1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 1 Homework, Page 575 Determine whether the ordered pair is a solution to the system. 1.

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 2 Homework, Page 575 Solve the system by substitution. 5.

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 3 Homework, Page 575 Solve the system by substitution. 9.

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 4 Homework, Page 575 Solve the system algebraically. Support your answer graphically. 13.

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 5 Homework, Page 575 Solve the system algebraically. Support your answer graphically. 17.

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 6 Homework, Page 575 Solve the system by elimination. 21.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 7 Homework, Page 575 Solve the system by elimination. 25.

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 8 Homework, Page 575 Use the graph to estimate any solutions of the system. 29. No solution, parallel lines

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 9 Homework, Page 575 Use graphs to determine the number of solutions the system has. 33. Infinitely many solutions.

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 10 Homework, Page 575 Solve the system graphically. Support numerically. 37.

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 11 Homework, Page 575 Solve the system graphically. Support numerically. 41.

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 12 Homework, Page 575 45. The table shows expenditures, in billions, from federal hospital and medical insurance trust funds. A. Find the quadratic regression equation and superimpose its graph on a scatter plot of the data. B. Find the logistic regression equation and superimpose its graph on the scatter plot of the data. C. When will the two models predict expenditures of 300 billion dollars? D. Explain the long range implications of using the quadratic regression model to predict future expenditures. E. Explain the long range implications of using the logistic regression model to predict future expenditures. YearAmountYearAmount 1990110.21999213.5 1995183.22000225.3 1997209.52001246.5 1998210.22002267.1

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 13 Homework, Page 575 45. A. Find the quadratic regression equation and superimpose its graph on a scatter plot of the data. B. Find the logistic regression equation and superimpose its graph on the scatter plot of the data.

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 14 Homework, Page 575 45. C. When will the two models predict expenditures of 300 billion dollars? The quadratic model predicts reaching $300-billion in 2006 and the logistic model predicts reaching $300-billion in 2007. D. Explain the long range implications of using the quadratic regression model to predict future expenditures. The quadratic model predicts expenditures reaching a maximum level of about $575-billion and then decreasing, eventually reaching zero, which is not realistic. E. Explain the long range implications of using the logistic regression model to predict future expenditures. The logistic model predicts expenditures leveling out at about $354-billion, which is also not realistic.

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 15 Homework, Page 575 49.Find the dimensions of a rectangle with a perimeter of 200 m and an area of 500m 2.

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 16 Homework, Page 575 53.The total cost of one medium and one large soda is $1.74. The large soda costs $0.16 more than the medium soda. Find the cost of each soda.

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 17 Homework, Page 575 57.Pedro has two plans to choose from to rent a van. Company A: a flat fee of $40 plus $0.10 per mile Company B: a flat fee of $25 plus $0.15 per mile (a) How many miles can Pedro drive in order to be charged the same amount by the two companies. (b) Give reasons why Pedro might choose one plan over the other. If Pedro is planning on driving more than 300 miles, Company A’s plan would be less expensive. If he is planning to drive less than 300 miles, Company B’s plan is less expensive.

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 18 Homework, Page 575 61.Which of the following is a solution of the system A. B. C. D. E.

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 19 Homework, Page 575 65. Consider the system of equations: (a) Solve the first equation in terms of x to determine the two implicit functions determined by the equation. (b) Solve the system of equations graphically.

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 20 Homework, Page 575 65. (c) Use substitution to confirm the solutions found in part (b).

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.2 Matrix Algebra

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 22 Quick Review

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 23 Quick Review Solutions

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 24 What you’ll learn about Matrices Matrix Addition and Subtraction Matrix Multiplication Identity and Inverse Matrices Determinant of a Square Matrix Applications … and why Matrix algebra provides a powerful technique to manipulate large data sets and solve the related problems that are modeled by the matrices.

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 25 Matrix

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 26 Matrix Vocabulary Each element, or entry, a ij, of the matrix uses double subscript notation. The row subscript is the first subscript i, and the column subscript is j. The element a ij is in the i th row and the j th column. In general, the order of an m × n matrix is m×n.

