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Chapter 6 Resource Masters StudentWorks PlusTM includes the entire Student Edition text along with the worksheets in this booklet. TeacherWorks PlusTM includes all of the materials found in this booklet for viewing, printing, and editing. Cover: Jason Reed/Photodisc/Getty Images Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Permission is granted to reproduce the material contained herein on the condition that such materials be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with the Glencoe Precalculus program. Any other reproduction, for sale or other use, is expressly prohibited. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240 - 4027 ISBN: 978-0-07-893807-8 MHID: 0-07-893807-4 Printed in the United States of America. 2 3 4 5 6 7 8 9 10 079 18 17 16 15 14 13 12 11 10 Contents Teacher’s Guide to Using the Chapter 6 Resource Masters ........................................... iv Lesson 6-4 Partial Fractions Study Guide and Intervention .......................... 22 Practice............................................................ 24 Word Problem Practice ................................... 25 Enrichment ...................................................... 26 Chapter Resources Student-Built Glossary ....................................... 1 Anticipation Guide (English) .............................. 3 Anticipation Guide (Spanish) ............................. 4 Lesson 6-5 Lesson 6-1 Linear Optimization Study Guide and Intervention .......................... 27 Practice............................................................ 29 Word Problem Practice ................................... 30 Enrichment ...................................................... 31 Multivariable Linear Systems and Row Operations Study Guide and Intervention ............................ 5 Practice.............................................................. 7 Word Problem Practice ..................................... 8 Enrichment ........................................................ 9 TI-Nspire Activity ............................................. 10 Assessment Chapter 6 Quizzes 1 and 2 ............................. 33 Chapter 6 Quizzes 3 and 4 ............................. 34 Chapter 6 Mid-Chapter Test ............................ 35 Chapter 6 Vocabulary Test ............................. 36 Chapter 6 Test, Form 1 ................................... 37 Chapter 6 Test, Form 2A................................. 39 Chapter 6 Test, Form 2B................................. 41 Chapter 6 Test, Form 2C ................................ 43 Chapter 6 Test, Form 2D ................................ 45 Chapter 6 Test, Form 3 ................................... 47 Chapter 6 Extended-Response Test ............... 49 Standardized Test Practice ............................. 50 Lesson 6-2 Matrix Multiplication, Inverses, and Determinants Study Guide and Intervention .......................... 11 Practice............................................................ 13 Word Problem Practice ................................... 14 Enrichment ...................................................... 15 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Lesson 6-3 Solving Linear Systems Using Inverses and Cramer’s Rule Study Guide and Intervention .......................... 16 Practice............................................................ 18 Word Problem Practice ................................... 19 Enrichment ...................................................... 20 Spreadsheet Activity ........................................ 21 Chapter 6 Answers ........................................... A1–A23 iii Glencoe Precalculus Teacher’s Guide to Using the Chapter 6 Resource Masters The Chapter 6 Resource Masters includes the core materials needed for Chapter 6. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet. Practice This master closely follows the types of problems found in the Exercises section of the Student Edition and includes word problems. Use as an additional practice option or as homework for second-day teaching of the lesson. Chapter Resources Student-Built Glossary (pages 1–2) These masters are a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to record definitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar. Give this to students before beginning Lesson 6-1. Encourage them to add these pages to their mathematics study notebooks. Remind them to complete the appropriate words as they study each lesson. Word Problem Practice This master includes additional practice in solving word problems that apply to the concepts of the lesson. Use as an additional practice or as homework for second-day teaching of the lesson. Enrichment These activities may extend the concepts of the lesson, offer an historical or multicultural look at the concepts, or widen students’ perspectives on the mathematics they are learning. They are written for use with all levels of students. Graphing Calculator, TI–Nspire, or Spreadsheet Activities These activities present ways in which technology can be used with the concepts in some lessons of this chapter. Use as an alternative approach to some concepts or as an integral part of your lesson presentation. Lesson Resources Study Guide and Intervention These masters provide vocabulary, key concepts, additional worked-out examples and Guided Practice exercises to use as a reteaching activity. It can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent. Chapter 6 iv Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Anticipation Guide (pages 3–4) This master, presented in both English and Spanish, is a survey used before beginning the chapter to pinpoint what students may or may not know about the concepts in the chapter. Students will revisit this survey after they complete the chapter to see if their perceptions have changed. Leveled Chapter Tests Assessment Options The assessment masters in the Chapter 6 Resource Masters offer a wide range of assessment tools for formative (monitoring) assessment and summative (final) assessment. • Form 1 contains multiple-choice questions and is intended for use with below grade level students. • Forms 2A and 2B contain multiple-choice questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations. • Forms 2C and 2D contain free-response questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations. • Form 3 is a free-response test for use with above grade level students. All of the above mentioned tests include a free-response Bonus question. Quizzes Four free-response quizzes offer assessment at appropriate intervals in the chapter. Mid-Chapter Test This 1-page test provides an option to assess the first half of the chapter. It parallels the timing of the Mid-Chapter Quiz in the Student Edition and includes both multiple-choice and free-response questions. Vocabulary Test This test is suitable for all students. It includes a list of vocabulary words and questions to assess students’ knowledge of those words. This can also be used in conjunction with one of the leveled chapter tests. Extended-Response Test Performance assessment tasks are suitable for all students. Sample answers are included for evaluation. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Standardized Test Practice These three pages are cumulative in nature. It includes two parts: multiple-choice questions with bubble-in answer format and short-answer free-response questions. Answers • The answers for the Anticipation Guide and Lesson Resources are provided as reduced pages. • Full-size answer keys are provided for the assessment masters. Chapter 6 v Glencoe Precalculus NAME DATE 6 PERIOD This is an alphabetical list of key vocabulary terms you will learn in Chapter 6. As you study this chapter, complete each term’s definition or description. Remember to add the page number where you found the term. Add these pages to your Precalculus Study Notebook to review vocabulary at the end of the chapter. Vocabulary Term Found on Page Definition/Description/Example augmented matrix coefficient matrix constraints Cramer’s Rule Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. determinant feasible solutions Gaussian elimination Gauss-Jordan elimination identity matrix inverse (continued on the next page) Chapter 6 1 Glencoe Precalculus Chapter Resources Student-Built Glossary NAME DATE 6 PERIOD Student-Built Glossary Vocabulary Term Found on Page Definition/Description/Example inverse matrix invertible linear programming multivariable linear system objective function optimization Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. partial fraction partial fraction decomposition reduced row-echelon form (esh-a-lon) row-echelon form singular matrix square system Chapter 6 2 Glencoe Precalculus NAME 6 DATE PERIOD Anticipation Guide Step 1 Before you begin Chapter 6 • Read each statement. • Decide whether you Agree (A) or Disagree (D) with the statement. • Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure). STEP 1 A, D, or NS STEP 2 A or D Statement 1. The augmented matrix of a system is derived from the coefficients and constant terms of the linear equations. 2. The row-echelon form of a matrix is unique. 3. If a matrix has an inverse, then it is a singular matrix. 4. The product of an m × r matrix and an r × n matrix results in an m × n matrix. 5. Cramer’s Rule uses inverses to solve systems. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 6. Cramer’s Rule applies when the determinant of the coefficient matrix is 0. 7. To find the partial fraction decomposition of an improper f(x) d(x) r(x) f(x) algorithm − = q(x) + − to rewrite it as the sum of a d(x) d(x) rational expression − , you must use the division polynomial and a proper rational expression. 8. Graphically, a rational function and its partial fraction decomposition are different. 9. Linear programming can be used to solve applications involving systems of equations. 10. In a linear programming problem, one evaluates the objective function at each vertex of the feasible region to maximize or minimize the function. Step 2 After you complete Chapter 6 • Reread each statement and complete the last column by entering an A or a D. • Did any of your opinions about the statements change from the first column? • For those statements that you mark with a D, use a piece of paper to write an example of why you disagree. Chapter 6 3 Glencoe Precalculus Chapter Resources Systems of Equations and Matrices NOMBRE 6 FECHA PERÍODO Ejercicios preparatorios Sistemas de ecuaciones y matrices Paso 1 Antes de que comiences el Capítulo 6 • Lee cada enunciado. • Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado. • Escribe A o D en la primera columna O si no estás seguro(a), escribe NS (no estoy seguro(a)). PASO 1 A, D o NS PASO 2 AoD Enunciado 1. La matriz aumentada de un sistema se deriva de los coeficientes y los términos constantes de las ecuaciones lineales. 2. La forma escalón por filas de una matriz es única. 3. Si una matriz tiene inversa, entonces es una matriz singular. 4. El producto de una matriz m × r por una matriz r × n es una matriz m × n. 5. La regla de Cramer permite resolver sistemas mediante el uso de inversas. 6. La regla de Cramer se aplica cuando el determinante de la matriz de coeficientes es cero. f(x) d(x) expresión racional impropia − , se debe usar el algoritmo de f(x) d(x) r(x) d(x) división − = q(x) + − para poder reformular dicha expresión como la suma de un polinomio y una expresión racional propia. 8. Una función racional y su descomposición en fracciones parciales son gráficamente distintas. 9. La programación lineal se usa para resolver aplicaciones que impliquen sistemas de ecuaciones. 10. En problemas de programación lineal, se evalúa la función objetivo en cada vértice de la región factible para maximizar o minimizar la función. Paso 2 Después de que termines el Capítulo 6 • Relee cada enunciado y escribe A o D en la última columna. • Compara la última columna con la primera. ¿Cambiaste de opinión sobre alguno de los enunciados? • En los casos en que hayas estado en desacuerdo con el enunciado, escribe en una hoja aparte un ejemplo de por qué no estás de acuerdo. Capítulo 6 4 Precálculo de Glencoe Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 7. Para calcular la descomposición en fracciones parciales de la NAME DATE 6-1 PERIOD Study Guide and Intervention Multivariable Linear Systems and Row Operations Gaussian Elimination You can solve a system of linear equations using matrices. Solving a system by transforming it into an equivalent system is called Gaussian elimination. First, create the augmented matrix. Then use elementary row operations to transform the matrix so that it is in row-echelon form. Then write the corresponding system of equations and use substitution to solve the system. Lesson 6-1 Example Solve the system of equations using Gaussian elimination with matrices. x - 2y + z = -1 2x + y - 3z = -7 3x - y + 2z = 0 Step 1 Write the augmented matrix. ⎡1 -2 1 -1⎤ 2 1 -3 -7 ⎣3 -1 2 0⎦ ⎢ Step 2 Apply elementary row operations to obtain a row-echelon form of the matrix. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. a. ⎡1 -2 1 ⎢ -1⎤ R2 - 2R1→ 0 5 -5 -5 ⎣3 -1 2 ⎢ 0⎦ b. d. e. ⎢ ⎢ ⎡ 1 -2 1 -1 ⎤ 0 1 -1 -1 R3 - 5R2→ ⎣ 0 0 4 8 ⎦ ⎢ ⎡ 1 -2 1 ⎢ -1 ⎤ 0 5 -5 -5 ⎣ R3 - 3R1→ 0 5 -1 ⎢ 3 ⎦ ⎢ ⎢ c. ⎡ 1 -2 1 -1 ⎤ 1 -1 -1 ⎣ 0 5 -1 3 ⎦ 1 − R→ 0 5 2 ⎢ ⎡ 1 -2 1 -1 ⎤ 0 1 -1 -1 1 − R3→ ⎣ 0 0 1 2 ⎦ 4 ⎢ Step 3 Write the corresponding system of equations and use substitution to solve the system. x - 2y + z = -1 y - z = -1 z=2 The solution of the system is x = -1, y = 1, and z = 2 or (-1, 1, 2). Exercises Solve each system of equations using Gaussian elimination with matrices. 1. -2x - y + z = 0 2. 5x - y = -13 3. -4x - y - z = -11 x + 2y - z = -3 -3x + 2y - z = 8 x-z=2 3x + y - 2z = -1 x - 4y + z = -10 2y + 4z = 0 Chapter 6 5 Glencoe Precalculus NAME DATE 6-1 Study Guide and Intervention PERIOD (continued) Multivariable Linear Systems and Row Operations Gauss-Jordan Elimination If you continue to apply elementary row operations to the row-echelon form of any augmented matrix, you can obtain a matrix in which every column has one element equal to 1 and the remaining elements equal to 0. This is called the reduced row-echelon form of the matrix. Solving a system by transforming an augmented matrix so that it is in reduced row-echelon form is called Gauss-Jordan elimination. Example ⎡1 0 0 a⎤ 0 1 0 b ⎣0 0 1 c⎦ ⎢ Solve the system of equations. x - 2y + z = 15 -2x - y + 2z = -1 -x + y = -9 Write the augmented matrix. Apply elementary row operations to obtain a row-echelon form. Then apply elementary row operations to obtain zeros above the leading 1s in each row. Augmented Matrix ⎡ 1 -2 1 15 ⎤ -2 -1 2 -1 ⎣ -1 1 0 -9 ⎦ ⎢ ⎡ 1 -2 1 15 ⎤ 2R1 + R2→ 0 -5 4 29 ⎣ -1 1 0 -9 ⎦ ⎢ ⎡ 1 -2 1 15 ⎤ 0 -5 4 29 R1 + R3→ ⎣ 0 -1 1 6 ⎦ Row-echelon form ⎢ ⎡ 1 -2 1 15 ⎤ R2 + 2R3→ 0 1 0 -5 ⎣0 0 1 1⎦ ⎢ ⎡ 1 -2 1 15 ⎤ 0 1 -2 -7 ⎣ -R3→ 0 0 1 1 ⎦ ⎢ R1 + 2R2→ ⎡ 1 0 1 5 ⎤ 0 1 0 -5 ⎣0 0 1 1⎦ ⎢ ⎢ ⎡ 1 -2 1 15 ⎤ 0 1 -2 -7 ⎣ R2 + R3→ 0 0 -1 -1 ⎦ Reduced row-echelon form R1 - R3→ ⎡ 1 0 0 4 ⎤ 0 1 0 -5 ⎣0 0 1 1⎦ ⎢ The solution of the system is x = 4, y = -5, and z = 1 or (4, -5, 1). Exercises Solve each system of equations using Gaussian or Gauss-Jordan elimination. 1. 3x - 2y + z = -22 2. x - 4z = 6 3. -2x - y - z = 1 -4x + z = 31 -2y + 3z = -2 -x + 3y - 2z = 24 2x - 5y = -24 2x - 5y = 6 4x + 2y + z = 2 Chapter 6 6 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. ⎡ 1 -2 1 15 ⎤ R2 - 6R3→ 0 1 -2 -7 ⎣ 0 -1 1 6 ⎦ ⎢ NAME 6-1 DATE PERIOD Practice Multivariable Linear Systems and Row Operations Write each system of equations in triangular form using Gaussian elimination. Then solve the system. ⎢ ⎡ -5 -3 0 -2 ⎤ 3. 0 -2 6 24 ⎣ 4 0 -7 2 ⎦ ⎢ Write the augmented matrix for each system of linear equations. 4. 5x - 2y = 14 5. 3x + 4y + 7z = -8 -3x + y = -7 6. -4x - 2y - z = 5 -2x - 3y + z = 6 2x - z = 8 5x - 2y + z = 4 y - 2z = -4 Solve each system of equations using Gauss-Jordan elimination. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 7. -4x - 2y = -6 8. -2x - 5y + z = 6 x + 3y = -11 9. 8x - y + 3z = -38 3x + 2y - 4z = -1 -2x + 5y - 4z = 32 5x - y + 2z = -6 x - y + z = -9 10. FRUIT Three customers bought fruit at Michael’s Groceries. The table shows the amount of fruit bought by each person. Write and solve a system of equations to determine the price of each type of fruit. Chapter 6 Name Apples Oranges Pears Total Cost ($) Rosario 5 4 3 13.50 Lindsay 7 2 4 14.20 Edwin 3 8 2 15.30 7 Glencoe Precalculus Lesson 6-1 ⎡ 2 3 1 -23 ⎤ 2. -3 -1 4 -5 ⎣ -1 5 -1 -19 ⎦ ⎡ 1 -1 -12 ⎤ 1. ⎢ ⎣ -3 2 32 ⎦ NAME 6-1 DATE PERIOD Word Problem Practice Multivariable Linear Systems and Row Operations 1. FOOD Mark bought 5 hamburgers and 3 bags of chips at a cost of $16.25. Henry bought 4 hamburgers and 8 bags of chips at a cost of $20. Write and solve a system of equations to determine the cost of a hamburger and a bag of chips. 4. MOVIES The table shows the number of individuals attending the movies over the weekend at the Majestic Theater. Determine the costs for a child, adult, and senior citizen to attend the movies. 