Transcript Slide 1
Homework, Page 604 Use substitution to solve the system of equations. 21 25 7 1. x 3 y z 0 x 2 0 x x 3 2 0 2 2 2 7 y 2 y 3 z 1 2 y 3 2 1 2 y 6 1 2 y 7 2 z 2 25 7 , , 2 2 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 1 Homework, Page 604 Use Gaussian elimination to solve the system of equations. x y z 3 5. 4 x y 5 3 x 2 y z 4 x y z 3 4 y 4 z 7 5 y 4 z 5 8 15 12 5 15 x x x y z 3 x 2 3 4 4 4 4 4 15 z 4 y 4 z 7 4 2 4 z 7 4z 15 4 y2 15 5 ,2, 4 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 2 Homework, Page 604 Perform the indicated elementary row operations on the matrix. 2 6 4 0 2 3 3 1 2 9. 3 2 R1 R3 2 6 4 0 2 3 3 1 2 3 2 R1 R3 6 4 2 6 4 2 0 2 3 0 2 3 3 3 9 1 6 2 0 8 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 3 Homework, Page 604 What elementary row operations applied to the matrix will yield the given matrix. 2 1 1 2 1 2 3 0 3 1 1 2 13. 1 2 3 0 2 1 1 2 3 1 1 2 R12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 4 Homework, Page 604 Find a row echelon form for the matrix. 17. 1 3 1 2 1 4 3 0 1 1 2 R1 R2 0 3 1 3 3R3 R2 0 20 0 9 1 9 R2 R3 0 0 3 1 1 3 1 7 6 3R1 R3 0 7 6 0 1 0 9 2 1 1 3 1 1 0 R2 0 1 0 20 2 0 9 2 3 1 1 3 1 1 1 0 R3 0 1 0 2 0 2 0 0 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 5 Homework, Page 604 Find the reduced row echelon form for the matrix. 21. 1 0 2 1 3 2 4 7 2 1 3 4 1 0 2 1 0 1 1 2 0 0 0 0 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 6 Homework, Page 604 Write the augmented matrix corresponding to the system of equations. 25. 2 x 3 y z 1 x y 4 z 3 3x z 2 2 3 1 1 1 1 4 3 3 0 1 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 7 Homework, Page 604 Write the system of equations corresponding to the augmented matrix. 29. 3 2 1 4 5 2 3x 2 y 1 4 x 5 y 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 8 Homework, Page 604 Solve the system of equations by finding a row echelon form for the augmented matrix. 33. x 2 y z 8 2 x y 3 z 9 3 x y 3z 5 1 2 1 8 2 1 3 9 3 1 3 5 1 0 0 2 0 1 0 1 2, 1, 4 0 0 1 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 9 Homework, Page 604 Solve the system of equations by finding the reduced row echelon form for the augmented matrix. x y 3z 2 37. 3 x 4 y 10 z 5 x 2 y 4z 3 1 1 3 2 3 4 10 5 1 2 4 3 1 0 2 0 0 1 1 0 No solution 0 0 0 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 10 Homework, Page 604 Solve the system of equations by finding the reduced row echelon form for the augmented matrix. x 2y 4 41. 3x 4 y 5 2x 3y 4 1 2 4 3 4 5 2 3 4 1 0 0 0 1 0 No solution 0 0 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 11 Homework, Page 604 Write the system of equations as a matrix equation AX = B, with A as the coefficient matrix of the system. 45. 2 x 5 y 3 x 2y 1 2 5 x 3 1 2 y 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 12 Homework, Page 604 Solve the system of equations by using an inverse matrix. 49. 2 x 3 y 13 4 x y 5 3 1 14 14 2 3 x 13 1 A AX B 2 1 4 1 y 5 7 7 3 1 x 14 14 13 2 1 X A B 2,3 2 1 y 5 3 7 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 13 Homework, Page 604 Solve the system of equations by using an inverse matrix. 2 x y z w 3 53. x 2 y 3z w 12 3x y z 2w 3 2 x 3 y z 3w 3 1, 2, 2,3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 14 Homework, Page 604 53. 2 1 1 1 2 3 AX B 3 1 1 2 3 1 5 12 x 13 y X A1 B 12 z 7 w 6 7 4 1 5 5 1 12 3 12 1 x 3 13 2 1 1 1 y 12 3 12 A1 12 7 2 z 3 1 1 1 3 w 3 3 6 6 7 3 1 2 4 4 1 5 1 3 12 2 1 3 1 1 3 12 12 2 1, 2, 2,3 3 1 1 1 2 3 6 3 3 3 1 2 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 15 Homework, Page 604 Use the method of your choice to solve the system of equations. 