧ ⎪⎨
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1.4 Function Notation 67 (b) How mu
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1.4 Function Notation 69 1.4.3 Answ
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1.4 Function Notation 71 18. For f(
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1.4 Function Notation 73 26. For f(
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1.4 Function Notation 75 69. C(0) =
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1.5 Function Arithmetic 77 ( ) 4. F
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1.5 Function Arithmetic 79 Next, we
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1.5 Function Arithmetic 81 r(x + h)
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1.5 Function Arithmetic 83 Solution
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1.5 Function Arithmetic 85 27. f(x)
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1.5 Function Arithmetic 87 1.5.2 An
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1.5 Function Arithmetic 89 13. For
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1.5 Function Arithmetic 91 37. 39.
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1.6 Graphs of Functions 93 1.6 Grap
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1.6 Graphs of Functions 95 In the p
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1.6 Graphs of Functions 97 The calc
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1.6 Graphs of Functions 99 The calc
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1.6 Graphs of Functions 101 on [−
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1.6 Graphs of Functions 103 1. Find
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1.6 Graphs of Functions 105 Example
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1.6 Graphs of Functions 107 1.6.2 E
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1.6 Graphs of Functions 109 In Exer
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1.6 Graphs of Functions 111 For Exe
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1.6 Graphs of Functions 113 y y 4 5
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1.6 Graphs of Functions 115 6. f(x)
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1.6 Graphs of Functions 117 17. y 1
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1.6 Graphs of Functions 119 90. h i
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1.7 Transformations 121 7 y 7 y (5,
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1.7 Transformations 123 2 to all of
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1.7 Transformations 125 2 1 (0, 0)
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1.7 Transformations 127 Solution. 1
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1.7 Transformations 129 y y 3 2 1 (
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1.7 Transformations 131 A few remar
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1.7 Transformations 133 6 y 6 y (4,
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1.7 Transformations 135 Finally, to
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1.7 Transformations 137 (a, f(a)) a
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1.7 Transformations 139 shift right
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1.7 Transformations 141 Thecomplete
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1.7 Transformations 143 64. The gra
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1.7 Transformations 145 25. y =2−
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1.7 Transformations 147 38. g(x) =f
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1.7 Transformations 149 50. y = S 1
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Chapter 2 Linear and Quadratic Func
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2.1 Linear Functions 153 P 2 5. m =
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2.1 Linear Functions 155 Equation 2
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2.1 Linear Functions 157 y 4 3 2 y
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2.1 Linear Functions 159 4. We find
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2.1 Linear Functions 161 average sp
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2.1 Linear Functions 163 2.1.1 Exer
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2.1 Linear Functions 165 40. A rest
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2.1 Linear Functions 167 61. y = 2
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2.1 Linear Functions 169 2.1.2 Answ
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2.1 Linear Functions 171 35. (a) F
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2.2 Absolute Value Functions 173 2.
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2.2 Absolute Value Functions 177 By
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2.2 Absolute Value Functions 179 Ex
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2.2 Absolute Value Functions 181 wh
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2.2 Absolute Value Functions 185 24
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2.2 Absolute Value Functions 187 31
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2.3 Quadratic Functions 189 y y (
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2.3 Quadratic Functions 191 shows t
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2.3 Quadratic Functions 193 From g(
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2.3 Quadratic Functions 195 ( a x +
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2.3 Quadratic Functions 197 To find
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2.3 Quadratic Functions 199 Neverth
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2.3 Quadratic Functions 201 15. The
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2.3 Quadratic Functions 203 2.3.2 A
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2.3 Quadratic Functions 205 8. f(x)
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2.3 Quadratic Functions 207 25. (a)
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2.4 Inequalities with Absolute Valu
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2.5 Regression 225 2.5 Regression W
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2.5 Regression 227 2. Find the leas
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2.5 Regression 229 The coefficient
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2.5 Regression 231 4. The chart bel
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2.5 Regression 233 2.5.2 Answers 1.
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Chapter 3 Polynomial Functions 3.1
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3.1 Graphs of Polynomials 237 of th
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3.1 Graphs of Polynomials 239 In or
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3.1 Graphs of Polynomials 241 Despi
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3.1 Graphs of Polynomials 243 at th
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3.1 Graphs of Polynomials 245 Theor
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3.1 Graphs of Polynomials 247 In Ex
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3.1 Graphs of Polynomials 249 37. T
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3.1 Graphs of Polynomials 251 9. f(
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3.1 Graphs of Polynomials 253 21. g
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3.1 Graphs of Polynomials 255 33. (
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3.2 The Factor Theorem and the Rema
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3.3 Real Zeros of Polynomials 269 3
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3.3 Real Zeros of Polynomials 275 f
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3.3 Real Zeros of Polynomials 277 T
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3.3 Real Zeros of Polynomials 279 (
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3.3 Real Zeros of Polynomials 281 I
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3.4 Complex Zeros and the Fundament
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Chapter 4 Rational Functions 4.1 In
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4.1 Introduction to Rational Functi
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4.2 Graphs of Rational Functions 32
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4.3 Rational Inequalities and Appli
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Chapter 5 Further Topics in Functio
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5.1 Function Composition 361 2. As
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5.1 Function Composition 363 (+)
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5.1 Function Composition 365 9. The
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5.1 Function Composition 367 Theore
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5.1 Function Composition 369 5.1.1
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5.1 Function Composition 371 50. (g
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5.1 Function Composition 373 7. For
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5.1 Function Composition 375 20. Fo
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5.1 Function Composition 377 56. (g
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5.2 Inverse Functions 379 The main
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5.2 Inverse Functions 381 y = f −
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5.2 Inverse Functions 383 (b) We ca
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5.2 Inverse Functions 385 y = f(x)
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5.2 Inverse Functions 387 = = 2x 2x
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5.2 Inverse Functions 389 We get g
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5.2 Inverse Functions 391 Using wha
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5.2 Inverse Functions 393 1. Explai
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5.2 Inverse Functions 395 26. Show
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5.3 Other Algebraic Functions 397 5
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5.3 Other Algebraic Functions 399 I
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5.3 Other Algebraic Functions 401 b
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5.3 Other Algebraic Functions 405
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5.3 Other Algebraic Functions 415 (
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Chapter 6 Exponential and Logarithm
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6.1 Introduction to Exponential and
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6.2 Properties of Logarithms 437 6.