2. MANUFACTURING A company manufactures tables, chairs, and stools. Last week, it built a total of 275 items. The number of chairs built was four times the total number of tables and stools built. The total value of these items is $42,125 with a chair selling for $150, a table for $200, and a stool for $75. Write and solve a system of equations to determine the number of each item built last week. Day Child Adult Senior Citizen Total Paid($) Fri 80 110 25 1755 Sat 100 175 40 2685 Sun 45 85 30 1385 a. Write a system of equations representing Mr. Wiley’s investment pattern. 3. COINS Tina has 31 nickels, dimes, and quarters in her purse. She has 5 more nickels then the total number of dimes and quarters. If the total value of the coins is $3.25, how many of each coin does Tina have in her purse? Write and solve a system to determine the number of coins. b. Write the augmented matrix for the system of equations that you wrote in part a. c. Solve the system that you wrote in part b using Gauss-Jordan elimination. Chapter 6 8 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 5. INVESTING Mr. Wiley invested $5000 in three different accounts at the beginning of last year, yielding him a total of $182.50 of interest at the end of the year. The three accounts were a simple savings account earning 1%, a certificate of deposit earning 3.5%, and municipal bonds earning 4.3%. His municipal bond investment was 5 times the amount of money invested in the simple savings account. NAME DATE 6-1 PERIOD Enrichment Circles The general form equation for a circle is Ax2 + Ay2 + Dx + Ey + F = 0, where A ≠ 0. Suppose you want to find the equation of a circle passing through the points (-1, 2), (3, 4), and (2, -1). How can you use the general form equation, systems of equations, and matrices to answer the question? Because A ≠ 0, divide both sides of the equation by A. The resulting D E F equation is x2 + y2 + Dx + Ey + F = 0, where D = − , E = − , and F = − . A A Lesson 6-1 A Because the three points are on the circle, they satisfy this equation. Use substitution to get the following system. 1 + 4 -1D + 2E + F = 0 9 + 16 + 3D + 4E + F = 0 4 + 1 + 2D -1E + F = 0 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. The augmented matrix for this system is -D + 2E + F = -5 3D + 4E + F = -25 2D + -E + F = -5 or ⎡ -1 2 1 -5 ⎤ 3 4 1 -25 . ⎣ 2 -1 1 -5 ⎦ ⎢ ⎡ 1 -2 -1 ⎢ 5⎤ Using Gaussian elimination, you can find the equivalent matrix to be 0 1 -5 5 . ⎣ 0 0 18 ⎢ -30 ⎦ ⎢ ⎢ 30 5 10 10 Using substitution, you can find that F = - − or - − , E = - − , and D = - − . 18 3 3 3 Substituting these values back into x2 + y2 + Dx + Ey + F = 0, you get 10 10 5 x-− y-− = 0. Multiplying both sides of the equation by 3 results x2 + y2 - − 3 3 3 in the equation of the circle in general form: 3x2 + 3y2 - 10x - 10y - 5 = 0. Exercises Find an equation of the circle passing through the given points. 1. (1, 0), (-1, 2), (3, 1) Chapter 6 2. (3, 6), (5, 4), (3, 2) 9 Glencoe Precalculus NAME 6-1 DATE PERIOD TI-Nspire Activity Reduced Row-Echelon Form You can solve a system of equations by entering the augmented matrix into a TI-Nspire and finding the reduced row-echelon form of the matrix. Example Solve the system of equations. x+y+z=5 2x + 3y - z = 55 -x + 4y + 2z = 4 Step 1: Add a CALCULATOR page. Enter the augmented matrix by pressing / and the multiplication key. Select the 3 by 3 matrix. Change the number of columns to 4. Type in the elements. Press ·. Step 2: Press menu and choose MATRIX & VECTOR > REDUCED ROW-ECHELON FORM. Press / v to enter the matrix above which is considered the previous answer. Press ·. Step 3: Use the matrix to solve the system. The solution is (8, 9, -12). Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. To solve another system of three equations in three variables, you can copy the left side of the previous line onto the current line by highlighting it and pressing ·. Then simply update the numbers in the cells and press ·. Exercises Solve each system of equations. 1. x + 2y + z = 39 2. x - y -z = -11 6x + y + z = 8 2x + 3y + 2z = 24 -x + 5y - 3z = -4 4x - y + 4z = -22 3. 2x - z = 19 4. x + y + z = 10 x + 3y = 29 x - y - 2z = 31 2x - y - z = 14 2x - 2y + 3z = 97 Chapter 6 10 Glencoe Precalculus NAME DATE 6-2 PERIOD Study Guide and Intervention Matrix Multiplication, Inverses, and Determinants Multiply Matrices To multiply matrix A by matrix B, the number of columns in A must be equal to the number of rows in B. If A has dimensions m × r and B has dimensions r × n, their product, AB, is an m × n matrix. If the number of columns in A does not equal the number of rows in B, the matrices cannot be multiplied. ⎡ a b ⎤ ⎡ e f ⎤ ⎡ae + bg af + bh ⎤ ⎢ =⎢ ·⎢ ⎣ c d ⎦ ⎣ g h ⎦ ⎣ ce + dg cf + dh⎦ Example if possible. ⎡ 4 -2 ⎤ ⎡ -1 Use matrices A = ⎢ and B = ⎢ ⎣ -1 3 ⎦ ⎣ -2 ⎡ 4 -2⎤ ⎡ -1 AB = ⎢ ·⎢ ⎣ -1 3⎦ ⎣ -2 2 3⎤ to find AB, 4 -1 ⎦ 2 3⎤ 4 -1⎦ To find the first entry in AB, write the sum of the products of the entries in row 1 of A and in column 1 of B. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. ⎡ 4 -2⎤ ⎡ -1 ⎢ ·⎢ ⎣ -1 3⎦ ⎣ -2 ⎤ ⎦ 2 3⎤ ⎡ 4(-1) + (-2)(-2) =⎢ 4 -1⎦ ⎣ Follow this same procedure to find the entry for row 1, column 2 of AB. ⎡ 4 -2⎤ ⎡ -1 ·⎢ ⎢ ⎣ -1 3⎦ ⎣ -2 2 3⎤ ⎡ 4(-1) + (-2)(-2) =⎢ 4 -1⎦ ⎣ 4(2) + (-2)(4) ⎤ ⎦ Continue multiplying each row by each column to find the sum for each entry. ⎡ 4 -2⎤ ⎡ -1 ⎢ ·⎢ ⎣ -1 3⎦ ⎣ -2 2 3⎤ ⎡ 4(-1) + (-2)(-2) 4(2) + (-2)(4) 4(3) + (-2)(-1) ⎤ =⎢ 4 -1⎦ ⎣ (-1)(-1) + 3(-2) (-1)(2) + 3(4) (-1)(3) + 3(-1) ⎦ Then simplify each sum. ⎡ 4 -2⎤ ⎡ -1 ·⎢ ⎢ ⎣ -1 3⎦ ⎣ -2 2 3⎤ ⎡ 0 0 14⎤ =⎢ 4 -1⎦ ⎣ -5 10 -6⎦ Exercises Find AB and BA, if possible. ⎡-1 5⎤ ⎡ 2 4⎤ 1. A = ⎢ , B = ⎢ ⎣ 0 -2⎦ ⎣-3 -1⎦ Chapter 6 ⎡-2 4 0⎤ ⎡-1 3⎤ 2. A = ⎢ , B = ⎢ ⎣ –3 –1 2⎦ ⎣-3 2⎦ 11 Glencoe Precalculus Lesson 6-2 A is a 2 × 2 matrix and B is a 2 × 3 matrix. Because the number of columns for A is equal to the number of rows for B, the product AB exists. NAME DATE 6-2 Study Guide and Intervention PERIOD (continued) Matrix Multiplication, Inverses, and Determinants Inverses and Determinants The identity matrix is an n × n matrix consisting of all 1s on its main diagonal, from upper left to lower right, and 0s for all other elements. Let In be the identity matrix of order n and let A be an n × n matrix. If there exists a matrix B such that AB = BA = In, then B is called the inverse of A and is written as A–1. If a matrix has an inverse, it is invertible. The determinant of a 2 × 2 matrix can be used to determine whether or not a matrix is invertible. ⎡a b⎤ ⎡ d -b⎤ 1 If A = ⎢ , det(A) = ad - cb. If ad - cb ≠ 0, then A-1 = − ⎢ . ad - cb ⎣-c a⎦ ⎣ c d⎦ Example 1 ⎡ 7 -4⎤ ⎡3 4 ⎤ Determine whether A = ⎢ and B = ⎢ are ⎣-5 3⎦ ⎣5 7⎦ inverse matrices. If A and B are inverse matrices, then AB = BA = I. ⎡ 7 -4⎤ ⎡3 4⎤ ⎡7(3) + (-4)(5) AB = ⎢ ·⎢ =⎢ ⎣-5 3⎦ ⎣5 7⎦ ⎣ -5(3) + 3(5) 7(4) + (-4)(7) ⎤ ⎡1 0⎤ or ⎢ ⎣0 1⎦ -5(4) + 3(7)⎦ ⎡3 4⎤ ⎡ 7 -4⎤ ⎡3(7) + 4(-5) 3(-4) + 4(3) ⎤ ⎡1 0⎤ BA = ⎢ ·⎢ =⎢ or ⎢ ⎣5 7⎦ ⎣-5 3⎦ ⎣ 5(7) + 7(-5) 5(-4) + 7(3)⎦ ⎣0 1⎦ Because AB = BA = I, B = A-1 and A = B-1. Example 2 ⎢2 -2⎢ det(A) = ⎢ ⎢ ⎢3 -6⎢ 1 ⎡-6 2⎤ A-1 = - − ⎢ 6 ⎣-3 2⎦ ⎡ 1⎤ ⎢1 -− 3 =⎢ 1 1 ⎢− -− 3⎦ ⎣2 = 2(-6) - 3(-2) or -6 Since det(A) ≠ 0, A is invertible. Exercises Determine whether A and B are inverse matrices. Explain your reasoning. ⎡11 5⎤ ⎡ 1 -5⎤ 1. A = ⎢ , B = ⎢ ⎣ 2 1⎦ ⎣-2 11⎦ ⎡3 2⎤ ⎡1 2⎤ 2. A = ⎢ , B = ⎢ ⎣4 1⎦ ⎣4 3⎦ ⎡ 5 -1⎤ 3. Find the determinant of A = ⎢ . Then find A-1, if it exists. ⎣-10 2⎦ ⎡3 2⎤ 4. Find the determinant of A = ⎢ . Then find A–1, if it exists. ⎣1 -1⎦ Chapter 6 12 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. ⎡ 2 -2⎤ Find the determinant of A = ⎢ . Then find A–1, if it exists. ⎣ 3 -6⎦ NAME DATE 6-2 PERIOD Practice Matrix Multiplication, Inverses, and Determinants Find AB and BA, if possible. ⎡-1 6 0⎤ ⎡ 2 -4⎤ 1. A = ⎢ , B = ⎢ ⎣ 3 -2 1⎦ ⎣ -1 3⎦ ⎡ 3 0⎤ ⎡ 3 5⎤ 2. A = ⎢ , B = ⎢ ⎣-1 2⎦ ⎣ -2 0⎦ Company Club Type and Quantity Club Club Value ($) 1-Wood 3-Wood 5-Wood Putter 1-Wood 210 A 600 520 310 300 3-Wood 170 B 210 400 450 400 5-Wood 150 Putter 120 Write each system of equations as a matrix equation, AX = B. Then use Gauss-Jordan elimination on the augmented matrix to solve for X. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 4. x1 - 2x2 + 3x3 = 4 5. 2x1 + x2 + 2x3 = 11 5x1 + 3x2 - x3 = 13 -5x1 - x2 + 4x3 = 1 4x1 - x2 + 4x3 = 11 3x1 - 2x2 + 8x3 = 28 Determine whether A and B are inverse matrices. ⎡1 2⎤ ⎡ 3 -2⎤ 6. A = ⎢ , B = ⎢ ⎣1 3⎦ ⎣-1 1⎦ ⎡5 -2⎤ ⎡-1 0⎤ 7. A = ⎢ , B = ⎢ ⎣4 -3⎦ ⎣ 2 -8⎦ Find the determinant of each matrix. Then find its inverse, if it exists. ⎡6 5⎤ 8. ⎢ ⎣2 2⎦ Evaluate. ⎡-1 5⎤ A =⎢ ⎣ 3 0⎦ 10. AB + C Chapter 6 ⎡-2 4⎤ 9. ⎢ ⎣ 3 -6⎦ ⎡-4 2 -1⎤ B =⎢ ⎣ 0 -5 3⎦ ⎡-1 0 -4⎤ C =⎢ ⎣ 3 -2 1⎦ 11. A(B - C) 13 Glencoe Precalculus Lesson 6-2 3. GOLF The number of golf clubs manufactured daily by two different companies is shown, as well as the selling price of each type of club. Use this information to determine which company’s daily production has the highest retail value. How much greater is the value? NAME DATE 6-2 PERIOD Word Problem Practice Matrix Multiplication, Inverses, and Determinants 1. INVENTORY A hardware company keeps three types of lawnmowers in stock at each of its three stores. The current inventory and retail price for each mower is shown. Determine which store’s inventory has the greatest value. What is this value? 3. LANDSCAPING Two dump trucks have capacities of 10 tons and 12 tons. They make a total of 20 round trips to haul 226 tons of topsoil for a landscaping project. How many round trips does each truck make? Store Mower Type 4 HP A B C 5 4 3 4.5 HP 3 5 4 5 HP 7 2 3 Mower Type 4 HP 4.5 HP 5 HP Retail Value ($) 250 300 350 4. CRAFTS A craft store orders beads from three different vendors, A, B, and C. One month, the store ordered a total of 150 units of beads from these vendors. The shipping charges are as shown. C I 6 4 2 Joelle 3 5 1 Luisa 2 4 6 System A System B SS 20% 40% C 50% 30% I 30% 30% C 40 30 b. Write the system of equations that you found in part a as a matrix equation, DX = E. One of two weighted systems shown below is used. Criteria B 35 c. Solve the system that you found in part b to determine how many units of beads were purchased from each of the vendors. Use matrices to determine which system favors each skater. a. Holly b. Joelle c. Luisa Chapter 6 14 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. SS Holly A Charge per unit ($) The total delivery cost was $5375. The store ordered twice the number of units of beads from vendor C than it ordered from vendor A. a. Write a system of equations representing this situation. 2. ICE SKATING Holly, Joelle, and Luisa are competitive skaters. Their routines are judged on skating skill (SS), choreography (C), and interpretation (I). In a recent competition, they received the following scores. Skater Vendor NAME DATE 6-2 PERIOD Enrichment Travel 1 Suppose three state parks P1, P2, and P3 are connected by roads as shown in the figure. As you can see, there are only two ways to travel from P1 to P2 without going through P3. There are only three ways to travel from P1 to P3 without going through P2. The matrix M represents the number of ways to go from one park to another without traveling through the third park. P1 P1 M= P2 P3 P2 1 1 P3 ⎡ 0 2 3⎤ 2 0 1 ⎣ 3 1 0⎦ ⎢ Lesson 6-2 Confirm the numbers in each cell. For example, the element in row 2, column 3 indicates that there is only one way to travel from P2 to P3 without going through P1. The element in row 1, column 3 indicates there are three ways to travel from P1 to P3 without going through P2. It can be shown that if we square matrix M, we can determine how many ways there are to travel from one park to another by traveling through the third park. P1 P2 P3 P1 P2 P3 P1 P2 P3 ⎡ 0 2 3⎤ P1 ⎡ 0 2 3⎤ P1 ⎡13 3 2⎤ 2 M = P2 2 0 1 P2 2 0 1 = P2 3 5 6 P3 ⎣ 3 1 0⎦ P3 ⎣ 3 1 0⎦ P3 ⎣ 2 6 10⎦ Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. P1 ⎢ ⎢ ⎢ Consider the product of row 1 and column 2. 0 P1→P1 × 2 P1→P2 + 2 P1→P2 × 0 P2→P2 + 3 P1→P3 × 1 P3→P2 = 3 The result indicates that there are three ways to travel from P1 to P2 by going through P3. Similarly, consider the product of row 3 and column 2. 3 P3→P1 × 2 P1→P2 + 1 P3→P2 × 0 P2→P2 + 0 P3→P3 × 1 P3→P2 = 6 The result indicates that there are six ways to travel from P3 to P2 by going through P1. Exercises 1. What does the product of row 1 and column 3 indicate? 2. What does the product of row 2 and column 1 indicate? Chapter 6 15 Glencoe Precalculus NAME DATE 6-3 PERIOD Study Guide and Intervention Solving Linear Systems Using Inverses and Cramer’s Rule Use Inverse Matrices A square system has the same number of equations as variables. If a square matrix has an inverse, the system has one unique solution. Example Use an inverse matrix to solve each system of equations, if possible. a. 3x - 7y = -16 -x + 2y = 8 b. -2x + y + z = 0 x + 2z = 9 x - 2y - 9z = -31 Write the system in matrix form. A · X = B Write the system in matrix form. A · X = B ⎡-2 1 1⎤ ⎡x⎤ ⎡ 0⎤ 9 1 0 2 · y = ⎣ 1 -2 -9⎦ ⎣ z⎦ ⎣-31⎦ Use a graphing calculator to find A-1. ⎡-16⎤ ⎡ 3 -7 ⎤ ⎡x⎤ ⎢ ·⎢ = ⎢ y ⎣ 8⎦ ⎣-1 2 ⎦ ⎣ ⎦ ⎢ Use the formula for an inverse of a 2 × 2 matrix to find the inverse A-1. ⎢ ⎢ ⎡ d -b⎤ 1 A-1 = − ⎢ a⎦ ad - cb ⎣-c ⎡2 1 = −− ⎢ (2)(3) - (-7)(-1) ⎣1 7⎤ 3⎦ Multiply A-1 by B to solve the system. X= A-1 · B ⎡ 4 7 2⎤ ⎡ 0⎤ ⎡ 1⎤ = 11 17 5 · 9 = -2 ⎣-2 -3 -1⎦ ⎣-3⎦ ⎣ 4⎦ So, the solution of the system is (1, -2, 4). ⎢ So, the solution of the system is (-24, -8). ⎢ ⎢ Exercises Use an inverse matrix to solve each system of equations, if possible. 1. -2x + 5y = 24 2. x - y + 2z = 5 3x - y = -10 x - z = -4 3x + 2y + z = 0 3. 3x + y = 7 4. x + y - z = -5 -2x - 5y = 43 2x - 3y + 2z = 20 y + 4z = 18 Chapter 6 16 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Multiply A-1 by B to solve the system. X = A-1 · B ⎡-24⎤ ⎡-2 -7 ⎤ ⎡-16⎤ = ⎢ ·⎢ = ⎢ ⎣ -8⎦ ⎣-1 -3 ⎦ ⎣ 8⎦ NAME DATE 6-3 Study Guide and Intervention PERIOD (continued) Solving Linear Systems Using Inverses and Cramer’s Rule Use Cramer’s Rule Another method, known as Cramer’s Rule, can be used to solve a square system of equations. Let A be the coefficient matrix of a system of n linear equations in n variables given by AX = B. If det(A) ≠ 0, then the unique solution of the system is given by |A | |A | |A | |A | |A| |A| |A| |A| 3 n 1 2 x1 = − , x2 = − , x3 = − , … , xn = − , where Ai is the matrix obtained by replacing the ith column of A with the column of constants B. If det(A) = 0, then AX = B has either no solution or infinitely many solutions. Example Use Cramer’s Rule to find the solution of the system of linear equations, if a unique solution exists. -2x1 + x2 = -7 5x1 - 2x2 = 17 ⎡-2 1⎤ The coefficient matrix is A = ⎢ . Calculate the determinant of A. ⎣ 5 -2⎦ ⎪-25 -21 ⎥ = (-2)(-2) - 5(1) or -1 Because the determinant of A does not equal zero, you can apply Cramer’s Rule. x1 -7 1 ⎪ -7(-2) - 17(1) 17 -2⎥ =−=−= − -3 =− or 3 x2 -2 -7 ⎪ -2(17) - 5(-7) 5 17⎥ =−=−= − 1 =− or -1 |A1| |A| -1 -1 |A2| |A| -1 -1 Lesson 6-3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. |A| = -1 -1 Therefore, the solution is x1 = 3 and x2 = -1 or (3, -1). Exercises Use Cramer’s Rule to find the solution of each system of linear equations, if a unique solution exists. 1. x - 2y = -5 2. 3x - 3y = -18 -2x - 5y = -8 3. 3x + y = 21 -x + 4y = 9 4. -2x - 4y = 2 -x + 2y = 14 Chapter 6 x + 3y = -3 17 Glencoe Precalculus NAME 6-3 DATE PERIOD Practice Solving Linear Systems Using Inverses and Cramer’s Rule Use an inverse matrix to solve each system of equations, if possible. 1. 4x - 7y = 30 2. -2x - 8y = -36 -6x + 2y = -11 4x + 3y = 7 3. x - 2y + 7z = -33 4. x + y - 2z = 5 -4x + 5y - z = 18 x + 2y + z = 8 5x - 3y = -11 2x + 3y - z = 1 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 5. TELEVISION During the summer, Manuel watches television M hours per day, Monday through Friday. Harry watches television H hours per day, Friday through Sunday. Ellen watches television E hours per day, Friday and Saturday. Altogether, they watch television 33 hours each week. On Fridays, they watch a total of 11 hours of television. If the number of hours Ellen spends watching television on any given day is twice the number of hours that Manuel spends watching television on any given day, how many hours of television does each of them watch each day? Use Cramer’s Rule to find the solution of each system of linear equations, if a unique solution exists. 6. -4x - 5y = 1 7. x + y + z = 8 -2x - 3y = -1 3x - z = -22 y + 2z = 20 8. PAPER ROUTE Payton, Santiago, and Queisha each have a paper route. Payton delivers 5 times as many papers as Santiago. Santiago delivers twice as many papers as Queisha. If 20 papers were added to Payton’s route, he would then deliver four times the total number of papers that Santiago and Queisha deliver. How many papers does each person deliver? Chapter 6 18 Glencoe Precalculus NAME DATE 6-3 PERIOD Word Problem Practice Solving Linear Systems Using Inverses and Cramer’s Rule 1. PERIMETER The perimeter of rectangle WXYZ is 92 centimeters. The perimeter of triangle WXZ is 80 centimeters. If the −− length of XZ is two more than twice the −−− length of WZ, what are the values of a, b, and c? B Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. ; C 9 D 4. NUTS A nut company makes three types of one-pound gift boxes: X, Y, and Z. The table shows the amount of each nut in each box. : Gift Box Cashews Almonds Hazelnuts X 0.2 0.3 0.5 Y 0.4 0 0.6 Z 0.4 0.4 0.2 The company has 10,000 pounds of cashews, 7000 pounds of almonds, and 12,000 pounds of hazelnuts in its gift boxes. 