57. x 2 y 2 z w 5 2x y 2z 5 3 x 3 y 3 z 2 w 12 x z w 1 1 2 3 1 5 1 5 0 3 3 2 12 0 1 1 1 1 0 2 2 1 1 2 0 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 3 3,3, 2,0 0 1 0 2 0 0 1 0 0 0 0 1 0 0 Slide 7- 16 Homework, Page 604 Use the method of your choice to solve the system of equations. 61. 2 x y z 4 w 1 x 2y z w 1 x y z 2w 0 2 1 1 0 1 1 4 1 1 2 1 1 1 0 1 1 2 0 0 0 0 0 0 0 0 0 2 1 1 0 1 1 1 2 w,1 w, w, w 0 1 1 0 0 0 0 0 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 17 Homework, Page 604 Use the method of your choice to solve the system of equations. x y z 2w 0 65. y z 2w 1 x y 3w 3 2 x 2 y z 5w 4 1 0 1 2 1 1 2 0 1 1 1 2 1 0 1 0 3 3 0 2 1 5 4 0 0 0 0 0 1 0 3 0 No solution 0 1 1 0 0 0 0 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 18 Homework, Page 604 Determine f so that its graph contains the given points. 69. Family of Curves f x ax 2 bx c 1, 4 , 1, 2 f x ax 2 bx c 1, 4 , 1, 2 4 a 1 b 1 c a b c 4 2 2 a 1 b 1 c 2 a b c 2 1 1 1 4 1 0 1 3 1 1 1 2 0 1 0 1 0 0 0 0 0 0 0 0 f x c 3 x 2 x c for any c Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 19 Homework, Page 604 73. Children ride a train for 25 cents, adults pay $1.00, and seniors pay 75 cents. On a given day, 1400 passengers paid $740 for their rides. There were 250 more children than all other passengers. Find the number of children, adults, and senior riders. a adults c children s seniors a c s 1400 a 0.25c 0.75s 740 a s c 250 a c s 250 1 0.25 0.75 740 1 0 0 410 1 1 1 1400 0 1 0 825 1 250 0 0 1 165 1 1 410 adults; 825 children; and 165 seniors Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 20 Homework, Page 604 77. Morgan has $50,000 to invest and wants to receive $5,000 interest the first year. He puts part in CDs earning 5.75% APY, part in bonds earning 8.7% APY, and the rest in a growth fund earning 14.6% APY. How much should he put in each fund if he puts the least amount possible in the growth fund. c CDs b bonds g growth fund b g 50000 0.087b 0.146 g 5000 CDs $0.00 Bonds $38,983.05 Growth fund $11,016.95 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 21 Homework, Page 604 Use inverse matrices to find the equilibrium point for the supply and demand curves. p 100 5 x Demand curve 81. p 20 10 x Supply curve p 100 5 x p 5 x 100 p 20 10 x p 10 x 20 AX B 1 2 3 1 5 p 100 A1 3 1 10 x 20 1 1 15 15 1 2 1 p 3 3 100 73 3 x 1 1 20 5 1 15 15 3 5 1 ,73 1 3 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 22 7.4 Partial Fractions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley What you’ll learn about Partial Fraction Decomposition Denominators with Linear Factors Denominators with Irreducible Quadratic Factors Applications … and why Partial fraction decompositions are used in calculus in integration and can be used to guide the sketch of the graph of a rational function. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 24 Partial Fraction Decomposition of f(x)/d(x) 1. Degree of f degree of d : Use the division algorithm to divide f by d to obtain the f ( x) r ( x) quotient q and remainder r and write q( x) . d ( x) d ( x) 2. Factor d ( x) into a product of factors of the form (mx n)u or (ax 2 bx c) v , where ax 2 bx c is irreducible. 3. For each factor (mx n)u : The partial fraction decomposition of r ( x) / d ( x) must Au A1 A2 include the sum ... , where A1 , A2 ,..., Au are real numbers. 