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6.2 Properties of Logarithms 439 lo
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6.2 Properties of Logarithms 441 ex
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6.2 Properties of Logarithms 443 as
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6.3 Exponential Equations and Inequ
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6.4 Logarithmic Equations and Inequ
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6.5 Applications of Exponential and
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Chapter 7 Hooked on Conics 7.1 Intr
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7.1 Introduction to Conics 497 If t
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7.2 Circles 499 4 y 3 2 1 −4 −3
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7.2 Circles 501 We close this secti
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7.2 Circles 503 7.2.2 Answers 1. (x
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7.3 Parabolas 505 7.3 Parabolas We
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Page 523 and 524: 7.3 Parabolas 513 7.3.2 Answers 1.
Page 525 and 526: 7.3 Parabolas 515 8. ( y + 2) 3 2 (
Page 527 and 528: 7.4 Ellipses 517 Minor Axis Major A
Page 529 and 530: 7.4 Ellipses 519 This equation is f
Page 531 and 532: 7.4 Ellipses 521 As with circles an
Page 533 and 534: 7.4 Ellipses 523 From this sketch,
Page 535 and 536: 7.4 Ellipses 525 7.4.1 Exercises In
Page 537 and 538: 7.4 Ellipses 527 7.4.2 Answers x 2
Page 539 and 540: 7.4 Ellipses 529 (x − 4) 2 (y −
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Page 543 and 544: 7.5 Hyperbolas 533 endpoints of the
Page 545 and 546: 7.5 Hyperbolas 535 from the center
Page 547 and 548: 7.5 Hyperbolas 537 As with the othe
Page 549 and 550: 7.5 Hyperbolas 539 y 6 5 4 3 2 Jeff
Page 551 and 552: 7.5 Hyperbolas 541 7.5.1 Exercises
Page 553 and 554: 7.5 Hyperbolas 543 is a hyperbola.
Page 555 and 556: 7.5 Hyperbolas 545 (y − 3) 2 (x
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Page 559 and 560: Chapter 8 Systems of Equations and
Page 561 and 562: 8.1 Systems of Linear Equations: Ga
Page 567: 8.1 Systems of Linear Equations: Ga
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Page 591 and 592: 8.3 Matrix Arithmetic 581 As did ma
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Page 595 and 596: 8.3 Matrix Arithmetic 585 Using Ri
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8.4 Systems of Linear Equations: Ma
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8.5 Determinants and Cramer’s Rul
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8.6 Partial Fraction Decomposition
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8.7 Systems of Non-Linear Equations
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20. Solve the following system ⎧
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Chapter 9 Sequences and the Binomia
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9.1 Sequences 653 3. From {2n − 1
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9.1 Sequences 655 Solution. A good
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9.1 Sequences 657 Looking at the de
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9.1 Sequences 659 28. 0.9, 0.99, 0.
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9.2 Summation Notation 661 9.2 Summ
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9.2 Summation Notation 663 5∑ (
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9.2 Summation Notation 665 S = n (2
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9.2 Summation Notation 667 payment
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9.2 Summation Notation 669 n∑ k=1
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9.2 Summation Notation 671 In Exerc
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9.3 Mathematical Induction 673 9.3
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9.3 Mathematical Induction 675 This
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9.3 Mathematical Induction 677 it i
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9.3 Mathematical Induction 679 9.3.
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9.4 The Binomial Theorem 681 9.4 Th
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9.4 The Binomial Theorem 683 songs
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9.4 The Binomial Theorem 685 (a + b
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9.4 The Binomial Theorem 687 ∑k
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9.4 The Binomial Theorem 689 ( 0 0)
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9.4 The Binomial Theorem 691 9.4.1
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Index n th root of a complex number
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Index 1071 central angle, 701 chang
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Index 1073 reflective property, 523
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Index 1075 matrix, multiplicative,
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Index 1077 midpoint definition of,
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Index 1079 instantaneous, 161, 472
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Index 1081 inconsistent, 553 indepe
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College Algebra, 2013a

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