2. BASEBALL In one season, Marty, Carlos, and Andrew hit a total of 108 home runs. Marty and Andrew together hit twice as many home runs as did Carlos, although Carlos had one more home run than Andrew. How many home runs did each player hit? a. Write a system of equations representing this situation. b. Solve the system of equations that you wrote in part a as a matrix equation, AX = B. c. Determine how many of each gift box the company has. Chapter 6 19 Glencoe Precalculus Lesson 6-3 8 3. INVESTING A total of $4000 is invested in three accounts paying 4%, 5%, and 3.5% simple interest. The combined annual interest is $173.75. If the interest earned at 5% is $70 more than the interest earned at 4%, how much money is invested in each account? NAME 6-3 DATE PERIOD Enrichment Pick’s Theorem Consider the simple polygon drawn on square dot paper, shown at the right. Pick’s Theorem states that the area of the polygon A is equal to half the number of boundary points b plus the number of b interior points n minus 1, or A = − + n - 1. In the figure, b = 10 2 10 + 2 - 1 or 6 square units. and n = 2. Therefore, A = − 2 You can use systems of equations and matrices to verify Pick’s Theorem. To verify that the 1 coefficients in the equation for A are − , 1, and -1, you can write a system of three equations 2 of the form A = bx + ny + z, where the values of A, b, and n vary from polygon to polygon. Start by drawing three simple polygons on square dot paper similar to the ones shown below. Be sure that the number of boundary points, interior points, and the area of the figures are different. The table shown below summarizes the information. Figure b n A square 8 1 4 rectangle 18 12 20 triangle 11 6 10.5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Use the table to write a system and matrix equation. 8x + y + z = 4 ⎡ 8 1 1⎤ ⎡x⎤ ⎡ 4 ⎤ 18x + 12y + z = 20 18 12 1 y = 20 11x + 6y + z = 10.5 ⎣11 6 1⎦ ⎣ z⎦ ⎣10.5⎦ 1 Solving this system, we can see that x = − , y = 1, and z = -1. ⎢ ⎢ ⎢ 2 Exercises 1. Write the matrix equation that would be used to verify Pick’s Theorem using the polygons at the right. 2. Verify Pick’s Theorem using three simple polygons of your choice. Chapter 6 20 Glencoe Precalculus NAME DATE 6-3 PERIOD Spreadsheet Activity Cramer’s Rule You can use a spreadsheet to solve systems of equations with Cramer's Rule. A 1 2 3 4 5 6 7 8 9 10 11 12 6 5 B 3 1 C To use the spreadsheet to solve a system of equations, write each equation in the form below. ax + by = c In the spreadsheet, the values of a, b, and c for the first equation are entered in cells A1, B1, and C1, respectively. The values of a, b, and c for the second equation are entered in cells A2, B2, and C2, respectively. The values for the system 6x + 3y = -12 and 5x + y = 8 are shown. D -12 8 = A1*B2 - B1*A2 = C1*B2 - B1*C2 = A1*C2 - C1*A2 x= y= = (A6/A4) = (A8/A4) Exercises 2. Write matrices whose determinants are found using the formulas in cells A6 and A8. Lesson 6-3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 1. Study the formula in cell A4. Write a matrix whose determinant is found using this formula. 3. Explain how the values of x and y are found using Cramer’s Rule. Use the spreadsheet to solve each system of equations. 4. 6x + 3y = -12 5x + y = 8 5. 5x - 3y = 19 7x + 2y = 8 6. 0.3x + 1.6y = 0.44 0.4x + 2.5y = 0.66 7. 3y = 4x + 28 5x + 7y = 8 Chapter 6 21 Glencoe Precalculus NAME 6-4 DATE PERIOD Study Guide and Intervention Partial Fractions Linear Factors The function g(x) shown below can be written as the sum of two fractions with denominators that are linear factors of the original denominator. 3x - 1 2 -1 =− +− g(x) = − 2 2x - 3x + 1 x-1 2x - 1 Each fraction in the sum is a partial fraction. The sum of these partial fractions make up the partial fraction decomposition of the original rational function. If the denominator of a rational expression contains a repeated linear factor, the partial fraction decomposition must include a partial fraction with its own constant numerator for each power of this factor. x + 11 x - 3x - 4 Example . Find the partial fraction decomposition of − 2 Rewrite the equation as partial fractions with constant numerators, A and B, and denominators that are the linear factors of the original denominator. x + 11 x - 3x - 4 A B − =− +− 2 x-4 Form a partial fraction decomposition. x+1 x + 11 = A(x + 1) + B(x - 4) Multiply each side by the LCD, x2 - 3x - 4. x + 11 = Ax + A + Bx - 4B Distributive Property Group like terms. 1x + 11 = (A + B)x + (A + (-4B)) A+B=1 A + (-4B) = 11 → C · X = D ⎡1 1⎤ ⎢ ⎣1 -4⎦ · ⎡ A⎤ ⎢ ⎣B⎦ = ⎡ 1⎤ ⎢ ⎣ 11 ⎦ x + 11 3 -2 Solving for X yields A = 3 and B = -2. Therefore, − =− +− . 2 x - 3x - 4 x-4 x+1 Exercises Find the partial fraction decomposition of each rational expression. 5x - 34 1. − 2 x - x - 12 x2 + 1 2x(x - 1) 3. −2 Chapter 6 -7x + 13 x - 5x - 14 2. − 2 2 -x-1 − 4. 5x 2 x (x - 1) 22 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Equate the coefficients on the left and right side of the equation to obtain a system of two equations. To solve the system, write it in matrix form CX = D and solve for X. NAME DATE 6-4 PERIOD Study Guide and Intervention (continued) Partial Fractions Irreducible Quadratic Factors Not all rational expressions can be written as the sum of partial fractions using only linear factors in the denominator. If the denominator of a rational expression contains an irreducible quadratic factor, the partial fraction decomposition must include a partial fraction with a linear numerator of the form Bx + C for each power of this factor. 4x4 - 2x3 - 13x2 + 7x + 9 x(x - 3) Example Find the partial fraction decomposition of −− . 2 2 This expression is proper. The denominator has one linear factor and one irreducible factor of multiplicity 2. 4x4 - 2x3 - 13x2 + 7x + 9 Bx + C Dx + E A −− =− +− +− x2 - 3 x(x2 - 3)2 (x2 - 3)2 x 4x4 - 2x3 -13x2 + 7x + 9 = A(x2 - 3)2 + (Bx + C)x(x2 - 3) + (Dx + E)x 4x4 - 2x3 -13x2 + 7x + 9 = Ax4 + Bx4 + Cx3- 6Ax2 - 3Bx2 + Dx2 - 3Cx + Ex + 9A 4x4 - 2x3 -13x2 + 7x + 9 = (A + B)x4 + Cx3 + (-6A - 3B + D)x2 + (-3C + E)x + 9A A+B=4 A = 1 C = -2 B = 3 C = -2 -3C + E = 7 D = 2 9A = 9 E = 1 -6A - 3B + D = -13 → 4x4 - 2x3 - 13x2 + 7x + 9 2x + 1 3x - 2 1 Therefore, −− =− +− +− . 2 2 2 2 2 x(x - 3) x x -3 (x - 3) Exercises Find the partial fraction decomposition of each rational expression. 5 1. − 3 3x3 - 2x2 - 8x + 5 (x - 3) x + 5x 2. −− 2 2 2x3 + x + 3 (x + 1) 4. − 2 2 3. − 2 2 Chapter 6 x3 + 2x2 + 2 (x + 1) 23 Glencoe Precalculus Lesson 6-4 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Write and solve the system of equations obtained by equating coefficients. NAME 6-4 DATE PERIOD Practice Partial Fractions Find the partial fraction decomposition of each rational expression. 3x - 7 1. − 2 2 - 10x - 2 − 2. 6x 3 2 9x + 15 3. − 2 x 4. − 2 x - 7x + 12 x + 3x + 2 x + x - 2x 2x - 9x + 9 Find the partial fraction decomposition of each improper rational expression. -5x2 - 11x + 54 x + 2x - 8 3x2 + 5x + 2 x + 2x 6. − 2 6x2 + 17x + 2 x +x 8. − 2 5. − 2 7. − 2 -8x2 + 22x - 10 (2x - 3) 5x4 - 7x3 - 12x2 + 6x + 21 (x - 3)(x - 2) -2x2 + 29x - 100 x - 10x + 25x 10. −− 2 2 2x2 + 5 x + 6x + 9x 12. −− 2 2 9. −− 3 2 11. − 3 2 4x4 + 8x3 + 6x2 + 6x + 5 (3x + 2)(x + 1) 13. GROWTH When working with exponential growth in calculus, it is often 1 x(50 - x) necessary to work with functions of the form f(x) = − and to decompose these functions into the sum of its partial fractions. Find the partial decomposition of f(x). Chapter 6 24 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Find the partial fraction decomposition of each rational expression with repeated factors. NAME 6-4 DATE PERIOD Word Problem Practice Partial Fractions 1. CHEMISTRY A chemist uses the 4. AREA Calculus can be used to find the area of the shaded region shown below. The shaded region is bounded 50(25 + x) function f(x) = − to determine 50 + x how much acid she must mix with a 25% acid solution to achieve the desired percentage. Find the partial fraction decomposition of f(x). x - 44 by the graphs of f(x) = − , y = 0, 2 x - 11x x = 2, and x = 8. To find the area, first find the partial fraction decomposition of f(x). 2. INFECTIONS A function of the form 1 , where a > 0 often g(x) = − (x + 1)(a - x) plays a role when studying the spread of an infection in certain populations. Find the partial fraction decomposition of g(x) when a = 350. [0, 10] scI: 1 by [-2, 10] scI: 1 3. VOLUME Consider the domain 0 ≤ x ≤ 10 for the graph of f(x) shown below. 10 y 8 6 b. Write the matrix form AX = B for the system of equations found in part a. f (x) = 60 x(x + 3)2 c. Find the partial fraction decomposition of the rational expression. 4 2 0 2 4 6 8 x 10 Suppose you were to revolve the graph of f(x) around the x-axis, creating a three-dimensional object. Using calculus, you could find the volume of the object. But first, you would need to find the partial fraction decomposition of f(x). Find the partial decomposition of f(x). 5. KAYAKING The total time it takes for a kayaker to travel 10 miles upstream and 10 miles downstream with a paddling rate of 4 miles per hour in still water is given by the function -80 f(x) = − . Find the partial fraction 2 x - 16 decomposition of f(x). Chapter 6 25 Glencoe Precalculus Lesson 6-4 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. a. Write the system of equations obtained by equating the coefficients. NAME 6-4 DATE PERIOD Enrichment Heaviside Method Another method that can be used to find the partial fraction decomposition of a rational expression is called the Heaviside Method. Consider the equation shown below. 2x - 47 A B − +− =− 2 x+4 x - 3x - 28 x-7 Multiply both sides of the equation by the least common denominator (x + 4)(x - 7). ⎡ A 2x - 47 B ⎤ = (x + 4)(x - 7) ⎢− (x + 4)(x - 7) − +− 2 x - 7⎦ x -3x - 28 ⎣x + 4 2x - 47 = A(x - 7) + B(x + 4) To solve for A, let x = -4. This eliminates B. 2(-4) - 47 = A(-4 - 7) + B(-4 + 4) -55 = -11A 5 = A To solve for B, let x = 7. This eliminates A. 2(7) - 47 = A(7 - 7) + B(7 + 4) -33 = 11B Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. -3 = B Substitute A = 5 and B = -3 into the original equation to find the partial fraction decomposition. 2x - 47 5 -3 − +− = − 2 x - 3x - 28 x+4 x-7 Exercises Use the Heaviside Method to write the partial fraction decomposition of each rational expression. x + 39 x - 3x - 18 1. − 2 2 - 8x - 4 − 2. 3x 2 3 13x - 51 3. − 2 x - 8x + 15 4. − 2 18x2 + 39x - 30 x - x - 6x -x - 114 6. − 2 5. − 3 2 Chapter 6 x + x - 2x -6x + 18 x - 10x + 24 x + 3x - 54 26 Glencoe Precalculus NAME DATE 6-5 PERIOD Study Guide and Intervention Linear Optimization Linear Programming Linear programming is a process for finding a minimum or maximum value for a specific quantity. The following steps can be used to solve a linear programming problem. Step Step Step Step 1 2 3 4 Write an objective function and a list of constraints to model the situation. Graph the region corresponding to the solution of the system of constraints. Find the coordinates of the vertices of the region formed. Evaluate the objective function at each vertex to find the minimum or maximum. A leather company wants to add belts and wallets to its product line. Belts require 2 hours of cutting time and 6 hours of sewing time. Wallets require 3 hours of cutting time and 3 hours of sewing time. The cutting machine is available 12 hours a week and the sewing machine is available 18 hours per week. Belts will net $18 in profit and wallets will net $12. How much of each product should be produced to achieve maximum profit? Let x represent the number of belts and y represent the number of wallets. The objective function is then given by f(x, y) = 18x + 12y. Write the constraints. x ≥ 0; y ≥ 0 Numbers of items cannot be negative. 2x + 3y ≤ 12 Cutting time 6x + 3y ≤ 18 Sewing time Graph the system. The solution is the shaded region, including its y boundary segments. Find the coordinates of the four vertices by solving the system of boundary equations for each point of intersection. The coordinates are (0, 0), (0, 4), (1.5, 3), and (3, 0). (1.5, 3) (0, 4) Evaluate the objective function for each ordered pair. Point f(x, y) = 18x + 12y Result (0, 0) f(0, 0) = 18(0) + 12(0) 0 (0, 4) f(0, 4) = 18(0) + 12(4) 48 (1.5, 3) f(1.5, 3) = 18(1.5) + 12(3) 63 (3, 0) f(3, 0) = 18(3) + 12(0) 54 0 (3, 0) x ← Maximum Since f is greatest at (1.5, 3), the company will maximize profit if it makes and sells 1.5 belts for every 3 wallets. Exercises Find the maximum and minimum values of the objective function f(x, y) and for what values of x and y they occur, subject to the given constraints. 1. f(x, y) = 3x - 2y 2x + y ≤ 10 x + 2y ≤ 8 x≥0 y≥0 Chapter 6 Lesson 6-5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Example 2. f(x, y) = x + 2y x+y≤4 x + 3y ≤ 6 x≥0 y≥0 27 Glencoe Precalculus NAME 6-5 DATE PERIOD Study Guide and Intervention (continued) Linear Optimization No or Multiple Optimal Solutions Linear programming models can have one, multiple, or no optimal solutions. If the graph of the objective function f to be optimized is coincident with one side of the region of feasible solutions, f has multiple optimal solutions. If the region does not form a polygon, but instead is unbounded, f may have no minimum value or maximum value. Example Find the maximum value of the objective function f(x, y) = 6x + 3y and for what values of x and y it occurs, subject to the following constraints. 2x + y ≤ 8 y≤4 x≤3 x≥0 y≥0 Graph the region bounded by the given constraints. Find the value of the objective function f(x, y) = 6x + 3y at each vertex. f(0, 0) = 6(0) + 3(0) or 0 y f(0, 4) = 6(0) + 3(4) or 12 (2, 4) f(2, 4) = 6(2) + 3(4) or 24 (3, 2) f(3, 2) = 6(3) + 3(2) or 24 x 0 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. f(3, 0) = 6(3) + 3(0) or 18 Because f(x, y) = 24 at (2, 4) and (3, 2), the problem has multiple optimal solutions. An equation of the line through these two vertices is y = -2x + 8. Therefore, f has a maximum value of 24 at every point on y = -2x + 8 for 2 ≤ x ≤ 3. Exercises Find the maximum and minimum values of the objective function f(x, y) and for what values of x and y they occur, subject to the given constraints. 1. f(x, y) = 2x + y 2. f(x, y) = 3x + 6y 2x + y ≤ 11 x - y ≥ -4 y≤5 x + 2y ≤ 20 y≥0 x≥0 x≤5 x≤8 x≥0 y≥0 Chapter 6 28 Glencoe Precalculus NAME DATE 6-5 PERIOD Practice Linear Optimization Find the maximum and minimum values of the objective function f(x, y) and for what values of x and y they occur, subject to the given constraints. 1. f(x, y) = 2x + 5y 2. f(x, y) = 4x + 3y x≥0 x≥0 y≥0 y≥0 x+y≤7 2x + 3y ≥ 6 2x + 3y ≤ 18 x+y≤8 4. f(x, y) = 3x + 3y x≥0 x≥0 x≤7 y≥0 y≥0 y≤8 y≤5 x + y ≤ 10 x + 2y ≥ 14 3x + 2y ≤ 24 5. SKATES A manufacturer produces roller skates and ice skates. Manufacturer Information Roller Skates Ice Skates Maximum Time Available Assembling 5 minutes 4 minutes 200 minutes Checking and Packaging 1 minute 4 minutes 120 minutes Profit per Skate $40 $30 a. Write an objective function and list the constraints that model the given situation. c. How many roller skates and ice skates should be manufactured to maximize profit? What is the maximum profit? d. Describe why the company would choose a number of roller skates and ice skates different from the answer in part c. Chapter 6 29 60 y 50 40 30 20 10 0 x 10 20 30 40 50 60 Roller Skates Glencoe Precalculus Lesson 6-5 b. Sketch a graph of the region determined by the constraints from part a to find the set of feasible solutions for the objective function. Ice Skates Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 3. f(x, y) = 2x - 3y NAME DATE 6-5 PERIOD Word Problem Practice Linear Optimization 1. FARMING Mr. Fields owns a 360-acre farm on which he plants corn and soybeans. The table shows the cost of labor and the profit per acre for each crop. Mr. Fields can spend up to $60,000 for spring planting. Per Acre Corn Soybeans Labor ($) 300 150 Profit ($) 340 300 2. NUTRITION A certain diet recommends at least 140 milligrams of Vitamin A and at least 145 milligrams of Vitamin B daily. These requirements can be obtained from two types of food. Type X contains 10 milligrams of Vitamin A and 20 milligrams of Vitamin B per pound. Type Y contains 30 milligrams of Vitamin A and 15 milligrams of Vitamin B per pound. Type X costs $12 per pound. Type Y costs $8 per pound. a. Write an objective function and list the constraints that model this situation. a. Write an objective function and list the constraints that model this situation. 