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 27 Example Determining the Order of a Matrix

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 28 Matrix Addition and Matrix Subtraction

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 29 Example Matrix Addition

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 30 Example Using Scalar Multiplication

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 31 The Zero Matrix

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 32 Additive Inverse

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 33 Matrix Multiplication

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 34 Example Matrix Multiplication

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 35 Identity Matrix

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 36 Inverse of a Square Matrix

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 37 Inverse of a 2 × 2 Matrix

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 38 Minors and Cofactors of an n × n Matrix

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 39 Determinant of a Square Matrix

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 40 Transpose of a Matrix

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 41 Example Using the Transpose of a Matrix

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 42 Inverses of n × n Matrices An n × n matrix A has an inverse if and only if det A ≠ 0.

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 43 Example Finding Inverse Matrices

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 44 Properties of Matrices Let A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined. 1. Community property Addition: A + B = B + A Multiplication: Does not hold in general 2. Associative property Addition: (A + B) + C = A + (B + C) Multiplication: (AB)C = A(BC) 3. Identity property Addition: A + 0 = A Multiplication: A·I n = I n ·A = A 4. Inverse property Addition: A + (-A) = 0 Multiplication: AA -1 = A -1 A = I n |A|≠0 5. Distributive property Multiplication over addition: A(B + C) = AB + AC (A + B)C = AC + BC Multiplication over subtraction: A(B - C) = AB - AC (A - B)C = AC - BC

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 45 Reflecting Points About a Coordinate Axis

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 46 Example Using Matrix Multiplication

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 47 Homework Homework Assignment #10 Read Section 7.3 Page 590, Exercises: 1 – 65 (EOO), skip 53

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.3 Multivariate Linear Systems and Row Operations

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 49 Quick Review

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 50 Quick Review Solutions

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 51 What you’ll learn about Triangular Forms for Linear Systems Gaussian Elimination Elementary Row Operations and Row Echelon Form Reduced Row Echelon Form Solving Systems with Inverse Matrices Applications … and why Many applications in business and science are modeled by systems of linear equations in three or more variables.

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 52 Triangular Form of a System of Equations A system of equations is said to be in triangular form, if it has as many equations as variables and if the equations are arranged in such a manner that the top equation has all variables, the next lacks one variable, the next lacks the first variable and a second and so on. For example,

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 53 Example Solving a System of Equations in Triangular Form by Substitution Solve the system.

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 54 Equivalent Systems of Linear Equations The following operations produce an equivalent system of linear equations. 1. Interchange any two equations of the system. 2. Multiply (or divide) one of the equations by any nonzero real number. 3. Add a multiple of one equation to any other equation in the system. These operations, when used to reduce a system to triangular form, are called Gaussian elimination.

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 55 Example Solving a System of Equations Using Gaussian Elimination Solve the system

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 56 Example Solving a System of Equations Using Gaussian Elimination Solve the system

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 57 Augmented Matrix An augmented matrix is one in which there is one more column than row and where the first columns are the coefficients of a system of equations and the last column contains the constants of the equations. For instance, the system may be represented by the augmented matrix Augmented matrices may be used to record the steps of the Gaussian elimination process.

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 58 Row Echelon Form of a Matrix A matrix is in row echelon form if the following conditions are satisfied. 1. Rows consisting entirely of 0’s (if there are any) occur at the bottom of the matrix. 2. The first entry in any row with nonzero entries is 1. 3. The column subscript of the leading 1 entries increases as the row subscript increases.

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 59 Elementary Row Operations on a Matrix A combination of the following operations will transform a matrix to row echelon form. 1. Interchange any two rows. 2. Multiply all elements of a row by a nonzero real number. 3. Add a multiple of one row to any other row.

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 60 Example Finding a Row Echelon Form

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 61 Reduced Row Echelon Form If we continue to apply elementary row operations to a row echelon form of a matrix, we can obtain a matrix in which every column that has a leading 1 has 0’s elsewhere. This is the reduced echelon form.

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 62 Example Solving a System Using Inverse Matrices

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 63 Example Solving a System Using Inverse Matrices Solve the system.

64. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 64 Example Solving a Word Problem 74. Stewart’s Metals has three silver alloys on hand. One is 22% silver, one is 30%, and the third is 42%. How many grams of each alloy are required to produce 80 grams of a new alloy that is 34% silver if the amount of the 30% alloy is twice the amount of the 22% alloy used?