2 u mx n mx n mx n 4. For each factor (ax 2 bx c)v : The partial fraction decomposition of r ( x) / d ( x) must Bv x Cv B1 x C1 B2 x C2 include the sum 2 ... , where B1 , B2 ,..., Bv v 2 ax bx c ax 2 bx c 2 ax bx c and C1 , C2 ,..., Cv are real numbers. The partial fraction decomposition of the original rational function is the sum of q( x) and the fractions in parts 3 and 4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 25 Example Decomposing a Fraction with Distinct Linear Factors 3x 3 Find the partial fraction decomposition of . x 1 x 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 26 Example Decomposing a Fraction with Repeated Linear Factors Find the partial fraction decomposition of Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3x 3 x 1 x 2 2 . Slide 7- 27 Example Decomposing a Fraction with an Irreducible Quadratic Factor Find the partial fraction decomposition of x 2 3x 1 x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 2 x 1 . Slide 7- 28 Example Reversing a Decomposed Fraction to Identify the Parent Function Find the function that yields the partial fraction decomposition: 2 1 x 3 . x 1 x 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 29 Homework Homework Assignment #12 Read Section 7.5 Page 614, Exercises: 1 – 49 (EOO), skip 45 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 30 7.5 Systems of Inequalities in Two Variables Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quick Review Find the x- and y -intercepts of the line. 1. 3 x 4 y 24 x y 2. 1 20 30 Find the point of intersection of the two lines. 3. x y 3 and 2 x y 5 4. x y 1 and y 3 x 1 5. 7 x 3 y 10 and x y 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 32 Quick Review Solutions Find the x- and y -intercepts of the line. 1. 3 x 4 y 24 (0,6) and (8,0) x y 2. 1 (0,30) and (20,0) 20 30 Find the point of intersection of the two lines. 3. x y 3 and 2 x y 5 (8/3,1/3) 4. x y 1 and y 3 x 1 (0,1) 5. 7 x 3 y 10 and x y 1 (1.3,0.3) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 33 What you’ll learn about Graph of an Inequality Systems of Inequalities Linear Programming … and why Linear programming is used in business and industry to maximize profits, minimize costs, and to help management make decisions. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 34 Steps for Drawing the Graph of an Inequality in Two Variables 1. Draw the graph of the equation obtained by replacing the inequality sign by an equal sign. Use a dashed line if the inequality is < or>. Use a solid line if the inequality is ≤ or ≥. 2. Check a point in each of the two regions of the plane determined by the graph of the equation. If the point satisfies the inequality, then shade the region containing the point. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 35 Example Graphing a Linear Inequality Draw the graph of y 2 x 4. State the boundary of the region. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 36 Example Solving a System of Inequalities Graphically Solve the system 2 x 3 y 4 and y x2 . Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 37 Example Solving a Word Problem 38. Paul’s diet is to contain at least 24 units of carbohydrates and 16 units of protein. Food substance A costs $1.40 per unit and each unit contains 3 units of carbohydrates and 4 units of protein. Food substance B costs $0.90 per unit and each unit contains 2 units of carbohydrates and 1 units of protein. How many units of each food substance should be purchased to minimize cost? What is the minimum cost? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 38