450 400 b. Sketch a graph of the region determined by the constraints from part a to find the feasible solutions for the objective function. b y 12 350 300 8 250 4 200 150 0 100 50 0 12 x c 50 100 150 200 c. How many pounds of each type of food should be purchased to satisfy the requirements at the minimum cost? What is the minimum cost? 250 Acres of Corn c. How can Mr. Fields maximize his profit? What is his maximum profit? Chapter 6 4 30 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Acres of Soybeans b. Sketch a graph of the region determined by the constraints from part a to find the feasible solutions for the objective function. NAME 6-5 DATE PERIOD Enrichment Convex Polygons You have already learned that over a closed convex polygonal region, the maximum and minimum values of any linear function occur at the vertices of the polygon. To see why the values of the function at any point on the boundary of the region must be between the values at the vertices, consider the convex polygon with vertices P and Q. −−− Let W be a point on PQ. y PW If W lies between P and Q, let − = w. PQ 8 x 0 Example If f(x, y) = 3x + 2y, find the maximum value of the function over the shaded region at the right. y The maximum value occurs at the vertex (6, 3). The minimum value occurs at (0, 0). The values of f(x, y) at W 1 and W 2 are between the maximum and minimum values. f(Q) = f(6, 3) = 3(6) + 2(3) or 24 f(W 1) = f(2, 1) = 3(2) + 2(1) or 8 1 f(W 2) = f(5, 2.5) = 3(5) + 2(2.5) or 20 0 f(P) = f(0, 0) = 3(0) + 2(0) or 0 3 2 8 8 x Exercises −− Let P and Q be vertices of a closed convex polygon, and let W lie on PQ. Let f(x, y) = ax + by. 1. If f(Q) = f(P), what is true of f ? of f(W)? Lesson 6-5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Then 0 < w < 1 and the coordinates of W are ((1 - w)x 1 + wx 2, (1 - w)y 1 + wy2). Now consider the function f(x, y) = 3x - 5y. f(W) = 3[(1 - w)x 1 + wx 2] - 5[(1 - w)y 1 + wy 2] = (1 - w)(3x 1) + 3wx 2 + (1 - w)(-5y1) - 5wy 2 = (1 - w)(3x1 - 5y1) + w(3x 2 - 5y2) = (1 - w)f(P) + wf(Q) This means that f(W) is between f(P) and f(Q), or that the greatest and least values of f(x, y) must occur at P or Q. 1 2 2. If f(Q) = f(P), find an equation of the line containing P and Q. Chapter 6 31 Glencoe Precalculus NAME 6 DATE PERIOD Chapter 6 Quiz 1 SCORE (Lessons 6-1 and 6-2) 1. 3x - 2y = 1 2. 2x - 5y + 7z = -9 -5x + y = -11 Assessment Write the augmented matrix for each system of linear equations. 1. -x - y + 2z = 1 -3x + 4y - z = 10 Solve each system of equations. 2. 3. -4x + 2y = 22 3. 4. -3x + y - 2z = 2 x - 3y = -8 2x - y = -5 -5x - 2z = 1 4. ⎡ 2 -4 ⎤ . 5. MULTIPLE CHOICE Find the determinant of ⎢ ⎣ -3 5 ⎦ A -22 B -2 C 2 D 22 6. Find A-1, if it exists. ⎡ 3 -7 ⎤ A=⎢ ⎣-2 5⎦ 6. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAME 6 5. DATE PERIOD Chapter 6 Quiz 2 SCORE (Lesson 6-3) Use an inverse matrix to solve each system of equations, if possible. 1. -4x + y = 18 2. -3x - y = 8 x - 5y = -14 1. 5x + 3z = -16 -4y - z = 10 2. Use Cramer’s Rule to find the solution of each system of linear equations, if a unique solution exists. 3. 2x + 5y = 3 4. 3x - y + 2z = -3 -x - 3y = -5 3. -x + 2y - z = 2 2x - 3y + z = - 1 4. 5. MULTIPLE CHOICE Find the inverse matrix required ⎡ -4 7 ⎤ ⎡ x ⎤ ⎡-19⎤ ·⎢ =⎢ to solve ⎢ . ⎣ -1 2 ⎦ ⎣ y ⎦ ⎣ -5⎦ ⎡2 7⎤ A ⎢ ⎣ 1 -4 ⎦ Chapter 6 ⎡ 2 -7 ⎤ B ⎢ ⎣ 1 -4 ⎦ ⎡ -4 7 ⎤ D ⎢ ⎣ -1 2 ⎦ ⎡-2 7 ⎤ C ⎢ ⎣-1 4 ⎦ 33 5. Glencoe Precalculus NAME DATE 6 PERIOD Chapter 6 Quiz 3 SCORE (Lesson 6-4) Find the partial fraction decomposition of each rational expression. x-3 1. − 2 2x + 24 x - x -6 1. 6x2 - x + 16 x + 4x 2. x -x 2. − 2 -x2 + x + 32 x - 8x + 16x 4. − 3 3. − 3 2 5. MULTIPLE CHOICE Which rational expression is improper? A 3x2 + 5x -4 − (x - 2)(x + 1) 5x -1 B − x2 - x -2 23x - 26 C − 2 x - 32x + 12 4. x+2 D −2 (x + 1) 5. NAME 6 3. DATE PERIOD Chapter 6 Quiz 4 SCORE (Lesson 6-5) A vertices C interior points B exterior points D constraints Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 1. MULTIPLE CHOICE If a linear programming problem can be optimized, it occurs at one of the ___________ of the region representing the solutions. 1. Find the maximum and minimum values of the objective function f(x, y) and for what values of x and y they occur, subject to the given constraints. 2. f(x, y) = 3x + 4y x≥0 y≥0 x+y≤7 x + 3y ≤ 15 3. f(x, y) = -2x + y x≥0 y≤7 y≤x 7x + 2y ≤ 84 2. 3. 4. CLOCKS It costs a clockmaker $30 to make a small clock and $50 to make a large clock. He has a budget of $1500 to build them. He will not make more than 40 small clocks. a. If he makes a profit of $60 on the small clocks and $45 on the large clocks, how many of each size clock must he make 4a. and sell to maximize his profit? 4b. b. What is his maximum profit? Chapter 6 34 Glencoe Precalculus NAME DATE 6 PERIOD Chapter 6 Mid-Chapter Test SCORE (Lessons 6-1 through 6-3) Assessment Part I Write the letter for the correct answer in the blank at the right of each question. 1. Solve the system of equations. 4x - 3y = -22 -2x - 5y = -28 B (-1, 6) A (-1, -6) C (1, -6) D (1, 6) 1. H (-2, 4, -1) J (2, 4, -1) 2. ⎡ 2 4⎤ D ⎢ ⎣ -3 -6 ⎦ 3. 2. Solve the system of equations. -4x + 2y - z = 17 x - 3y + 2z = -16 2x + y - 4z = 4 F (2, -4, 1) G (-2, -4, 1) 3. Which of the following matrices is singular? ⎡ 0 1⎤ A ⎢ ⎣ -1 2 ⎦ ⎡ 7 -1 ⎤ B ⎢ ⎣4 4⎦ ⎡ 4 -3 ⎤ C ⎢ ⎣5 2⎦ Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 4. Use an inverse matrix to solve the system of equations, if possible. -4x - 2y = -14 -8x - 4y = 4 F no solution G (-1, 1) H (1, -2) J (2, 3) 4. Part II 5. FRUIT Katie bought 4 apples and 6 pears for $9.80. Sylvia bought 3 apples and 9 pears for $10.95. a. Write a set of linear equations for this situation. 5a. b. Determine the cost of one apple and one pear. 5b. ⎡ -1 1 6. Find the determinant of 1 -1 ⎣ 1 -1 ⎢ 0⎤ 1 . 1⎦ 6. 7. Find AB if possible. ⎡ -1 ⎤ ⎡ -4 3 2 ⎤ A= 0 -1 1 , B = 1 ⎣ -2 ⎦ ⎣ -2 1 0 ⎦ ⎢ Chapter 6 ⎢ 7. 35 Glencoe Precalculus NAME DATE PERIOD 6 Chapter 6 Vocabulary Test SCORE augmented matrix coefficient matrix constraint Cramer’s Rule determinant feasible solutions Gaussian elimination Gauss-Jordan elimination identity matrix inverse inverse matrix invertible linear programming multiple optimal solutions multivariable linear system nonsingular matrix objective function optimization partial fraction partial fraction decomposition reduced row-echelon form row-echelon form singular matrix square system unbounded Choose a term from the vocabulary list above to complete each sentence. 1. 2. Transferring a system into an equivalent system is called _______________. 2. 3. The multiplicative inverse of a square matrix is called its _______________. 3. 4. The process for finding a minimum or maximum value for a specific quantity is known as _______________. 4. 5. If a matrix has an inverse, then the matrix is said to be a(n) _______________ matrix. 5. 6. A method for solving square systems using determinants instead of row reduction or inverses is known as _______________. 6. 7. When a rational function is written as the sum of its partial fractions, this sum is called the _______________ of the rational function. 7. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 1. A system of linear equations that has the same number of equations as variables is called a(n) _______________. Define each term in your own words. 8. identity matrix 9. objective function Chapter 6 36 Glencoe Precalculus DATE 6 PERIOD Chapter 6 Test, Form 1 SCORE Write the letter for the correct answer in the blank at the right of each question. 1. What is the augmented matrix for the given system? 2x - 3y = -16 x + 5y = 18 ⎡2 −3⎤ ⎡ 5 3⎤ ⎡2 -3 ⎡ 5 3 -16⎤ A ⎢ B ⎢ C ⎢ D ⎢ ⎣1 5⎦ ⎣−1 2⎦ ⎣1 5 ⎣-1 2 18⎦ -16⎤ 18⎦ 1. 2⎤ -1 5⎦ 2. 2. Which matrix is not in row-echelon form? ⎡ 1 -3 F ⎢0 ⎣0 1 0 2⎤ -6 0⎦ G ⎡1 2 ⎢ ⎣0 1 4⎤ -2 ⎦ ⎡1 7 H ⎢ ⎣0 1 ⎡ 1 -2 J 1 -4 ⎣0 1 -2 ⎤ 5⎦ ⎢ 3. Solve the system of equations using Gaussian elimination. -3x - 5y = 2 2x + 3y = -2 A (4, 2) B (-4, -2) C (4, -2) D (-4, 2) 3. 4. FOOD The table shows several boxes of assorted candy available at a candy shop. What is the price per pound for each candy? Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Box Chocolate Taffy Nougat Price ($) Grand Edition 10 5 0 12.25 Special Edition 10 5 5 16.25 Deluxe Edition 15 10 5 24.25 F ($0.85, $0.75, $0.80) H ($0.80, $0.75, $0.85) G ($0.75, $0.80, $0.85) J ($0.75, $0.85, $0.80) 4. C 12 D 20 5. ⎡ -2 0 18⎤ J ⎢ ⎣-10 28 4⎦ 6. ⎡–1 1⎤ D ⎢ ⎣ 4 –3⎦ 7. ⎡3 -2⎤ 5. What is the determinant of ⎢ ? ⎣4 0⎦ A -8 B 8 ⎡-2 4 6 ⎤ 6. Find DE if D = ⎢ and E = ⎣ 5 -7 -1⎦ ⎡-20 44⎤ F ⎢ ⎣ 8 -42⎦ ⎡ -2 -10⎤ G 0 -28 ⎣-18 -4⎦ ⎢ ⎡ 1 -2⎤ ⎢ 0 ⎣-3 4. 4⎦ ⎡-20 8⎤ H ⎢ ⎣ 44 -42⎦ ⎡ 3 -1 ⎤ 7. Find the inverse of ⎢ , if it exists. ⎣-4 1⎦ A does not exist Chapter 6 ⎡-1 -1⎤ B ⎢ ⎣-4 -3⎦ ⎡1 1⎤ C ⎢ ⎣4 3⎦ 37 Glencoe Precalculus Assessment NAME NAME DATE 6 Chapter 6 Test, Form 1 PERIOD (continued) ⎡–4 –6 ⎤ ⎡ 3 –2 ⎤ 8. What is B if A = ⎢ and AB = ⎢ ? ⎣ 1 –4 ⎦ ⎣ 2 –12 ⎦ ⎡ 2 0⎤ ⎡-2 0⎤ ⎡-2 0⎤ ⎡-2 0⎤ H ⎢ J ⎢ F ⎢ G ⎢ ⎣ 1 3⎦ ⎣-1 -3⎦ ⎣-1 3⎦ ⎣-1 3 ⎦ 9. Solve the following system of equations using an inverse matrix. -4x - 2y + z = 6 A (1, 0, -2) -x - y - 2z = -3 B (-1, 0, -2) 8. 2x + 3y - z = -4 C (-1, 0, 2) D (1, 0, 2) 9. 10. Use Cramer’s Rule to solve the system of equations. -3x + 7y = 78 -2x + 5y = 55 G (5, 9) F (5, -9) H (-5, 9) J (-5, -9) 10. 11. FUNDRAISING The cheerleading squad is raising money for new uniforms by selling popcorn balls and calendars. Tanya raised $70 by selling 25 popcorn balls and 30 calendars. Nichole raised $53 by selling 20 popcorn balls and 22 calendars. What is the cost of one calendar? A $1 B $1.25 C $1.50 D $1.75 11. –2x + 10 (x – 1) (x + 3) 12. Find the partial fraction decomposition of − . 4 –2 H − +− 2 –4 +− G − 2 -4 J − +− x -1 x+1 x+3 x+1 x–3 x-1 x–3 12. x+3 13. Find the maximum value of the objective function f(x, y) = 2x + 4y, subject to the constraints x ≥ 0, x ≤ 8, y ≥ 0, and x + y ≤ 10. A 40 B 24 C 20 D 16 13. 14. Find the minimum value of the objective function f(x, y) = -2x - 3y, and for what values of x and y, subject to the constraints x ≥ 0, x ≤ 5, y ≥ 0, y ≤ 5, and 5y - 2x ≥ 0. F -40, (12.5, 5) G -25, (5, 5) H -15, (0, 5) J -10, (5, 0) 14. 15. COLLECTIONS A scouting troop is collecting aluminum cans and paper to recycle. The total weight collected cannot exceed 30 pounds. The troop cannot collect more than 20 pounds of paper or 15 pounds of aluminum. If the troop earns $2.50 per pound for aluminum and $0.50 for paper, what is the maximum profit? A $35 Bonus Chapter 6 B $45 C $55 ⎡ 2 -3⎤ If A = ⎢ , find (A-1)-1. ⎣-5 8⎦ D $75 15. B: 38 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. -2 4 +− F − DATE 6 PERIOD Chapter 6 Test, Form 2A SCORE Write the letter for the correct answer in the blank at the right of each question. 1. What is the augmented matrix for the system? 4x + 2y = 6 -3x - 2y = -4 ⎡ 4 2 A ⎢ ⎣ -3 -2 6⎤ -4⎦ ⎡ 4 -3 B ⎢ ⎣ 2 -2 6⎤ -4⎦ ⎡ 4 2⎤ C ⎢ ⎣ -3 -2⎦ 2. Which matrix is not in row-echelon form? ⎡ 1 4 0 -2 ⎤ ⎡ 1 -2 3 5 ⎤ ⎡ 1 -2 G 0 0 0 0 H ⎢ F 0 1 3 1 ⎣0 1 5 ⎦ ⎣0 0 1 ⎣0 0 1 2⎦ ⎢ ⎢ -7 ⎤ 2⎦ ⎡-2 2 ⎤ D ⎢ ⎣ 3 4⎦ 1. ⎡1 5 J ⎢ ⎣0 1 2. 3⎤ -4 ⎦ 3. Solve the system of equations using Gaussian elimination. -2x - 3y + z = 4 4x + y - 2z = -13 -x + 2y - 4z = -8 A (2, 1, 3) B (-2, 1, 3) C (2, -1, 3) D (-2, -1, -3) 3. 4. SEWING The table shows several packages of assorted spools of thread available at a store. What is the price per spool of each kind of thread? Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Package Red White Blue Price ($) Bargain Spools 5 5 0 5.50 Simply Spools 5 2 5 7.50 Mega Spools 10 10 10 18.00 F ($0.50, $0.60, $0.70) H ($0.70, $0.60, $0.50) G ($0.70, $0.50, $0.60) J ($0.60, $0.50, $0.70) 4. ⎡ 1 -2 4 ⎤ 5. What is the determinant of 0 5 3 ? ⎣-5 -2 1 ⎦ ⎢ A -151 B -141 C 141 D 151 5. ⎡ -1 ⎡-0.4 1.2⎤ 3⎤ 6. Find AB if A = ⎢ and B = ⎢ . ⎣0.5 -0.2⎦ ⎣ 5 -0.1⎦ ⎡ 0.62 -1.2⎤ F ⎢ ⎣-1.5 15.4⎦ ⎡-0.62 ⎡ 15.4 -1.5⎤ 1.2⎤ G ⎢ H ⎢ ⎣ 1.5 -15.4⎦ ⎣-1.2 0.62⎦ ⎡ 6 8⎤ 7. Find the inverse of ⎢ , if it exists. ⎣-3 -4⎦ ⎡-4 -8 ⎤ ⎡-6 -3 ⎤ ⎡ 6 3⎤ A ⎢ B ⎢ C ⎢ ⎣ 3 6⎦ ⎣ 8 4⎦ ⎣ -8 -4 ⎦ Chapter 6 39 ⎡-15.4 1.5⎤ J ⎢ 6. ⎣ 1.2 -0.62⎦ D does not exist 7. Glencoe Precalculus Assessment NAME NAME DATE 6 Chapter 6 Test, Form 2A PERIOD (continued) ⎡-4 -1 ⎤ ⎡-8 13 ⎤ and DE = ⎢ , find E. 8. Given D = ⎢ ⎣ 2 3⎦ ⎣ 4 -9 ⎦ ⎡ 2 -3 ⎤ ⎡-2 3 ⎤ ⎡-1 0 ⎤ G ⎢ H ⎢ F ⎢ ⎣ 0 -1 ⎦ ⎣ 0 1⎦ ⎣-3 2 ⎦ ⎡1 0⎤ J ⎢ ⎣ 3 -2 ⎦ 8. 9. Solve the following system of equations using an inverse matrix. -x + 2y - 3z = 11 A (-3, -4, 2) 2x + z = 4 B (3, 4, -2) x - y + 2z = -5 C (3, -4, 2) D (-3, 4, -2) 9. 10. Use Cramer’s Rule to solve the system of equations. -4x + 2y = 30 F (4, 7) -x - y = -3 G (4, -7) J (-4, 7) H (-4, -7) 10. 11. FOOD On Friday, Lila raised $147.50 by selling 20 hamburgers and 35 hot dogs. On Saturday, she raised $107.50 by selling 15 hamburgers and 25 hot dogs. What was the selling price of one hot dog? A $2.50 B $2.75 C $3.00 D $3.25 11. 19x -1 12. Find the partial fraction decomposition of − . 2 3x -10x + 3 7 -2 +− H − -7 2 +− G − 7 -2 +− J − x -3 x+3 3x - 1 x+3 x -3 3x + 1 3x + 1 12. 3x - 1 13. Find the maximum value of the objective function f(x, y) = 3x - y subject to the constraints x ≥ 0, y ≥ 0, x + y ≤ 8, and x + 6y ≤ 24. A -4 B 15 C 24 D 48 13. 14. Find the minimum value of the objective function f(x, y) = 2x + 3y and for what values of x and y, subject to the constraints y ≥ 0, y ≤ 6, and y - x ≤ 3. F 0, (0, 0) G 9, (0, 3) H 18, (0, 6) J 24, (3, 6) 14. 15. SALES Tony can sell a maximum of 20 boxes of birthday cards and holiday cards. He cannot sell more than 12 boxes of birthday cards or 15 boxes of holiday cards. If he earns $3.50 per box of birthday cards and $2.50 per box of holiday cards, what is his maximum profit? A $79.50 Bonus Chapter 6 B $62 C $55 D $37.50 If A and B are inverse 2 × 2 matrices, what matrix represents the product of A and B? 40 15. B: Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. -7 2 +− F − DATE 6 PERIOD Chapter 6 Test, Form 2B SCORE Write the letter for the correct answer in the blank at the right of each question. 1. What is the augmented matrix for the given system? -5x + 2y = -21 3x - 4y = 21 ⎡-5 2⎤ ⎡-4 -2⎤ ⎡-5 2 -21⎤ ⎡-5 3 A ⎢ B ⎢ C ⎢ D ⎢ ⎣ 3 -4⎦ ⎣-3 -5⎦ ⎣ 3 -4 ⎣ 2 -4 21⎦ 2. Which matrix is not in row-echelon form? ⎡ 1 -2 0 ⎡ 1 -1 0 5⎤ 3⎤ ⎡ 1 -5 F 0 1 -3 2 G 0 1 -2 4 H ⎢ ⎣0 1 ⎣ 1 0 0 -1 ⎦ ⎣ 0 0 1 -2 ⎦ ⎢ ⎢ ⎡1 0 3⎤ J ⎢ ⎣0 1 6⎦ -21⎤ 1. 21⎦ -7 ⎤ 2⎦ 2. 3. Solve the system of equations using Gaussian elimination. 2x - y + 3z = -4 -x + 2y - z = 0 x + 2y + z = -4 A (2, 2, 2) B (1, 1, 1) C (-2, -2, -2) D (-1, -1, -1) 3. 4. CRAFTS The table shows several packages of assorted shapes available at a craft store. What is the price per shape? Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Package Hearts Stars Diamonds Price ($) Mini Shapes 20 0 20 3.00 More Shapes 40 20 10 6.00 Mighty Shapes 50 50 50 15.00 F ($0.05, $0.10, $0.15) H ($0.05, $0.15, $0.10) G ($0.10, $0.15, $0.05) J ($0.15, $0.10, $0.05) ⎡-2 4. 0⎤ 5. What is the determinant of 3 -4 -1 ? ⎣ 2 -1 0⎦ ⎢ A -12 4 B -6 D 12 5. ⎡ 0.3 -13⎤ H ⎢ ⎣-0.1 -4⎦ ⎡-0.3 -13⎤ J ⎢ ⎣-0.1 -4⎦ 6. ⎡ 5 4⎤ C ⎢ ⎣-1 -1⎦ D does not exist 7. C 6 ⎡-3 2⎤ ⎡-0.1 3⎤ 6. Find EF if E = ⎢ and F = ⎢ . ⎣-1 0.5⎦ ⎣ 0 -2⎦ ⎡-0.3 -13⎤ F ⎢ ⎣ 0.1 -4⎦ ⎡ 0.3 -13⎤ G ⎢ ⎣ 0.1 -4⎦ ⎡-5 4⎤ 7. Find the inverse of ⎢ , if it exists. ⎣ 1 -1⎦ ⎡-1 -4⎤ A ⎢ ⎣-1 -5⎦ Chapter 6 ⎡ 5 -4⎤ B ⎢ ⎣-1 1⎦ 41 Glencoe Precalculus Assessment NAME NAME DATE 6 Chapter 6 Test, Form 2B PERIOD (continued) ⎡ 24 -4 ⎤ ⎡-4 -1 ⎤ and AC = ⎢ , find C. 8. Given A = ⎢ ⎣ -18 -14 ⎦ ⎣ 3 5⎦ ⎡ 6 -2 ⎤ ⎡-4 -2 ⎤ ⎡ 6 0⎤ G ⎢ H ⎢ F ⎢ ⎣0 4⎦ ⎣ 0 -6 ⎦ ⎣ -2 4 ⎦ ⎡-6 2 ⎤ J ⎢ ⎣ 0 -4 ⎦ 8. 9. Solve the following system of equations using an inverse matrix. -3x + 2y - z = 19 A (4, -2, 3) x - 4y + 2z = -18 B (4, 2, 3) –x + 3z = -5 C (-4, 2, -3) D (-4, -2, -3) 9. 10. Use Cramer’s Rule to solve the system of equations. 6x - 2y = 28 -x + y = -8 G (-3, 5) F (3, -5) H (-3, -5) J (3, 5) 10. 11. PETS Willy’s Pet Store sells puppies and kittens. Last week, there were 5 puppies and 4 kittens sold, earning $388. This week, there were 8 puppies and 7 kittens sold, earning $643. What was the selling price of one kitten? A $51 B $48 C $42 D $37 11. 9x -16 . 12. Find the partial fraction decomposition of − 3x2 + 11x - 4 3 -4 H − +− 3 -4 G − +− -3 4 J − +− x+4 x -4 3x - 1 x+4 x -4 3x + 1 3x - 1 3x + 1 12. 13. Find the maximum value of the objective function f(x, y) = 2x + 3y subject to the constraints x ≥ 0, y ≥ 0, x + y ≤ 6, and y ≤ 3. A 9 B 12 C 15 D 24 13. 14. Find the minimum value of the objective function f(x, y) = -x + 2y and for what values of x and y, subject to the constraints x ≥ 0, x ≤ 3, y ≥ 0, y ≤ 2, and x + y ≥ 1. F -7, (3, -2) G -3, (3, 0) H 2, (0, 1) J 4, (0, 2) 14. 15. DRINKS Homer can buy at most 14 bottles of water and juice, but he cannot buy more than 8 bottles of water or 10 bottles of juice. If a bottle of water cost $2 and a bottle of juice costs $3.50, what is the most money he can spend? A $47 B $43 C $35 D $28 15. ⎡ 1 0⎤ Bonus What is the inverse of matrix A, if A = ⎢ ? ⎣ 0 1⎦ Chapter 6 42 B: Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. -3 4 F − +− NAME 6 DATE PERIOD Chapter 6 Test, Form 2C SCORE -4x + y = 12 3x - 2y = -14 Assessment 1. Write the augmented matrix for the given system of equations. 1. 2. Give an example of a matrix that is not in row-echelon form. 2. 3. Solve the system of equations using Gaussian elimination. -4x + 2y - 3z = 8 -x + y + 2z = 3 x - 3y - z = -7 3. 4. FISHING Ned, Reed, and Mel bought hooks, bobbers, and sinkers at the same fishing store. The table shows how many of each they bought and what they each paid. Determine the price per item. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Fishermen Hooks Bobbers Sinkers Paid ($) Ned 4 3 0 8.15 Reed 5 2 1 8.90 Mel 1 3 4 8.45 5. Find the determinant of ⎡-1 5 3 ⎤ 0 2 -4 . ⎣ 3 -2 1 ⎦ ⎢ 4. 5. ⎡-1 9 ⎤ ⎡-4 3 ⎤ and B = ⎢ . 6. Find AB if A = ⎢ ⎣ 0 -4 ⎦ ⎣ 5 -2 ⎦ 6. ⎡3 -6 ⎤ 7. Find the inverse of ⎢ , if it exists. ⎣1 -2⎦ 7. 8. Given M and MN, find N. ⎡4 -3 ⎤ ⎡-13 36 ⎤ MN = ⎢ M=⎢ ⎣5 2 ⎦ ⎣ 1 22 ⎦ 8. 9. Solve the system of equations using an inverse matrix. -3x + y + z = 2 5x + 2y - 4z = 21 x - 3y - 7z = -10 9. 10. Use Cramer’s Rule to solve the system of equations. 2x + y = 7 x+y=5 Chapter 6 10. 43 Glencoe Precalculus NAME 6 DATE Chapter 6 Test, Form 2C PERIOD (continued) 11. DINER Last week, the owner of a diner spent $91.25 on 15 gallons of milk and 11 pounds of butter. This week, he spent $70.40 on 12 gallons of milk and 8 pounds of butter. Find the cost of one pound of butter. 11. 12. Find all values of n such that the system represented by the given augmented matrix cannot be solved using an inverse matrix. ⎡-4 -n -5 ⎤ ⎢ ⎣ 4n 9 2⎦ 12. Find the partial decomposition of each rational expression. 5x - 41 13. − 2 13. x -2x -15 x2 - 13x + 61 (x + 1)(x - 4) 14. 8x2 - 4x + 18 x + 3x 15. 14. −2 15. − 3 10 16. GROWTH The function f(x) = − is a logistic growth 2 100x - x function. Find the partial decomposition of f(x). 16. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Find the maximum value of the objective function f(x, y) subject to the given constraints. 17. f(x, y) = 2x - y x≥0 y≥0 y≤5 x+y≤8 y≥0 -x + y ≤ 6 17. 18. f(x, y) = 3x - 5y x≥0 x≤4 2x + y ≤ 12 18. 19. Find the minimum value of the objective function f(x, y) and for what values of x and y it occurs, subject to the given constraints. f(x, y) = 3x + y x≥0 x≤5 y≥0 y≤7 y-x≤4 19. 20. VOLUNTEERING Tanya packs baskets at a local food pantry. She can put at most 20 canned and bottled items in a basket, but she cannot put more than 12 canned items or more than 14 bottled items in a basket. If a canned item costs the pantry $0.40 and a bottled item costs the pantry $0.60, what is the 20. most expensive basket Tanya can pack? Bonus Matrix D is the product of invertible matrices A, B, and C. In terms of A, B, C, and/or D, what does A-1D equal? B: Chapter 6 44 Glencoe Precalculus NAME DATE 6 PERIOD Chapter 6 Test, Form 2D SCORE Assessment 1. Write the augmented matrix for the given system of equations. 5x - 2y = -3 1. 4x + 7y = -11 2. Give an example of a matrix that is in row-echelon form. 2. 3. Solve the system of equations using Gaussian elimination. 2x - 3y + z = 14 -2x - z = -5 3. x + 4y = -10 4. SALES The table shows how many of each type of magazine Romeo sold each day and the money he received from the sales. Determine the price per magazine. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Day Sports News Eduational Money Earned ($) Saturday 3 0 2 19.00 Sunday 5 1 3 34.50 Monday 1 1 4 27.50 ⎡ -2 5. Find the determinant of ⎢ ⎣ 5 3 -4 1 -1 0⎤ 1 . 0⎦ ⎡ 5 ⎡2 -4⎤ 6. Find CD if C = ⎢ and D = ⎢ ⎣5 -6⎦ ⎣-4 4. 5. 0⎤ . 1⎦ ⎡ 12 -2⎤ 7. Find the inverse of ⎢ , if it exists. ⎣-11 2⎦ 6. 7. 8. Given U and UV, find V. ⎡-6 3⎤ U=⎢ ⎣ 1 -2⎦ ⎡ 15 -9⎤ UV = ⎢ ⎣-1 -6⎦ 8. 9. Solve the system of equations using an inverse matrix. x - 2y + z = -5 3x - 2y + z = 3 9. 2x - y + 2z = -7 10. Use Cramer’s Rule to solve the system of equations. -3x + 3y = -15 6x - 9y = 39 Chapter 6 45 10. Glencoe Precalculus NAME DATE 6 Chapter 6 Test, Form 2D PERIOD (continued) 11. FOOD Last week, Gina spent $24.75 on 8 cups of coffee and 5 bagels. This week, she spent $28.50 on 9 cups of coffee and 6 bagels. What is the cost of one bagel? 11. 12. Find all values of n such that the system represented by the given augmented matrix cannot be solved using an inverse matrix. ⎡ n2 -6 3⎤ ⎢ ⎣ -6 4 -1⎦ 12. Find the partial decomposition of each rational expression. 2x + 6 x - 3x 13. − 2 13. 2 - 2x - 5 − 14. 5x 2 3 14. x +x -7x2 + x - 10 (x - 2)(2x + 1) 15. − 2 15. 11 is a logistic 16. GROWTH The function f(x) = − 2 121x - x growth function. Find the partial decomposition of f(x). 16. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Find the maximum value of the objective function f(x, y) subject to the given constraints. 17. f(x, y) = -2x + 3y x≥0 x≤7 y≥0 x + 7y ≤ 28 17. 18. f(x, y) = x + 2y x≥0 y≥0 y ≤ 10 2x + y ≤ 18 y - 2x ≤ -6 18. 19. Find the minimum value of the objective function f(x, y) and for what values of x and y it occurs, subject to the given constraints. f(x, y) = -x + 2y x≥0 x ≤ 15 y≥0 y ≤ 13 2x + y ≥ 4 19. 20. BAKERY A baker can make at most 100 cupcakes and pies in one hour, but he cannot make more than 70 cupcakes or more than 60 pies in an hour. A cupcake costs $1.50 and a pie costs $6. What is the maximum value of baked goods that he can produce in one hour? 20. Bonus Matrix D is the product of invertible matrices A, B, and C. In terms of A, B, C, and/or D, what does (AB)-1D equal? Chapter 6 46 B: Glencoe Precalculus NAME 6 DATE PERIOD Chapter 6 Test, Form 3 SCORE 3x = 2y + 5 -y = -4x - 7 Assessment 1. Write the augmented matrix for the given system of equations. 1. 2. Give an example of a matrix that is in row-echelon form but contains one row with all zeros. 2. 3. Solve the system of equations using Gaussian elimination. 4x - 6z = 18 3y = 7x + 33 -8y + 12 = 4z 3. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 4. RENTALS Julie rents bicycles in a tourist town. The table shows how many of each type of bicycle Julie rented over the weekend and the money she received from the rentals. Determine the cost to rent each kind of bicycle. Day Tandem Child Adult Rental ($) Friday 2 1 7 269 Saturday 4 0 9 370 Sunday 3 2 6 296 ⎡ -1 ⎢ 5. Find the determinant of -4 ⎣ 2 3 0⎤ 5 -6 . 1 0⎦ 4. 5. ⎡ x 4⎤ ⎡1 -3⎤ ⎡2 22⎤ 6. Solve ⎢ ⎢ =⎢ for x and y. ⎣3 -1⎦ ⎣1 y⎦ ⎣2 -13⎦ 6. ⎡ 2 -1 2 ⎤ 7. Find the inverse of 1 0 -1 , if it exists. ⎣ 3 2 -1 ⎦ 7. ⎡4 -1⎤ ⎡-12 6⎤ 8. Given A = ⎢ and AB = ⎢ , find B. ⎣3 5⎦ ⎣ -9 -7⎦ 8. ⎢ 9. Solve the system of equations using an inverse matrix. -3x - y + z - 2w = -1 x + y - w = -1 2x - z + w = 0 3y + 2z = 1 10. Use Cramer’s Rule to solve the system of equations. 11x - 5y = 59 3x - y = 15 Chapter 6 47 9. 10. Glencoe Precalculus NAME 6 DATE Chapter 6 Test, Form 3 PERIOD (continued) 11. GARDENING Tayshia spent $13.60 on 4 tomato plants, 2 pepper plants, and 1 squash plant. Kiaya spent $16.80 on 6 tomato plants and 3 squash plants. LaTeesha spent $16.45 on 4 tomato plants, 3 pepper plants and 2 squash plants. What is the cost of one squash plant? 12. Find all values of n such that the system represented by ⎡ n 3 -1⎤ cannot be solved using an inverse matrix. ⎢ 4⎦ ⎣ -2 n - 5 11. 12. Find the partial decomposition of each rational expression. 15x2 - 2x - 59 13. − 2 13. -2x2 + 8x + 12 x + 4x + 4x 14. 4x2 + 7x - 1 x +x 15. (x - 6)(2x - 5) 14. − 3 2 15. − 2 13 is a logistic 16. GROWTH The function f(x) = − 2 169x - x growth function. Find the partial decomposition of f(x). 16. Find the maximum value of the objective function f(x, y) subject to the given constraints. x≥0 y-x≤2 y≥0 2x + y ≤ 20 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 17. f(x, y) = 3x - y 17. 18. f(x, y) = -x + 2y x≥0 y≤8 y≥0 x + y ≤ 12 y - 2x ≤ 4 18. 19. Find the minimum and maximum value of the objective function f(x, y) and for what values of x and y it occurs, subject to the given constraints. f(x, y) = 8x + 2y x≥0 x≤8 y≥0 y ≤ 10 4x + y ≤ 34 19. 20. CLOTHING Ginny sews dresses and gowns. Each dress sells for $200 and each gown sells for $650. It takes her 2 weeks to sew a dress and 5 weeks to sew a gown. She accepts orders for at least three times as many dresses as gowns. In the next 22 weeks, what is the maximum amount of money she can expect to earn? 20. Bonus Use a matrix equation to find the value of x for the given system of equations. ax + by = c dx + ey = f B: Chapter 6 48 Glencoe Precalculus 6 DATE PERIOD Extended-Response Test SCORE Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solutions in more than one way or investigate beyond the requirements of the problem. 1. AMUSEMENT PARKS Gala Amusement Corporation owns three amusement parks. The fraction of visitors at each park by quarter is shown in the first table. The total number of visitors for all three parks combined is shown in the second table, differentiated by quarter and time of day. Amusement Park 1 2 3 4 Ocean Side 0.40 0.25 0.15 0.35 Big Mountain 0.25 0.35 0.40 0.35 Lazy River 0.35 0.40 0.45 0.30 Quarter Morning Afternoon Evening 1 2000 2500 1500 2 3500 4000 2000 3 3000 4200 1800 4 2500 2000 1000 a. Write a 3 × 4 matrix representing the data found in the first table. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. b. Write a 4 × 3 matrix representing the data found in the second table. c. Use these two matrices to determine if there were more visitors to Lazy River Park in the morning or to Big Mountain Park in the afternoon. 2. Consider the following system of equations. 3x - y = -5 -x + 2z = 3 y-z=1 a. Write the system in matrix form. b. Find the inverse of the coefficient matrix. c. Use the matrix found in part b to solve the system. 3. CONSTRUCTION A builder constructs two styles of houses: the Executive, on which he makes a $30,000 profit, and the Suburban, on which he makes a $25,000 profit. He can complete up to 10 houses each year, with no more than 6 of them in the Executive style. a. How many houses of each style should he build to maximize his profit? Explain. b. If he keeps the original limit on Executives but raises the limit on the total number of houses he builds, what would be the effect on the number of each style built to maximize profit? Why? Chapter 6 49 Glencoe Precalculus Assessment NAME NAME DATE PERIOD 6 Standardized Test Practice SCORE (Chapters 1–6) Part 1: Multiple Choice Instructions: Fill in the appropriate circle for the best answer. -7 1. State the domain of f(x) = − . 2 x - x - 12 A B C D {x|x {x|x {x|x {x|x ≠ ≠ ≠ ≠ -4, 3 , x ∈ } -4, -3 , x ∈ } -3, 4 , x ∈ } 3, 4 , x ∈ } 1. A B C D 1 2. Evaluate − log64 x. 3 F 262,144 G 256 64 H − J 4 2. F G H J π C − D π 3. A B C D H sec x J tan x 4. F G H J D (1, -2, 1) 5. A B C D 6. F G H J 3 √2 2 3. Find the exact value of sin-1 −. π A − π B − 6 4 3 4. Simplify sin x tan x + cos x. F csc x G cot x 5. What is the solution of the system of equations shown? -4y + z = -7 2x - 3y + z = -7 A (-1, 2, 1) B (1, -2, -1) C (-1, 2, -1) 6. What is [g º f ](x) if f(x) = 3x2 - 1 and g(x) = x - 4? F [g º f ](x) = 3x2 - 5 H [g º f ](x) = 3x2 - 24x + 48 G [g º f ](x) = x2 - 8x + 16 J [g º f ](x) = 3x2 - 24x + 47 7. Solve 5x + 2 = 25x - 4. A 5 B 6 ⎡3 -5⎤ 8. What is the determinant of ⎢ ? ⎣1 3⎦ F -14 G -4 C 8 D 10 7. A B C D H 4 J 14 8. F G H J 9. A B C D 9. Which of the following is the inverse of f(x) = 2x + 3? A f -1(x) = -2x - 3 B f 1 (x) = − x-3 2 -1 Chapter 6 x-3 C f -1(x) = − D f 50 2 x+3 (x) = − 2 -1 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. -x + 3y + 2z = 9 NAME DATE 6 Standardized Test Practice PERIOD (continued) 10. Choose the radian measure that is equal to 405°. 9π G − 11π F − 4 4 7π H − 5π J − 10. F G H J ⎡ 1 -4 -3⎤ C ⎢ ⎣-2 0 5⎦ ⎡18 26⎤ D ⎢ ⎣-2 0⎦ 11. A B C D 12. F G H J 13. A B C D J 6 14. F G H J 3 D − 15. A B C D 16. F G H J 17. A B C D 4 4 11. Find AB, if possible. ⎡-1 4 3 ⎤ ⎡-1 0⎤ A=⎢ , B = ⎢ ⎣ 2 0 -5⎦ ⎣ 4 7⎦ A not possible ⎡7 -4 ⎤ B ⎢ ⎣0 -1⎦ 12. What is the effect on the graph of f(x) = log x when the equation is changed to g(x) = log (x + 4)? F G H J The The The The graph graph graph graph is is is is translated translated translated translated 4 4 4 4 units units units units up. to the left. to the right. down. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 1 13. Find the value of sin 2θ in the interval (0, 90°) given that cos θ = − . 15 A − 64 4 √ 15 D − 4 √ 15 8 15 B − C − 32 5 1 14. What is the solution of − x +x-4=− x? F 2 G 3 H 5 5π 15. Find the exact value of tan − . 4 1 A − 2 4 B − 5 C 1 2 16. Which is closest to the value of x? F 35.2 H 45.2 G 44.8 J 54.8 12 x° 17 17. The graph of an odd function is symmetric with respect to which of the following? A the x-axis B the y-axis Chapter 6 C the origin D none of these 51 Glencoe Precalculus Assessment (Chapters 1–6) NAME DATE 6 Standardized Test Practice PERIOD (continued) (Chapters 1–6) Part 2: Short Response Instructions: Write your answers in the space provided. 18. Find the vertical asymptotes in the interval [-2π, 2π] for x the graph of y = 3 tan − . 18. 2x - 4 19. Find the partial fraction decomposition of − . 2 19. 3 3x - 2x 20. List all of the possible rational zeros of f(x) = 2x4 - 3x2 + 5x - 3. 20. 21. Condense 3 ln (x - 2) - 4 ln x. 21. 22. Find all solutions to the given equation in the interval [0, 2π]. 2 cos2 θ = cos θ 22. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 23. State the amplitude, period, frequency, phase shift, and vertical π + 2. shift of y = 3 cos x - − ( 2 ) 23. 3x . 24. Consider g(x) = − x-1 a. Find the x- and y-intercepts. 24a. b. Find the vertical asymptote. 24b. c. Find the horizontal asymptote. 24c. 25. Consider f(x) = x 5 + 3x 4 - x 3 - 6x 2 + 8. a. Determine the consecutive integer values of x between which each real zero is located. 25a. b. Estimate the x-coordinates at which the relative maxima and relative minima occur. 25b. Chapter 6 52 Glencoe Precalculus Chapter 6 A1 Glencoe Precalculus DATE Before you begin Chapter 6 Systems of Equations and Matrices Anticipation Guide PERIOD D D 2. The row-echelon form of a matrix is unique. After you complete Chapter 6 Chapter 6 3 Chapter Resources 3/23/09 3:48:07 PM Glencoe Precalculus • For those statements that you mark with a D, use a piece of paper to write an example of why you disagree. • Did any of your opinions about the statements change from the first column? A A 9. Linear programming can be used to solve applications involving systems of equations. 10. In a linear programming problem, one evaluates the objective function at each vertex of the feasible region to maximize or minimize the function. D A 8. Graphically, a rational function and its partial fraction decomposition are different. polynomial and a proper rational expression. f(x) rational expression − , you must use the division d(x) r(x) f(x) algorithm − = q(x) + − to rewrite it as the sum of a d(x) d(x) 7. To find the partial fraction decomposition of an improper D D 5. Cramer’s Rule uses inverses to solve systems. 6. Cramer’s Rule applies when the determinant of the coefficient matrix is 0. A 4. The product of an m × r matrix and an r × n matrix results in an m × n matrix. 3. If a matrix has an inverse, then it is a singular matrix. A STEP 2 A or D 1. The augmented matrix of a system is derived from the coefficients and constant terms of the linear equations. Statement • Reread each statement and complete the last column by entering an A or a D. Step 2 STEP 1 A, D, or NS • Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure). • Decide whether you Agree (A) or Disagree (D) with the statement. • Read each statement. Step 1 6 NAME 0ii_004_PCCRMC06_893807.indd Sec1:3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. DATE PERIOD ⎢ ⎢ ⎢ ⎡ 1 -2 1 -1 ⎤ 0 1 -1 -1 1 − R → ⎣0 0 1 2⎦ 4 3 ⎢ Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC06_893807.indd 5 Chapter 6 Answers 5 2y + 4z = 0 (-2, 3, 4) (2, -1, 3) Lesson 6-1 11/17/09 4:38:23 PM Glencoe Precalculus (3, -2, 1) x-z=2 x - 4y + z = -10 3x + y - 2z = -1 3. -4x - y - z = -11 -3x + 2y - z = 8 2. 5x - y = -13 x + 2y - z = -3 1. -2x - y + z = 0 Solve each system of equations using Gaussian elimination with matrices. Exercises ⎡ 1 -2 1 -1 ⎤ 1 − R → 0 1 -1 -1 5 2 ⎣ 0 5 -1 3 ⎦ c. Step 3 Write the corresponding system of equations and use substitution to solve the system. x - 2y + z = -1 y - z = -1 z=2 The solution of the system is x = -1, y = 1, and z = 2 or (-1, 1, 2). ⎡ 1 -2 1 -1 ⎤ 0 1 -1 -1 R3 - 5R2→ ⎣ 0 0 4 8 ⎦ ⎢ e. b. d. ⎢ ⎡ 1 -2 1 ⎢ -1 ⎤ 0 5 -5 -5 R3 - 3R1→ ⎣ 0 5 -1 ⎢ 3 ⎦ ⎢ ⎡1 -2 1 ⎢ -1⎤ R2 - 2R1→ 0 5 -5 -5 ⎣3 -1 2 ⎢ 0⎦ a. Step 2 Apply elementary row operations to obtain a row-echelon form of the matrix. ⎢ ⎡1 -2 1 -1⎤ 2 1 -3 -7 ⎣3 -1 2 0⎦ Step 1 Write the augmented matrix. Example Solve the system of equations using Gaussian elimination with matrices. x - 2y + z = -1 2x + y - 3z = -7 3x - y + 2z = 0 You can solve a system of linear equations using matrices. Solving a system by transforming it into an equivalent system is called Gaussian elimination. First, create the augmented matrix. Then use elementary row operations to transform the matrix so that it is in row-echelon form. Then write the corresponding system of equations and use substitution to solve the system. Multivariable Linear Systems and Row Operations Study Guide and Intervention Gaussian Elimination 6-1 NAME Answers (Anticipation Guide and Lesson 6-1) (continued) PERIOD A2 ⎢ ⎢ ⎢ R1 - R3→ ⎡ 1 0 0 4 ⎤ 0 1 0 -5 ⎣0 0 1 1⎦ Reduced row-echelon form ⎢ ⎡ 1 -2 1 15 ⎤ 0 1 -2 -7 -R3→ ⎣ 0 0 1 1 ⎦ Glencoe Precalculus 005_032_PCCRMC06_893807.indd 6 6 (-2, -2, -2) (-7, 2, 3) Chapter 6 4x + 2y + z = 2 2x - 5y = 6 2x - 5y = -24 Glencoe Precalculus (-1, 5, -4) -x + 3y - 2z = 24 3. -2x - y - z = 1 -2y + 3z = -2 2. x - 4z = 6 -4x + z = 31 1. 3x - 2y + z = -22 Solve each system of equations using Gaussian or Gauss-Jordan elimination. Exercises ⎢ ⎢ ⎡ 1 -2 1 15 ⎤ 0 -5 4 29 R1 + R3→ ⎣ 0 -1 1 6 ⎦ The solution of the system is x = 4, y = -5, and z = 1 or (4, -5, 1). ⎢ R1 + 2R2→ ⎡ 1 0 1 5 ⎤ 0 1 0 -5 ⎣0 0 1 1⎦ ⎡ 1 -2 1 15 ⎤ R2 + 2R3→ 0 1 0 -5 ⎣0 0 1 1⎦ ⎢ ⎢ Row-echelon form ⎡ 1 -2 1 15 ⎤ 0 1 -2 -7 R2 + R3→ ⎣ 0 0 -1 -1 ⎦ ⎡ 1 -2 1 15 ⎤ R2 - 6R3→ 0 1 -2 -7 ⎣ 0 -1 1 6 ⎦ ⎢ ⎡ 1 -2 1 15 ⎤ ⎡ 1 -2 1 15 ⎤ 0 -5 4 29 -2 -1 2 -1 2R1 + R2→ ⎣ -1 1 0 -9 ⎦ ⎣ -1 1 0 -9 ⎦ Augmented Matrix Example Solve the system of equations. x - 2y + z = 15 -2x - y + 2z = -1 -x + y = -9 Write the augmented matrix. Apply elementary row operations to obtain a row-echelon form. Then apply elementary row operations to obtain zeros above the leading 1s in each row. If you continue to apply elementary row operations to the row-echelon form of any augmented matrix, you can obtain a matrix in which every column has one element equal to 1 and the remaining elements equal to 0. This is called the reduced row-echelon form of the matrix. Solving a system by transforming an augmented matrix so that it is in reduced row-echelon form is called Gauss-Jordan elimination. ⎡1 0 0 a⎤ 0 1 0 b ⎣0 0 1 c⎦ Multivariable Linear Systems and Row Operations Study Guide and Intervention DATE 11/17/09 4:38:36 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 6 Gauss-Jordan Elimination 6-1 NAME PERIOD (-2, -5, -4) ⎢ ⎡ 2 3 1 -23 ⎤ 2. -3 -1 4 -5 ⎣ -1 5 -1 -19 ⎦ 3. (4, -6, 2) ⎢ ⎡ -5 -3 0 -2 ⎤ 0 -2 6 24 ⎣ 4 0 -7 2 ⎦ ⎣ ⎢ ⎢ 3 4 7 ⎢ -8 ⎤ 6 -2 -3 1 5 -2 1 ⎢ 4 ⎦ ⎣ ⎢ ⎡ -4 (-1, -1, -1) 5x - y + 2z = -6 3x + 2y - 4z = -1 8. -2x - 5y + z = 6 (-3, 2, -4) x - y + z = -9 -2x + 5y - 4z = 32 9. 8x - y + 3z = -38 ⎢ ⎢ 7 3 Lindsay Edwin Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC06_893807.indd 7 Chapter 6 5 8 2 4 Oranges 2 15.30 13.50 14.20 4 Total Cost ($) 3 Pears 7 Lesson 6-1 3/23/09 3:45:49 PM Glencoe Precalculus 4 3 ⎢ 13.50 ⎤ 7 2 4 14.20 ; (1.1, 1.25, 1); apples: $1.10, oranges: $1.25, pears: $1 ⎣ 3 8 2 ⎢ 15.30 ⎦ ⎡5 Apples Name Rosario 10. FRUIT Three customers bought fruit at Michael’s Groceries. The table shows the amount of fruit bought by each person. Write and solve a system of equations to determine the price of each type of fruit. (4, -5) x + 3y = -11 7. -4x - 2y = -6 -2 -1 5 ⎤ 2 0 -1 8 0 1 -2 -4 ⎦ y - 2z = -4 ⎡ 2x - z = 8 5x - 2y + z = 4 6. -4x - 2y - z = 5 -2x - 3y + z = 6 5. 3x + 4y + 7z = -8 Solve each system of equations using Gauss-Jordan elimination. ⎡ 5 -2 14 ⎤ ⎢ ⎣ -3 1 -7 ⎦ -3x + y = -7 4. 5x - 2y = 14 Write the augmented matrix for each system of linear equations. (-8, 4) ⎡ 1 -1 -12 ⎤ 1. ⎢ ⎣ -3 2 32 ⎦ Multivariable Linear Systems and Row Operations Practice DATE Write each system of equations in triangular form using Gaussian elimination. Then solve the system. 6-1 NAME Answers (Lesson 6-1) Chapter 6 A3 1 1 ⎢ 31 ⎤ PERIOD 8 85 30 40 25 Senior Citizen 1385 2685 1755 Total Paid($) Glencoe Precalculus $500: simple savings, $2000: certificate of deposit, $2500: municipal bonds c. Solve the system that you wrote in part b using Gauss-Jordan elimination. ⎢ ⎤ 1 1 5000 0.035 0.043 0.01 182.50 0 ⎦ 0 -1 ⎣5 ⎡1 b. Write the augmented matrix for the system of equations that you wrote in part a. s + c + m = 5000 0.01s + 0.035c + 0.043m = 182.50 5s - m = 0 a. Write a system of equations representing Mr. Wiley’s investment pattern. 5. INVESTING Mr. Wiley invested $5000 in three different accounts at the beginning of last year, yielding him a total of $182.50 of interest at the end of the year. The three accounts were a simple savings account earning 1%, a certificate of deposit earning 3.5%, and municipal bonds earning 4.3%. His municipal bond investment was 5 times the amount of money invested in the simple savings account. $4: child, $11: adult, $9: senior citizen 45 Sun 175 110 80 100 Fri Adult Child Sat Day 4. MOVIES The table shows the number of individuals attending the movies over the weekend at the Majestic Theater. Determine the costs for a child, adult, and senior citizen to attend the movies. 10/23/09 11:55:32 AM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Precalculus 005_032_PCCRMC06_893807.indd 8 Chapter 6 ⎢ 3.25 ⎦ ⎣ 18 nickels, 6 dimes, 7 quarters ⎢ 10.05 -10.10 -10.25 ⎢ 5 ; ⎡1 3. COINS Tina has 31 nickels, dimes, and quarters in her purse. She has 5 more nickels then the total number of dimes and quarters. If the total value of the coins is $3.25, how many of each coin does Tina have in her purse? Write and solve a system to determine the number of coins. ⎢ ⎢ 1 1 1 ⎢ 275 ⎤ -4 1 -4 0 ; 40 tables, ⎣ 200 150 75 ⎢ 42,125 ⎦ 220 chairs, 15 stools 2. MANUFACTURING A company manufactures tables, chairs, and stools. Last week, it built a total of 275 items. The number of chairs built was four times the total number of tables and stools built. The total value of these items is $42,125 with a chair selling for $150, a table for $200, and a stool for $75. Write and solve a system of equations to determine the number of each item built last week. hamburger: $2.50, chips: $1.25 ⎡ 5 3 16.25 ⎤ ; (2.5, 1.25); ⎢ ⎣ 4 8 20 ⎦ ⎡ DATE Multivariable Linear Systems and Row Operations Word Problem Practice 1. FOOD Mark bought 5 hamburgers and 3 bags of chips at a cost of $16.25. Henry bought 4 hamburgers and 8 bags of chips at a cost of $20. Write and solve a system of equations to determine the cost of a hamburger and a bag of chips. 6-1 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Enrichment DATE PERIOD A A ⎢ ⎢ -D + 2E + F = -5 3D + 4E + F = -25 2D + -E + F = -5 ⎡ -1 2 1 -5 ⎤ 3 4 1 -25 . ⎣ 2 -1 1 -5 ⎦ or ⎢ 18 3 3 3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC06_893807.indd 9 Chapter 6 9 Lesson 6-1 10/23/09 12:00:51 PM Glencoe Precalculus x2 + y2 - 6x - 8y + 21 = 0 2. (3, 6), (5, 4), (3, 2) Answers 3x2 + 3y2 - 7x - 13y + 4 = 0 1. (1, 0), (-1, 2), (3, 1) Find an equation of the circle passing through the given points. Exercises in the equation of the circle in general form: 3x2 + 3y2 - 10x - 10y - 5 = 0. 10 10 5 x-− y-− = 0. Multiplying both sides of the equation by 3 results x2 + y2 - − 3 3 3 Substituting these values back into x2 + y2 + Dx + Ey + F = 0, you get 30 5 10 10 Using substitution, you can find that F = - − or - − , E = - − , and D = - − . ⎡ 1 -2 -1 ⎢ 5 ⎤ Using Gaussian elimination, you can find the equivalent matrix to be 0 1 -5 5 . ⎣ 0 0 18 ⎢ -30 ⎦ The augmented matrix for this system is 1 + 4 -1D + 2E + F = 0 9 + 16 + 3D + 4E + F = 0 4 + 1 + 2D -1E + F = 0 Because the three points are on the circle, they satisfy this equation. Use substitution to get the following system. A D E F equation is x2 + y2 + Dx + Ey + F = 0, where D = − , E = − , and F = − . Because A ≠ 0, divide both sides of the equation by A. The resulting The general form equation for a circle is Ax2 + Ay2 + Dx + Ey + F = 0, where A ≠ 0. Suppose you want to find the equation of a circle passing through the points (-1, 2), (3, 4), and (2, -1). How can you use the general form equation, systems of equations, and matrices to answer the question? Circles 6-1 NAME Answers (Lesson 6-1) TI-Nspire Activity Solve the system of equations. x+y+z=5 2x + 3y - z = 55 -x + 4y + 2z = 4 A4 10 (23, -18, 5) (14, 5, 9) Glencoe Precalculus 005_032_PCCRMC06_893807.indd 10 Chapter 6 2x - 2y + 3z = 97 x - y - 2z = 31 2x - y - z = 14 x + 3y = 29 4. x + y + z = 10 (-2, 10, -1) (-4, 11, 21) 3. 2x - z = 19 2x + 3y + 2z = 24 4x - y + 4z = -22 6x + y + z = 8 2. x - y -z = -11 -x + 5y - 3z = -4 1. x + 2y + z = 39 Solve each system of equations. Exercises To solve another system of three equations in three variables, you can copy the left side of the previous line onto the current line by highlighting it and pressing ·. Then simply update the numbers in the cells and press ·. Step 3: Use the matrix to solve the system. The solution is (8, 9, -12). Step 2: Press menu and choose MATRIX & VECTOR > REDUCED ROW-ECHELON FORM. Press / v to enter the matrix above which is considered the previous answer. Press ·. Step 1: Add a CALCULATOR page. Enter the augmented matrix by pressing / and the multiplication key. Select the 3 by 3 matrix. Change the number of columns to 4. Type in the elements. Example Glencoe Precalculus PERIOD You can solve a system of equations by entering the augmented matrix into a TI-Nspire and finding the reduced row-echelon form of the matrix. DATE 3/23/09 3:46:02 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 6 Reduced Row-Echelon Form 6-1 NAME DATE PERIOD 2 3⎤ to find AB, 4 -1 ⎦ 2 3⎤ ⎡ 4(-1) + (-2)(-2) =⎢ 4 -1⎦ ⎣ 2 3⎤ ⎡ 4(-1) + (-2)(-2) 4(2) + (-2)(4) =⎢ 4 -1⎦ ⎣ ⎤ ⎦ ⎤ ⎦ Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC06_893807.indd 11 Chapter 6 ⎡ -17 -9 ⎤ ⎡ -2 2⎤ ; BA = ⎢ ⎣ ⎣ 3 -13⎦ 6 2⎦ AB = ⎢ ⎡-1 5⎤ ⎡ 2 4⎤ 1. A = ⎢ , B = ⎢ ⎣ 0 -2⎦ ⎣-3 -1⎦ 11 2 3⎤ ⎡ 0 0 14⎤ =⎢ 4 -1⎦ ⎣ -5 10 -6⎦ Find AB and BA, if possible. Exercises ⎡ 4 -2⎤ ⎡ -1 ⎢ ·⎢ ⎣ -1 3⎦ ⎣ -2 Lesson 6-2 3/23/09 3:46:06 PM Glencoe Precalculus ⎡ -7 -7 6 ⎤ ; BA is undefined. ⎣ 0 -14 4 ⎦ AB = ⎢ ⎡-1 3⎤ ⎡-2 4 0⎤ 2. A = ⎢ , B = ⎢ ⎣ –3 –1 2⎦ ⎣-3 2⎦ 2 3⎤ ⎡ 4(-1) + (-2)(-2) 4(2) + (-2)(4) 4(3) + (-2)(-1) ⎤ =⎢ 4 -1⎦ ⎣ (-1)(-1) + 3(-2) (-1)(2) + 3(4) (-1)(3) + 3(-1) ⎦ Then simplify each sum. ⎡ 4 -2⎤ ⎡ -1 ⎢ ·⎢ ⎣ -1 3⎦ ⎣ -2 Continue multiplying each row by each column to find the sum for each entry. ⎡ 4 -2⎤ ⎡ -1 ⎢ ·⎢ ⎣ -1 3⎦ ⎣ -2 Follow this same procedure to find the entry for row 1, column 2 of AB. ⎡ 4 -2⎤ ⎡ -1 ⎢ ·⎢ ⎣ -1 3⎦ ⎣ -2 To find the first entry in AB, write the sum of the products of the entries in row 1 of A and in column 1 of B. A is a 2 × 2 matrix and B is a 2 × 3 matrix. Because the number of columns for A is equal to the number of rows for B, the product AB exists. 2 3⎤ 4 -1⎦ ⎡ 4 -2 ⎤ ⎡ -1 Use matrices A = ⎢ and B = ⎢ ⎣ -1 3 ⎦ ⎣ -2 ⎡ 4 -2⎤ ⎡ -1 AB = ⎢ ·⎢ ⎣ -1 3⎦ ⎣ -2 if possible. Example To multiply matrix A by matrix B, the number of columns in A must be equal to the number of rows in B. If A has dimensions m × r and B has dimensions r × n, their product, AB, is an m × n matrix. If the number of columns in A does not equal the number of rows in B, the matrices cannot be multiplied. ⎡ a b ⎤ ⎡ e f ⎤ ⎡ae + bg af + bh ⎤ ⎢ =⎢ ·⎢ ⎣ c d ⎦ ⎣ g h ⎦ ⎣ ce + dg cf + dh⎦ Matrix Multiplication, Inverses, and Determinants Study Guide and Intervention Multiply Matrices 6-2 NAME Answers (Lesson 6-1 and Lesson 6-2) Chapter 6 DATE (continued) PERIOD Matrix Multiplication, Inverses, and Determinants Study Guide and Intervention 7(4) + (-4)(7) ⎤ ⎡1 0⎤ or ⎢ -5(4) + 3(7)⎦ ⎣0 1⎦ A5 no; AB ≠ BA ≠ I2 yes; AB = BA = I2 12 Glencoe Precalculus 10/23/09 12:02:43 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Precalculus 005_032_PCCRMC06_893807.indd 12 Chapter 6 ⎡3 2⎤ 4. Find the determinant of A = ⎢ . Then find A–1, if it exists. ⎣1 -1⎦ ⎡ 0.2 0.4⎤ det(A) = -5, A-1 = ⎢ ⎣ 0.2 -0.6⎦ ⎡ 5 -1⎤ 3. Find the determinant of A = ⎢ . Then find A-1, if it exists. det(A) = 0; ⎣-10 2⎦ A-1 does not exist. ⎡3 2⎤ ⎡1 2⎤ 2. A = ⎢ , B = ⎢ ⎣4 1⎦ ⎣4 3⎦ ⎡11 5⎤ ⎡ 1 -5⎤ 1. A = ⎢ , B = ⎢ ⎣ 2 1⎦ ⎣-2 11⎦ Determine whether A and B are inverse matrices. Explain your reasoning. Exercises Since det(A) ≠ 0, A is invertible. = 2(-6) - 3(-2) or -6 ⎡ 1⎤ ⎢1 -− 3 =⎢ 1 1 ⎢− - − 3⎦ ⎣2 1 ⎡-6 2⎤ A-1 = - − ⎢ 6 ⎣-3 2⎦ ⎡ 2 -2⎤ Find the determinant of A = ⎢ . Then find A–1, if it exists. ⎣ 3 -6⎦ ⎢2 -2⎢ det(A) = ⎢ ⎢ ⎢3 -6⎢ Example 2 ⎡3 4⎤ ⎡ 7 -4⎤ ⎡3(7) + 4(-5) 3(-4) + 4(3) ⎤ ⎡1 0⎤ BA = ⎢ ·⎢ =⎢ or ⎢ ⎣5 7⎦ ⎣-5 3⎦ ⎣ 5(7) + 7(-5) 5(-4) + 7(3)⎦ ⎣0 1⎦ Because AB = BA = I, B = A-1 and A = B-1. ⎡ 7 -4⎤ ⎡3 4⎤ ⎡7(3) + (-4)(5) AB = ⎢ ·⎢ =⎢ ⎣-5 3⎦ ⎣5 7⎦ ⎣ -5(3) + 3(5) If A and B are inverse matrices, then AB = BA = I. Example 1 ⎡ 7 -4⎤ ⎡3 4⎤ Determine whether A = ⎢ and B = ⎢ are ⎣-5 3⎦ ⎣5 7⎦ inverse matrices. The identity matrix is an n × n matrix consisting of all 1s on its main diagonal, from upper left to lower right, and 0s for all other elements. Let In be the identity matrix of order n and let A be an n × n matrix. If there exists a matrix B such that AB = BA = In, then B is called the inverse of A and is written as A–1. If a matrix has an inverse, it is invertible. The determinant of a 2 × 2 matrix can be used to determine whether or not a matrix is invertible. ⎡a b ⎤ ⎡ d -b⎤ 1 If A = ⎢ , det(A) = ad - cb. If ad - cb ≠ 0, then A-1 = − ⎢ . ad - cb ⎣-c a⎦ ⎣ c d⎦ Inverses and Determinants 6-2 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. DATE PERIOD 210 B 400 520 3-Wood 450 310 5-Wood 400 300 Putter 150 120 170 Putter 210 3-Wood 5-Wood Club Value ($) Club 1-Wood ⎡ ⎤ ⎡ ⎢ ⎢ ⎢ no ⎡5 -2⎤ ⎡-1 0⎤ 7. A = ⎢ , B = ⎢ ⎣4 -3⎦ ⎣ 2 -8⎦ Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC06_893807.indd 13 Chapter 6 ⎡ 3 -27 12⎤ ⎢ ⎣-9 4 -2⎦ 10. AB + C Evaluate. ⎡-1 5⎤ A =⎢ ⎣ 3 0⎦ Answers 13 Lesson 6-2 10/23/09 12:03:23 PM Glencoe Precalculus ⎡-1 0 -4⎤ C =⎢ ⎣ 3 -2 1⎦ ⎡-12 -17 7⎤ ⎢ ⎣ -9 6 9⎦ 11. A(B - C) ⎡-4 2 -1⎤ B =⎢ ⎣ 0 -5 3⎦ Find the determinant of each matrix. Then find its inverse, if it exists. ⎡ 1 -2.5⎤ ⎡6 5⎤ ⎡-2 4⎤ 8. ⎢ 9. ⎢ 2; ⎢ 0; singular ⎣2 2⎦ ⎣ 3 -6⎦ ⎣-1 3⎦ yes ⎡1 2⎤ ⎡ 3 -2⎤ 6. A = ⎢ , B = ⎢ ⎣1 3⎦ ⎣-1 1⎦ Determine whether A and B are inverse matrices. ⎢ ⎢ ⎢ 3x1 - 2x2 + 8x3 = 28 3 ⎤ ⎡ x1 ⎤ 4 1 -2 2 1 2⎤ ⎡ x1 ⎤ ⎡ 11 ⎤ x · = (4, -3, -2) 5 3 -1 -5 -1 4 · x2 = 1 (2, 1, 3) 13 2 x 4 4 -1 11 ⎣ ⎦ ⎣ 3 -2 8⎦ ⎣ x3 ⎦ ⎣ 28 ⎦ ⎦ ⎣ 3⎦ ⎣ ⎡ -5x1 - x2 + 4x3 = 1 4x1 - x2 + 4x3 = 11 5. 2x1 + x2 + 2x3 = 11 5x1 + 3x2 - x3 = 13 4. x1 - 2x2 + 3x3 = 4 Write each system of equations as a matrix equation, AX = B. Then use Gauss-Jordan elimination on the augmented matrix to solve for X. 600 1-Wood A Company Club Type and Quantity ⎡ 9 15⎤ ⎡ 4 10⎤ 20 -4⎤ AB = ⎢ ; BA = ⎢ ⎣ 10 -12 3⎦ ⎣-7 -5⎦ ⎣–6 0⎦ 3. GOLF The number of golf clubs manufactured daily by two different companies is shown, as well as the selling price of each type of club. Use this information to determine which company’s daily production has the highest retail value. How much greater is the value? Company A; $69,300 AB is undefined; BA = ⎢ ⎡-14 ⎡-1 6 0⎤ ⎡ 2 -4⎤ 1. A = ⎢ , B = ⎢ ⎣ 3 -2 1⎦ ⎣ -1 3⎦ ⎡ 3 0⎤ ⎡ 3 5⎤ 2. A = ⎢ , B = ⎢ ⎣-1 2⎦ ⎣ -2 0⎦ Matrix Multiplication, Inverses, and Determinants Practice Find AB and BA, if possible. 6-2 NAME Answers (Lesson 6-2) A6 300 4 HP 250 Mower Type Retail Value ($) 2 350 5 HP 3 4 3 C 6 3 2 Holly Joelle Luisa 4 5 4 C I 6 1 2 20% 50% 30% SS C I 30% 30% 40% System B c. Luisa system A b. Joelle system A a. Holly system B Use matrices to determine which system favors each skater. System A Criteria One of two weighted systems shown below is used. SS Skater 2. ICE SKATING Holly, Joelle, and Luisa are competitive skaters. Their routines are judged on skating skill (SS), choreography (C), and interpretation (I). In a recent competition, they received the following scores. Store A; $4600 4.5 HP 7 5 HP 5 3 4.5 HP 4 B Store 5 A 4 HP Mower Type Glencoe Precalculus 005_032_PCCRMC06_893807.indd 14 Chapter 6 PERIOD 14 Charge per unit ($) B 40 C 30 ⎢ ⎢ Glencoe Precalculus 25 units from vendor A, 75 units from vendor B, 50 units from vendor C c. Solve the system that you found in part b to determine how many units of beads were purchased from each of the vendors. ⎢ ⎡ 1 1 1⎤ ⎡ A ⎤ ⎡ 150 ⎤ 35 40 30 · B = 5375 ⎣ –2 0 1⎦ ⎣ C ⎦ ⎣ 0⎦ b. Write the system of equations that you found in part a as a matrix equation, DX = E. A + B + C = 150 35A + 40B + 30C = 5375 -2A + C = 0 The total delivery cost was $5375. The store ordered twice the number of units of beads from vendor C than it ordered from vendor A. a. Write a system of equations representing this situation. A 35 Vendor 4. CRAFTS A craft store orders beads from three different vendors, A, B, and C. One month, the store ordered a total of 150 units of beads from these vendors. The shipping charges are as shown. 7 trips by the 10-ton truck, 13 trips by the 12-ton truck 3. LANDSCAPING Two dump trucks have capacities of 10 tons and 12 tons. They make a total of 20 round trips to haul 226 tons of topsoil for a landscaping project. How many round trips does each truck make? Matrix Multiplication, Inverses, and Determinants Word Problem Practice DATE 3/23/09 3:46:23 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 6 1. INVENTORY A hardware company keeps three types of lawnmowers in stock at each of its three stores. The current inventory and retail price for each mower is shown. Determine which store’s inventory has the greatest value. What is this value? 6-2 NAME Enrichment DATE ⎢ P1 P2 P3 ⎡ 0 2 3⎤ 2 0 1 ⎣ 3 1 0⎦ 1 1 PERIOD P2 P3 P1 ⎢ ⎢ P2 P3 ⎢ P1 P2 P3 3 1 6 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC06_893807.indd 15 Chapter 6 15 travel from P2 to P1 by going through P3. Lesson 6-2 3/23/09 3:46:27 PM Glencoe Precalculus 2. What does the product of row 2 and column 1 indicate? There are three ways to travel from P1 to P3 by going through P2. 1. What does the product of row 1 and column 3 indicate? There are two ways to Exercises The result indicates that there are six ways to travel from P3 to P2 by going through P1. 3 2 1 0 0 1 × + × + × = P2→P2 P3→P3 P3→P2 P3→P1 P1→P2 P3→P2 Similarly, consider the product of row 3 and column 2. The result indicates that there are three ways to travel from P1 to P2 by going through P3. 0 2 2 0 3 1 × + × + × = P1→P1 P1→P2 P1→P2 P2→P2 P1→P3 P3→P2 Consider the product of row 1 and column 2. P1 ⎡ 0 2 3⎤ P1 ⎡ 0 2 3⎤ P1 ⎡13 3 2⎤ M2 = P2 2 0 1 P2 2 0 1 = P2 3 5 6 P3 ⎣ 3 1 0⎦ P3 ⎣ 3 1 0⎦ P3 ⎣ 2 6 10⎦ P1 It can be shown that if we square matrix M, we can determine how many ways there are to travel from one park to another by traveling through the third park. Confirm the numbers in each cell. For example, the element in row 2, column 3 indicates that there is only one way to travel from P2 to P3 without going through P1. The element in row 1, column 3 indicates there are three ways to travel from P1 to P3 without going through P2. P3 M = P2 P1 Suppose three state parks P1, P2, and P3 are connected by roads as shown in the figure. As you can see, there are only two ways to travel from P1 to P2 without going through P3. There are only three ways to travel from P1 to P3 without going through P2. The matrix M represents the number of ways to go from one park to another without traveling through the third park. Travel 6-2 NAME Answers (Lesson 6-2) Chapter 6 DATE A7 7⎤ 3⎦ ⎢ ⎢ ⎢ x - z = -4 3x - y = -10 16 (2, -2, 5) 10/23/09 12:09:41 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Precalculus 005_032_PCCRMC06_893807.indd 16 Chapter 6 (6, -11) y + 4z = 18 2x - 3y + 2z = 20 4. x + y - z = -5 3. 3x + y = 7 -2x - 5y = 43 (-1, 0, 3) (-2, 4) 3x + 2y + z = 0 2. x - y + 2z = 5 1. -2x + 5y = 24 Glencoe Precalculus ⎢ ⎢ ⎡ 4 7 2⎤ ⎡ 0⎤ ⎡ 1⎤ = 11 17 5 · 9 = -2 ⎣-2 -3 -1⎦ ⎣-3⎦ ⎣ 4⎦ So, the solution of the system is (1, -2, 4). Multiply A-1 by B to solve the system. · B X= A-1 ⎢ ⎡-2 1 1⎤ ⎡x⎤ ⎡ 0⎤ 9 1 0 2 · y = ⎣ 1 -2 -9⎦ ⎣ z⎦ ⎣-31⎦ Use a graphing calculator to find A-1. b. -2x + y + z = 0 x + 2z = 9 x - 2y - 9z = -31 Write the system in matrix form. A · X = B Use an inverse matrix to solve each system of equations, if possible. Exercises So, the solution of the system is (-24, -8). Multiply A-1 by B to solve the system. X = A-1 · B ⎡-24⎤ ⎡-2 -7 ⎤ ⎡-16⎤ = ⎢ ·⎢ = ⎢ ⎣ -8⎦ ⎣-1 -3 ⎦ ⎣ 8⎦ ⎡2 1 = −− ⎢ (2)(3) - (-7)(-1) ⎣1 ⎡ d -b⎤ 1 A-1 = − ⎢ ad - cb ⎣-c a⎦ Use the formula for an inverse of a 2 × 2 matrix to find the inverse A-1. ⎡-16⎤ ⎡ 3 -7 ⎤ ⎡x⎤ ⎢ ·⎢ = ⎢ ⎣ 8⎦ ⎣-1 2 ⎦ ⎣y⎦ Write the system in matrix form. A · X = B a. 3x - 7y = -16 -x + 2y = 8 Use an inverse matrix to solve each system of equations, if possible. A square system has the same number of equations as variables. If a square matrix has an inverse, the system has one unique solution. Example PERIOD Solving Linear Systems Using Inverses and Cramer’s Rule Study Guide and Intervention Use Inverse Matrices 6-3 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. DATE (continued) PERIOD |A | |A| |A| |A | |A | |A| ⎪-25 -21 ⎥ = (-2)(-2) - 5(1) or -1 ⎪-717 -21⎥ -1 |A| -7(-2) - 17(1) -1 -1 ⎥ Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC06_893807.indd 17 Chapter 6 -x + 2y = 14 3. 3x + y = 21 -2x - 5y = -8 1. x - 2y = -5 (4, 9) (-1, 2) Answers 17 x + 3y = -3 4. -2x - 4y = 2 -x + 4y = 9 2. 3x - 3y = -18 Lesson 6-3 10/23/09 12:11:22 PM Glencoe Precalculus (3, -2) (-5, 1) Use Cramer’s Rule to find the solution of each system of linear equations, if a unique solution exists. Exercises Therefore, the solution is x1 = 3 and x2 = -1 or (3, -1). ⎪ -2 -7 |A2| -2(17) - 5(-7) 5 17 1 x2 = − =−= − =− or -1 -1 -1 -1 |A| -3 1 x1 = − =−= − =− or 3 |A | Because the determinant of A does not equal zero, you can apply Cramer’s Rule. |A| = ⎡-2 1⎤ The coefficient matrix is A = ⎢ . Calculate the determinant of A. ⎣ 5 -2⎦ Example Use Cramer’s Rule to find the solution of the system of linear equations, if a unique solution exists. -2x1 + x2 = -7 5x1 - 2x2 = 17 where Ai is the matrix obtained by replacing the ith column of A with the column of constants B. If det(A) = 0, then AX = B has either no solution or infinitely many solutions. |A| 3 n 1 2 x1 = − , x2 = − , x3 = − , … , xn = − , |A | Let A be the coefficient matrix of a system of n linear equations in n variables given by AX = B. If det(A) ≠ 0, then the unique solution of the system is given by Another method, known as Cramer’s Rule, can be used to solve a square system of equations. Solving Linear Systems Using Inverses and Cramer’s Rule Study Guide and Intervention Use Cramer’s Rule 6-3 NAME Answers (Lesson 6-3) PERIOD ) x + 2y + z = 8 2x + 3y - z = 1 no solution -4x + 5y - z = 18 5x - 3y = -11 (-1, 2, -4) A8 y + 2z = 20 (-5, 6, 7) 3x - z = -22 7. x + y + z = 8 Glencoe Precalculus 005_032_PCCRMC06_893807.indd 18 Chapter 6 18 Payton delivers 100; Santiago delivers 20; Queisha delivers 10 8. PAPER ROUTE Payton, Santiago, and Queisha each have a paper route. Payton delivers 5 times as many papers as Santiago. Santiago delivers twice as many papers as Queisha. If 20 papers were added to Payton’s route, he would then deliver four times the number of papers that Santiago and Queisha deliver. How many papers does each person deliver? -2x - 3y = -1 (-4, 3) 6. -4x - 5y = 1 Use Cramer’s Rule to find the solution of each system of linear equations, if a unique solution exists. Manuel: 3 hours, Harry: 2 hours, Ellen: 6 hours 5. TELEVISION During the summer, Manuel watches television M hours per day, Monday through Friday. Harry watches television H hours per day, Friday through Sunday. Ellen watches television E hours per day, Friday and Saturday. Altogether, they watch television 33 hours each week. On Fridays, they watch a total of 11 hours of television. If the number of hours Ellen spends watching television on any given day is twice the number of hours that Manuel spends watching television on any given day, how many hours of television does each of them watch each day? 4. x + y - 2z = 5 (-2, 5) 4x + 3y = 7 2. -2x - 8y = -36 3. x - 2y + 7z = -33 1 − , -4 2 ( -6x + 2y = -11 1. 4x - 7y = 30 Glencoe Precalculus Solving Linear Systems Using Inverses and Cramer’s Rule Practice DATE 3/23/09 3:46:41 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 6 Use an inverse matrix to solve each system of equations, if possible. 6-3 NAME D C : 9 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC06_893807.indd 19 Chapter 6 Marty hit 37 home runs; Carlos hit 36 home runs; Andrew hit 35 home runs. 2. BASEBALL In one season, Marty, Carlos, and Andrew hit a total of 108 home runs. Marty and Andrew hit twice as many home runs as did Carlos, although Carlos had one more home run than Andrew. How many home runs did each player hit? a = 16 centimeters, b = 30 centimeters, c = 34 centimeters ; B 8 DATE PERIOD 19 0.2 0.4 0.4 X Y Z 0.4 0 0.3 Almonds 0.2 0.6 0.5 Hazelnuts ⎢ ⎢ Lesson 6-3 3/23/09 3:46:44 PM Glencoe Precalculus 8000 X gift boxes, 9500 Y gift boxes, 11,500 Z gift boxes c. Determine how many of each gift box the company has. ⎢ ⎡ 0.2 0.4 0.4 ⎤ ⎡ X ⎤ ⎡ 10,000 ⎤ 0.3 0 0.4 · Y = 7000 ⎣ 0.5 0.6 0.2 ⎦ ⎣ Z ⎦ ⎣ 12,000 ⎦ b. Solve the system of equations that you wrote in part a as a matrix equation, AX = B. 0.2X + 0.4Y + 0.4Z = 10,000 0.3X + 0.4Z = 7000 0.5X + 0.6Y + 0.2Z = 12,000 a. Write a system of equations representing this situation. The company has 10,000 pounds of cashews, 7000 pounds of almonds, and 12,000 pounds of hazelnuts in its gift boxes. Cashews Gift Box 4. NUTS A nut company makes three types of one-pound gift boxes: X, Y, and Z. The table shows the amount of each nut in each box. $750 at 4%, $2000 at 5%, $1250 at 3.5% 3. INVESTING A total of $4000 is invested in three accounts paying 4%, 5%, and 3.5% simple interest. The combined annual interest is $173.75. If the interest earned at 5% is $70 more than the interest earned at 4%, how much money is invested in each account? Solving Linear Systems Using Inverses and Cramer’s Rule Word Problem Practice 1. PERIMETER The perimeter of rectangle WXYZ is 92 centimeters. The perimeter of triangle WXZ is 80 centimeters. If the −− length of XZ is two more than twice the −−− length of WZ, what are the values of a, b, and c? 6-3 NAME Answers (Lesson 6-3) Chapter 6 Enrichment DATE PERIOD A9 ⎢ ⎢ 2 ⎢ ⎢ 20 18 11 rectangle triangle 6 12 1 n 10.5 20 4 A Glencoe Precalculus 8 3/23/09 3:46:49 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Precalculus 005_032_PCCRMC06_893807.indd 20 Chapter 6 See students’ work. 2. Verify Pick’s Theorem using three simple polygons of your choice. ⎢ ⎡ 11 7 1 ⎤ ⎡ x ⎤ ⎡ 11.5 ⎤ 4 3 1 y = 4 ⎣ 11 3 1 ⎦ ⎣ z ⎦ ⎣ 7.5 ⎦ 1. Write the matrix equation that would be used to verify Pick’s Theorem using the polygons at the right. Exercises 1 Solving this system, we can see that x = − , y = 1, and z = -1. ⎢ Use the table to write a system and matrix equation. 8x + y + z = 4 ⎡ 8 1 1⎤ ⎡x⎤ ⎡ 4 ⎤ 18x + 12y + z = 20 18 12 1 y = 20 11x + 6y + z = 10.5 ⎣11 6 1⎦ ⎣ z⎦ ⎣10.5⎦ b Figure square Start by drawing three simple polygons on square dot paper similar to the ones shown below. Be sure that the number of boundary points, interior points, and the area of the figures are different. The table shown below summarizes the information. You can use systems of equations and matrices to verify Pick’s Theorem. To verify that the 1 coefficients in the equation for A are − , 1, and -1, you can write a system of three equations 2 of the form A = bx + ny + z, where the values of A, b, and n vary from polygon to polygon. 2 10 + 2 - 1 or 6 square units. and n = 2. Therefore, A = − 2 b interior points n minus 1, or A = − + n - 1. In the figure, b = 10 Consider the simple polygon drawn on square dot paper, shown at the right. Pick’s Theorem states that the area of the polygon A is equal to half the number of boundary points b plus the number of Pick’s Theorem 6-3 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Spreadsheet Activity DATE PERIOD A x= y= = A1*C2 - C1*A2 = C1*B2 - B1*C2 = A1*B2 - B1*A2 6 5 B = (A6/A4) = (A8/A4) 3 1 -12 8 C D Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC06_893807.indd 21 Chapter 6 (0.4, 0.2) Answers 21 (-4, 4) 7. 3y = 4x + 28 5x + 7y = 8 (2, -3) (4, -12) 6. 0.3x + 1.6y = 0.44 0.4x + 2.5y = 0.66 5. 5x - 3y = 19 7x + 2y = 8 4. 6x + 3y = -12 5x + y = 8 Use the spreadsheet to solve each system of equations. C1 B1 A1 C1 ⎪ ⎪ A2 C2⎥ C2 B2⎥ x = −; y = − B1 A1 B1 ⎪A2 B2⎥ ⎪A1 A2 B2⎥ 3. Explain how the values of x and y are found using Cramer’s Rule. ⎡C1 B1⎤ ⎡A1 C1⎤ ⎢ ; ⎢ ⎣C2 B2⎦ ⎣A2 C2⎦ 2. Write matrices whose determinants are found using the formulas in cells A6 and A8. ⎡A1 B1⎤ ⎢ ⎣A2 B2⎦ Lesson 6-3 3/23/09 3:46:54 PM Glencoe Precalculus To use the spreadsheet to solve a system of equations, write each equation in the form below. ax + by = c In the spreadsheet, the values of a, b, and c for the first equation are entered in cells A1, B1, and C1, respectively. The values of a, b, and c for the second equation are entered in cells A2, B2, and C2, respectively. The values for the system 6x + 3y = -12 and 5x + y = 8 are shown. 1. Study the formula in cell A4. Write a matrix whose determinant is found using this formula. Exercises 1 2 3 4 5 6 7 8 9 10 11 12 You can use a spreadsheet to solve systems of equations with Cramer's Rule. Cramer’s Rule 6-3 NAME Answers (Lesson 6-3) Partial Fractions Study Guide and Intervention DATE x-1 2x - 1 x + 11 . Find the partial fraction decomposition of − x2 - 3x - 4 x-4 x+1 A10 · ⎡1 1 ⎤ ⎢ ⎣1 -4⎦ ⎡A⎤ ⎢ ⎣B⎦ X = = ⎡ 1⎤ ⎢ ⎣ 11 ⎦ D -4 x-7 -3 x+2 2 Glencoe Precalculus 005_032_PCCRMC06_893807.indd 22 Chapter 6 2x(x - 1) 2x 1 (x - 1) 22 x (x - 1) x x Glencoe Precalculus x-1 2 3 1 -x-1 2 − + −2 + − − 4. 5x 2 -7x + 13 x - 5x - 14 x +1 1 3. −2 − + −2 -2 x-4 2. − −+− 2 x - x - 12 5x - 34 1. − −+− 2 7 x+3 Find the partial fraction decomposition of each rational expression. Exercises x + 11 3 -2 Solving for X yields A = 3 and B = -2. Therefore, − =− +− . x-4 x+1 x2 - 3x - 4 A+B=1 → A + (-4B) = 11 · C Equate the coefficients on the left and right side of the equation to obtain a system of two equations. To solve the system, write it in matrix form CX = D and solve for X. Group like terms. Distributive Property 1x + 11 = (A + B)x + (A + (-4B)) Multiply each side by the LCD, x2 - 3x - 4. x + 11 = Ax + A + Bx - 4B Form a partial fraction decomposition. x + 11 = A(x + 1) + B(x - 4) A B − =− +− 2 x + 11 x - 3x - 4 Rewrite the equation as partial fractions with constant numerators, A and B, and denominators that are the linear factors of the original denominator. Example Each fraction in the sum is a partial fraction. The sum of these partial fractions make up the partial fraction decomposition of the original rational function. If the denominator of a rational expression contains a repeated linear factor, the partial fraction decomposition must include a partial fraction with its own constant numerator for each power of this factor. 2x - 3x + 1 3x - 1 2 -1 =− +− g(x) = − 2 The function g(x) shown below can be written as the sum of two fractions with denominators that are linear factors of the original denominator. PERIOD 10/23/09 12:20:21 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 6 Linear Factors 6-4 NAME DATE Partial Fractions Study Guide and Intervention (continued) PERIOD 4x4 - 2x3 - 13x2 + 7x + 9 x(x - 3) Find the partial fraction decomposition of −− . 2 2 2 x x -3 (x - 3) x x +5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC06_893807.indd 23 Chapter 6 2x x +1 3-x (x + 1) 3. − +− − 2 2 2 2 2 2x3 + x + 3 (x + 1) x + 5x 5 1. − −-− 3 2 1 x 23 x+2 x +1 x-1 (x - 3) Lesson 6-4 11/17/09 4:38:50 PM Glencoe Precalculus -x (x + 1) 3x - 2 x -3 4. − + − − 2 2 2 2 2 x3 + 2x2 + 2 (x + 1) 3x3 - 2x2 - 8x + 5 (x - 3) 2. −− +− − 2 2 2 2 2 Find the partial fraction decomposition of each rational expression. Exercises x(x - 3) 4x - 2x - 13x + 7x + 9 2x + 1 3x - 2 1 Therefore, −− =− +− +− . 2 2 2 2 2 1 = E 9A = 9 3 = D -3C + E = 7 4 = -2 2 = C 3 1 B → = C = -2 A -6A - 3B + D = -13 A+B=4 Write and solve the system of equations obtained by equating coefficients. 4x4 - 2x3 -13x2 + 7x + 9 = (A + B)x4 + Cx3 + (-6A - 3B + D)x2 + (-3C + E)x + 9A 4x4 - 2x3 -13x2 + 7x + 9 = Ax4 + Bx4 + Cx3- 6Ax2 - 3Bx2 + Dx2 - 3Cx + Ex + 9A 4x4 - 2x3 -13x2 + 7x + 9 = A(x2 - 3)2 + (Bx + C)x(x2 - 3) + (Dx + E)x 4x4 - 2x3 - 13x2 + 7x + 9 Bx + C Dx + E A −− =− +− +− x2 - 3 x(x2 - 3)2 (x2 - 3)2 x This expression is proper. The denominator has one linear factor and one irreducible factor of multiplicity 2. Example Not all rational expressions can be written as the sum of partial fractions using only linear factors in the denominator. If the denominator of a rational expression contains an irreducible quadratic factor, the partial fraction decomposition must include a partial fraction with a linear numerator of the form Bx + C for each power of this factor. Irreducible Quadratic Factors 6-4 NAME Answers (Lesson 6-4) Chapter 6 Partial Fractions Practice DATE PERIOD -2 x-3 x+1 2x - 9x + 9 1 x-3 x x+2 -1 2x - 3 x-1 x 4. − −+− 2 x + x - 2x 2 -2 7 - 10x - 2 1 − 2. 6x −+−+− 3 2 x+2 A11 x+1 -5x2 - 11x + 54 x + 2x - 8 x+4 2x - 3 (2x - 3) 5 1 -2 - − +− 2 -8x2 + 22x - 10 8. − (2x - 3)2 x-2 -3 2 -5 + − +− 6. − 2 2 4x4 + 8x3 + 6x2 + 6x + 5 (3x + 2)(x + 1) x+2 -x 1 − +− +− 3x + 2 x2 + 1 (x2 + 1)2 1 − 50 - x x 50x 50(50 - x) 24 Glencoe Precalculus 3/23/09 3:47:07 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Precalculus 005_032_PCCRMC06_893807.indd 24 Chapter 6 50 50 1 1 − or − +− +− 1 − partial decomposition of f(x). decompose these functions into the sum of its partial fractions. Find the necessary to work with functions of the form f(x) = − and to 1 x(50 - x) 13. GROWTH When working with exponential growth in calculus, it is often –23 5 13 − +− +− 9x 9(x + 3) 3(x + 3)2 12. −− 2 2 2x2 + 5 x + 6x + 9x 11. − 3 2 3 x-5 2x - 1 − +− +− x-3 x2 - 2 (x2 - 2)2 5x - 7x - 12x + 6x + 21 (x - 3)(x - 2) 3 10. −− 2 2 4 -4 2 -1 − +− +− x-5 x (x - 5)2 9. −− 3 2 -2x + 29x - 100 x - 10x + 25x 2 Find the partial fraction decomposition of each rational expression with repeated factors. x 9 2 6+− +− 6x2 + 17x + 2 7. − x2 + x x 1 -2 3+− +− 5. − 2 3x2 + 5x + 2 x + 2x Find the partial fraction decomposition of each improper rational expression. x + 3x + 2 6 9x + 15 3. − +− − 2 3 x+2 3x - 7 1. − −+− 2 5 x - 7x + 12 x - 4 Find the partial fraction decomposition of each rational expression. 6-4 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Partial Fractions 50(25 + x) 50 + x x + 50 350 - x y 2 6 60 x(x + 3)2 4 f(x) = 8 x 10 –20 − Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC06_893807.indd 25 Chapter 6 x+3 x (x + 3) 20 –20 –20 − +− +− 3x 3(x + 3) (x + 3)2 25 PERIOD x+4 x-4 10 -10 − +− decomposition of f(x). x - 16 Lesson 6-4 10/23/09 12:32:31 PM Glencoe Precalculus -80 f(x) = − . Find the partial fraction 2 5. KAYAKING The total time it takes for a kayaker to travel 10 miles upstream and 10 miles downstream with a paddling rate of 4 miles per hour in still water is given by the function x - 11 -3 4 − − x + c. Find the partial fraction decomposition of the rational expression. ⎡ 1 1⎤ ⎡ A⎤ ⎡ 1⎤ ⎢ ·⎢ =⎢ ⎣-11 0⎦ ⎣B⎦ ⎣ -44⎦ b. Write the matrix form AX = B for the system of equations found in part a. A+B=1 -11A + 0B = -44 a. Write the system of equations obtained by equating the coefficients. [0, 10] scI: 1 by [-2, 10] scI: 1 x = 2, and x = 8. To find the area, first find the partial fraction decomposition of f(x). x - 11x x - 44 by the graphs of f(x) = − , y = 0, 2 Answers –20 3 3 − +− +− or 2 20 − Suppose you were to revolve the graph of f(x) around the x-axis, creating a three-dimensional object. Using calculus, you could find the volume of the object. But first, you would need to find the partial fraction decomposition of f(x). Find the partial decomposition of f(x). 0 2 4 6 8 10 0 ≤ x ≤ 10 for the graph of f(x) shown below. 351(x + 1) 351(350 - x) 3. VOLUME Consider the domain 1 1 − + − x+1 − + − or plays a role when studying the spread of an infection in certain populations. Find the partial fraction decomposition of g(x) when a = 350. 1 1 − − 351 351 (x + 1)(a - x) 1 , where a > 0 often g(x) = − 2. INFECTIONS A function of the form 1250 50 - − how much acid she must mix with a 25% acid solution to achieve the desired percentage. Find the partial fraction decomposition of f(x). function f(x) = − to determine DATE 4. AREA Calculus can be used to find the area of the shaded region shown below. The shaded region is bounded Word Problem Practice 1. CHEMISTRY A chemist uses the 6-4 NAME Answers (Lesson 6-4) Enrichment PERIOD x+4 x-7 A12 x+4 x-7 x-1 x+2 12 x-2 Glencoe Precalculus 005_032_PCCRMC06_893807.indd 26 Chapter 6 26 x + 3x - 54 Glencoe Precalculus -x - 114 6. − −+− 2 7 x+9 -8 x-6 x 1 x+3 5 x 5. − −+−+− 3 2 18x + 39x - 30 x - x - 6x 2 x + x - 2x -9 3 -6x + 18 +− 4. − − x2 - 10x + 24 x - 4 x-6 5 x-6 7 6 13x - 51 +− 3. − − x2 - 8x + 15 x - 3 x-5 -4 x+3 2 -3 4 - 8x - 4 2 − 2. 3x −+−+− 3 2 x + 39 x - 3x - 18 1. − −+− 2 Use the Heaviside Method to write the partial fraction decomposition of each rational expression. Exercises x - 3x - 28 2x - 47 5 -3 − +− = − 2 Substitute A = 5 and B = -3 into the original equation to find the partial fraction decomposition. -3 = B -33 = 11B 2(7) - 47 = A(7 - 7) + B(7 + 4) To solve for B, let x = 7. This eliminates A. 5 = A -55 = -11A 2(-4) - 47 = A(-4 - 7) + B(-4 + 4) To solve for A, let x = -4. This eliminates B. 2x - 47 = A(x - 7) + B(x + 4) Multiply both sides of the equation by the least common denominator (x + 4)(x - 7). ⎡ A 2x - 47 B ⎤ = (x + 4)(x - 7) ⎢− (x + 4)(x - 7) − +− x2 -3x - 28 ⎣x + 4 x - 7⎦ x - 3x - 28 2x - 47 A B − =− +− 2 Another method that can be used to find the partial fraction decomposition of a rational expression is called the Heaviside Method. Consider the equation shown below. DATE 3/23/09 3:47:17 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 6 Heaviside Method 6-4 NAME DATE PERIOD Write an objective function and a list of constraints to model the situation. Graph the region corresponding to the solution of the system of constraints. Find the coordinates of the vertices of the region formed. Evaluate the objective function at each vertex to find the minimum or maximum. f(3, 0) = 18(3) + 12(0) (3, 0) 63 54 ← Maximum 0 (3, 0) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC06_893807.indd 27 Chapter 6 1. f(x, y) = 3x - 2y 2x + y ≤ 10 max at (5, 0) = 15, x + 2y ≤ 8 min at (0, 4) = -8 x≥0 y≥0 27 x Lesson 6-5 3/23/09 3:47:22 PM Glencoe Precalculus 2. f(x, y) = x + 2y x+y≤4 max at (3, 1) = 5, x + 3y ≤ 6 min at (0, 0) = 0 x≥0 y≥0 Find the maximum and minimum values of the objective function f(x, y) and for what values of x and y they occur, subject to the given constraints. Exercises Since f is greatest at (1.5, 3), the company will maximize profit if it makes and sells 1.5 belts for every 3 wallets. f(1.5, 3) = 18(1.5) + 12(3) (1.5, 3) 0 f(0, 4) = 18(0) + 12(4) (0, 4) 48 Result f(x, y) = 18x + 12y f(0, 0) = 18(0) + 12(0) Point (0, 0) Example A leather company wants to add belts and wallets to its product line. Belts require 2 hours of cutting time and 6 hours of sewing time. Wallets require 3 hours of cutting time and 3 hours of sewing time. The cutting machine is available 12 hours a week and the sewing machine is available 18 hours per week. Belts will net $18 in profit and wallets will net $12. How much of each product should be produced to achieve maximum profit? Let x represent the number of belts and y represent the number of wallets. The objective function is then given by f(x, y) = 18x + 12y. Write the constraints. x ≥ 0; y ≥ 0 Numbers of items cannot be negative. 2x + 3y ≤ 12 Cutting time 6x + 3y ≤ 18 Sewing time Graph the system. The solution is the shaded region, including its y boundary segments. Find the coordinates of the four vertices by solving the system of boundary equations for each point of intersection. The coordinates are (0, 0), (0, 4), (1.5, 3), and (3, 0). (1.5, 3) (0, 4) Evaluate the objective function for each ordered pair. Step 1 Step 2 Step 3 Step 4 Linear programming is a process for finding a minimum or maximum value for a specific quantity. The following steps can be used to solve a linear programming problem. Linear Optimization Study Guide and Intervention Linear Programming Applications 6-5 NAME Answers (Lesson 6-4 and Lesson 6-5) Chapter 6 A13 Linear Optimization Study Guide and Intervention DATE (continued) PERIOD 0 y (3, 2) (2, 4) x x≤8 y≥0 for 3 ≤ x ≤ 5; min. at (0, 0) = 0 x≤5 x≥0 28 x≥0 point on y = -2x + 11 y≥0 2 Glencoe Precalculus min. at (0, 0) = 0 for 4 ≤ x ≤ 8; 1 point on y = - − x + 10 max. of 60 at every 11/17/09 4:39:09 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Precalculus 005_032_PCCRMC06_893807.indd 28 Chapter 6 x + 2y ≤ 20 max. of 11 at every x - y ≥ -4 2. f(x, y) = 3x + 6y y≤5 2x + y ≤ 11 1. f(x, y) = 2x + y Find the maximum and minimum values of the objective function f(x, y) and for what values of x and y they occur, subject to the given constraints. Exercises Because f(x, y) = 24 at (2, 4) and (3, 2), the problem has multiple optimal solutions. An equation of the line through these two vertices is y = -2x + 8. Therefore, f has a maximum value of 24 at every point on y = -2x + 8 for 2 ≤ x ≤ 3. f(3, 0) = 6(3) + 3(0) or 18 f(3, 2) = 6(3) + 3(2) or 24 f(2, 4) = 6(2) + 3(4) or 24 f(0, 4) = 6(0) + 3(4) or 12 f(0, 0) = 6(0) + 3(0) or 0 Graph the region bounded by the given constraints. Find the value of the objective function f(x, y) = 6x + 3y at each vertex. Example Find the maximum value of the objective function f(x, y) = 6x + 3y and for what values of x and y it occurs, subject to the following constraints. 2x + y ≤ 8 y≤4 x≤3 x≥0 y≥0 No or Multiple Optimal Solutions Linear programming models can have one, multiple, or no optimal solutions. If the graph of the objective function f to be optimized is coincident with one side of the region of feasible solutions, f has multiple optimal solutions. If the region does not form a polygon, but instead is unbounded, f may have no minimum value or maximum value. 6-5 NAME Linear Optimization Practice DATE PERIOD 2x + 3y ≤ 18 max. at (0, 6) = 30, min. at (0, 0) = 0 max. of 30 at every point on 3 y = -− x + 12 for 2 ≤ x ≤4, 2 min. at (0, 0) = 0 max. at (7, 3.5) = 3.5, min. at (4, 5) = -7 $40 Profit per Skate $30 4 minutes 4 minutes Ice Skates 120 minutes 200 minutes Maximum Time Available Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC06_893807.indd 29 Chapter 6 Answers 29 Sample answer: If customers cannot get ice skates, they might go somewhere else. They should combine the math model with customer needs. d. Describe why the company would choose a number of roller skates and ice skates different from the answer in part c. c. How many roller skates and ice skates should be manufactured to maximize profit? What is the maximum profit? 40 roller skates and no ice skates; $1600 b. Sketch a graph of the region determined by the constraints from part a to find the set of feasible solutions for the objective function. 0 10 20 30 40 50 60 y x Lesson 6-5 10/23/09 12:39:08 PM Glencoe Precalculus Roller Skates (40,0) (20,25) 10 20 30 40 50 60 (0,30) f(x, y) = 40x + 30y; x ≥ 0; y ≥ 0; 5x + 4y ≤ 200; x + 4y ≤ 120 a. Write an objective function and list the constraints that model the given situation. 1 minute 5 minutes Assembling Checking and Packaging Roller Skates Manufacturer Information 5. SKATES A manufacturer produces roller skates and ice skates. x + y ≤ 10 3x + 2y ≤ 24 y≥0 y≤5 y≥0 y≤8 x≤7 x + 2y ≥ 14 x≥0 x≥0 4. f(x, y) = 3x + 3y x+y≤8 max. at (8, 0) = 32, min. at (0, 2) = 6 x+y≤7 3. f(x, y) = 2x - 3y y≥0 2x + 3y ≥ 6 y≥0 x≥0 2. f(x, y) = 4x + 3y x≥0 1. f(x, y) = 2x + 5y Find the maximum and minimum values of the objective function f(x, y) and for what values of x and y they occur, subject to the given constraints. 6-5 NAME Ice Skates Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Answers (Lesson 6-5) A14 Linear Optimization 300 340 Profit ($) 300 150 Soybeans b 50 150 Acres of Corn 100 (40, 320) (0, 360) 200 250 (200, 0) c Glencoe Precalculus 005_032_PCCRMC06_893807.indd 30 Chapter 6 plant 40 acres of corn and 320 acres of soybeans; $109,600 c. How can Mr. Fields maximize his profit? What is his maximum profit? 0 50 100 150 200 250 300 350 450 400 b. Sketch a graph of the region determined by the constraints from part a to find the feasible solutions for the objective function. f(c, b) = 340c + 300b c + b ≤ 360 300c + 150b ≤ 60,000 c ≥ 0, b ≥ 0 a. Write an objective function and list the constraints that model this situation. Corn Per Acre 30 PERIOD 4 (5, 3) 2⎞ ⎝0, 9 3⎠ 12 x (14, 0) of X; about $77.33 3 Glencoe Precalculus 2 9− pounds of Y and 0 pounds c. How many pounds of each type of food should be purchased to satisfy the requirements at the minimum cost? What is the minimum cost? 0 4 8 12 ⎛ y b. Sketch a graph of the region determined by the constraints from part a to find the feasible solutions for the objective function. f(x, y) = 12x + 8y 10x + 30y ≥ 140 20x + 15y ≥ 145 x ≥ 0, y ≥ 0 a. Write an objective function and list the constraints that model this situation. 2. NUTRITION A certain diet recommends at least 140 milligrams of Vitamin A and at least 145 milligrams of Vitamin B daily. These requirements can be obtained from two types of food. Type X contains 10 milligrams of Vitamin A and 20 milligrams of Vitamin B per pound. Type Y contains 30 milligrams of Vitamin A and 15 milligrams of Vitamin B per pound. Type X costs $12 per pound. Type Y costs $8 per pound. Word Problem Practice Labor ($) Acres of Soybeans DATE 3/23/09 3:47:35 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 6 1. FARMING Mr. Fields owns a 360-acre farm on which he plants corn and soybeans. The table shows the cost of labor and the profit per acre for each crop. Mr. Fields can spend up to $60,000 for spring planting. 6-5 NAME DATE 8 PERIOD 0 1 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_032_PCCRMC06_893807.indd 31 Chapter 6 Sample answer: ax + by = f(P) 31 2. If f(Q) = f(P), find an equation of the line containing P and Q. 2 8 x x Lesson 6-5 3/23/09 3:47:40 PM Glencoe Precalculus −− It produces alternate optimal solutions all along PQ; f(W) = f(P). 1. If f(Q) = f(P), what is true of f ? of f(W)? 2 8 3 −− Let P and Q be vertices of a closed convex polygon, and let W lie on PQ. Let f(x, y) = ax + by. Exercises Example If f(x, y) = 3x + 2y, find the maximum value of the function over the shaded region at the right. y The maximum value occurs at the vertex (6, 3). The minimum value occurs at (0, 0). The values of f(x, y) at W 1 and W 2 are between the maximum and minimum values. f(Q) = f(6, 3) = 3(6) + 2(3) or 24 f(W 1) = f(2, 1) = 3(2) + 2(1) or 8 1 f(W 2) = f(5, 2.5) = 3(5) + 2(2.5) or 20 0 f(P) = f(0, 0) = 3(0) + 2(0) or 0 Then 0 < w < 1 and the coordinates of W are ((1 - w)x 1 + wx 2, (1 - w)y 1 + wy2). Now consider the function f(x, y) = 3x - 5y. f(W) = 3[(1 - w)x 1 + wx 2] - 5[(1 - w)y 1 + wy 2] = (1 - w)(3x 1) + 3wx 2 + (1 - w)(-5y1) - 5wy 2 = (1 - w)(3x1 - 5y1) + w(3x 2 - 5y2) = (1 - w)f(P) + wf(Q) This means that f(W) is between f(P) and f(Q), or that the greatest and least values of f(x, y) must occur at P or Q. You have already learned that over a closed convex polygonal region, the maximum and minimum values of any linear function occur at the vertices of the polygon. To see why the values of the function at any point on the boundary of the region must be between the values at the vertices, consider the convex polygon with vertices P and Q. −−− Let W be a point on PQ. y PW If W lies between P and Q, let − = w. PQ Enrichment Convex Polygons 6-5 NAME Answers (Lesson 6-5) Chapter 6 Assessment Answer Key (Lessons 6-1 and 6-2) Page 33 ⎡ (-5, 1) 3. (-1, 3, 2) 4. x 1. 2 -5 7 -1 -1 2 1 ⎣ -3 4 -1 10 ⎦ 4. x-1 x+2 x-3 -3 x-4 2 x 5 (x - 4) − + − + −2 x 6. B 2. H 3. D 4. F x +4 A B ⎡5 7⎤ ⎢ ⎣2 3⎦ 5. 1. 4 2x - 1 − +− 2 5. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Page 35 6 -4 − +− 2. 3. Mid-Chapter Test 3 -2 − +− -9 ⎤ ⎢ 2. (Lesson 6-4) Page 34 ⎡ 3 -2 1⎤ ⎢ ⎣ -5 1 -11 ⎦ 1. Quiz 3 Answers Quiz 1 Quiz 4 (Lesson 6-5) Quiz 2 (Lesson 6-3) Page 34 Page 33 1. (-4, 2) 1. 2. A (-2, -2, -2) 2. 3. 4. (-16, 7) max. at (3, 4) of 25; min. at (0, 0) of 0 max. at (0, 0) of 0; min. at (12, 0) 3. of -24 (1, 0, -3) 4a + 6p = 9.80 5a. 3a + 9p = 10.95; 5b. $1.25, $0.80 0 6. ⎡ 5. Chapter 6 C 40 small and 4a. 6 large $2670 4b. A15 3⎤ -3 ⎣ 3⎦ ⎢ 7. Glencoe Precalculus Chapter 6 Assessment Answer Key Vocabulary Test Page 36 Form 1 Page 37 1. 2. 1. Page 38 C 8. G 9. C 10. H 11. C 12. J 13. A 14. G 15. B J square system 2. Gaussian elimination 3. D 3. inverse matrix 4. optimization 6. Cramer’s Rule partial fraction 7. decomposition 4. F 5. B 8. Sample answer: the multiplicative identity for a square matrix 6. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 5. invertible F 9. Sample answer: the function in a linear programming problem that is to be optimized Chapter 6 7. ⎡ 2 -3⎤ ⎢ ⎦ ⎣ 8 -5 B: B A16 Glencoe Precalculus Chapter 6 Assessment Answer Key Page 40 8. 1. 2. G 1. 9. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 12. 4. 6. 7. C 14. 5. H D Chapter 6 6. 15. B B: ⎡1 0 ⎤ ⎢ ⎣0 1 ⎦ 7. A17 9. C 10. F 11. D 12. F 13. C 14. G 15. B H C G J D J J 13. 5. 3. A 8. F J B Page 42 C B 2. 11. 4. F A 10. 3. Form 2B Page 41 B G A ⎡1 0⎤ ⎢ 0 1⎦ ⎣ B: Glencoe Precalculus Answers Form 2A Page 39 Chapter 6 Assessment Answer Key Form 2C Page 43 ⎡ -4 ⎣ 3 1. Page 44 1 12 ⎤ -2 -14 ⎦ 11. $3.25 12. ±3 2. Sample answer: ⎡1 ⎣1 -3 5 ⎤ 0 -2 ⎦ (-1, 2, 0) 3. -2 7 − +− x-5 13. x+3 3 5 -2 − +− +− 2 14. x + 1 x -72 6. ⎡ 4 -48 ⎤ ⎣ -5 53 ⎦ 7. does not exist 8. ⎡ -1 6 ⎤ ⎣ 3 -4 ⎦ 9. 10. Chapter 6 1 − 10 x +3 1 − 10 −+ − or 100 - x 1 1 − + − 10x 10(100 - x) 16. x 17. 16 18. 12 19. 0, (0, 0) 20. $10.80 B: BC Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 5. (x - 4) 6 2x - 4 − +− 2 15. hooks: $1.10 bobbers: $1.25 4. sinkers: $0.90 x-4 (1, 6, -1) (2, 3) A18 Glencoe Precalculus Chapter 6 Assessment Answer Key Form 2D Page 45 2. ⎡ 5 -2 -3 ⎤ ⎣ 4 7 -11 ⎦ Sample answer: ⎡ 1 -3 5 ⎤ ⎣ 0 1 -2 ⎦ 3. (2, -3, 1) 11. $1.75 12. ±3 Answers 1. Page 46 -2 4 +− − x-3 x 13. 3 -5 2 − +− +− 2 x+1 x+3 -4 − +− x-2 2x2 + 1 14. x 15. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. sports: $3.00 news: $4.50 4. educational: $5.00 5. 3 6. ⎡ 26 -4 ⎤ ⎣ 49 -6 ⎦ 7. ⎡1 1⎤ ⎣ 5.5 6 ⎦ 8. ⎡ -3 4 ⎤ ⎣ -1 5 ⎦ 9. (4, 1, -7) 10. (2, -3) Chapter 6 x 1 − 11 16. 1 − 11 −+− or 121 - x x 1 1 −+ − 11x 11(121 - x) 17. 12 18. 24 19. -15, (15, 0) 20. $420 B: C A19 Glencoe Precalculus Chapter 6 Assessment Answer Key Form 3 Page 47 1. Page 48 ⎡ 3 -2 5 ⎤ ⎣ 4 -1 -7 ⎦ 11. $1.10 12. 2, 3 2. Sample answer: ⎡1 ⎣0 0 -2 ⎤ 0 0⎦ x+4 x-6 2x - 5 13. 3 -5 6 −+−+− x+2 (x + 2)2 14. x -1 4 +− 4+− x + 1 x 15. 7 +− − 2 (-3, 4, -5) 3. 1 − 1 − 13 13 − or +− 5. -42 6. 16. x 169 - x 1 1 − + − 13x 13(169 - x) 17. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 4. tandem: $34 child: $19 adult: $26 30 x = -2, y = 4 ⎡ 0.2 0.3 0.1 ⎤ -0.2 -0.8 0.4 7. ⎣ 0.2 -0.7 0.1 ⎦ 8. ⎡ -3 1 ⎤ ⎣ 0 -2 ⎦ 9. (-1, 1, -1, 1) 10. (4, -3) Chapter 6 14 min of 0 at (0, 0); max of 68 for every point on 4x + y = 34 where 6≤x≤8 19. 18. 20. B: A20 $2500 ce - bf − ae - bd Glencoe Precalculus Chapter 6 Assessment Answer Key ⎡0.40 0.25 0.15 0.35⎤ 1a. 0.25 0.35 0.40 0.35 ⎣0.35 0.40 0.45 0.30⎦ ⎢ ⎡ ⎢2000 ⎢3500 1b. ⎢3000 ⎢ ⎣2500 2500 4000 4200 2000 1500 2000 1800 1000 3a. The objective function for this situation is f(E, S) = 30,000E + 25,000S. The vertices of the feasibility region are (0, 0), (6, 0), (6, 4), and (0, 10). ⎤ ⎦ f(0, 0) = $0 f(6, 0) = $180,000 f(6, 4) = $280,000 f(0, 10) = $250,000 The point (6, 4) maximizes the builder’s profit at $280,000, so the builder should construct 6 Executive houses and 4 Suburban houses. ⎡0.40 0.25 0.15 0.35⎤ 1c. 0.25 0.35 0.40 0.35 × ⎣0.35 0.40 0.45 0.30⎦ ⎢ ⎡ ⎢2000 ⎢3500 ⎢3000 ⎢ ⎣2500 2500 4000 4200 2000 ⎤ 1500 2000 = 1800 1000⎦ ⎡3000 3330 1720⎤ 3800 4405 2145 ⎣4200 4965 2435⎦ Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. ⎢ 3b. Sample answer: He should max out the number of Executive houses built, which would be 6 houses, and then build Suburban houses to complete his construction for the year. This would maximize his profit. 4405 people visited Big Mountain Park in the afternoon while 4200 people visited Lazy River Park in the morning. More people visited Big Mountain Park. ⎡ 3 -1 0⎤ ⎡x⎤ ⎡-5⎤ 2a. -1 0 2 · y = 3 ⎣ 0 1 -1⎦ ⎣z ⎦ ⎣ 1⎦ ⎢ ⎢ ⎢ ⎡0.4 0.2 0.4⎤ 2b. 0.2 0.6 1.2 ⎣0.2 0.6 0.2⎦ ⎢ ⎡-1⎤ ⎡0.4 0.2 0.4⎤ ⎡-5⎤ 2c. 0.2 0.6 1.2 · 3 = 2 ⎣ 1⎦ ⎣0.2 0.6 0.2⎦ ⎣ 1⎦ ⎢ Chapter 6 ⎢ ⎢ A21 Glencoe Precalculus Answers Page 49, Extended-Response Test Sample Answers Chapter 6 Assessment Answer Key Standardized Test Practice Page 50 1. A B C F G H J 3. A B C D F G H F G H J 11. A B C D 12. F G H J 13. A B C D 14. F G H J 15. A B C D 16. F G H J 17. A B C D J A B C D 6. F G H J 7. A B C D 8. F G H J 9. A B C D A22 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 5. Chapter 6 10. D 2. 4. Page 51 Glencoe Precalculus Chapter 6 Assessment Answer Key Standardized Test Practice (continued) Answers Page 52 -3π 3π x=− ,x=− 2 18. 2 2 -4 − +− 3x - 2 x 19. 3 1 ±− , ±1, ± − , ±3 20. 21. 2 2 (x - 2)3 x ln − 4 π π 3π 5π − , −, −, − Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 22. 3 2 2 3 1 π , −, 2 3, 2π, − 23. 2π 2 24a. 0, 0 24b. x=1 24c. y=3 between -3 and -2 25a. rel. max. at x ≈ -2 and x ≈ 0; rel. min. at x ≈ -1 and x ≈ 1 25b. Chapter 6 A23 Glencoe Precalculus