1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.1 Solving Systems of Two Equations

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 2 What you’ll learn about The Method of Substitution Solving Systems Graphically The Method of Elimination Applications … and why Many applications in business and science can be modeled using systems of equations.

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 3 Solution of a System A solution of a system of two equations in two variables is an ordered pair of real numbers that is a solution of each equation. A system is solved when all of its solutions are found.

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 4 Example Using the Substitution Method

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 5 Example Solving a Nonlinear System by Substitution

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 6 Example Solving a Nonlinear System Algebraically

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 7 Example Solving a Nonlinear System Graphically

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 8 Example Using the Elimination Method

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 9 Example Finding No Solution

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 10 Example Finding Infinitely Many Solutions

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 11 Example Solving Word Problems with Systems Find the dimensions of a rectangular cornfield with a perimeter of 220 yd and an area of 3000 yd 2.

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 12 Homework Homework Assignment #9 Read Section 7.2 Page 575, Exercises: 1 – 65 (EOO)

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

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 14 Quick Review

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 15 Quick Review Solutions

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 16 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.

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 17 Matrix

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 18 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.

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 19 Example Determining the Order of a Matrix

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 20 Matrix Addition and Matrix Subtraction

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 21 Example Matrix Addition

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 22 Example Using Scalar Multiplication

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 23 The Zero Matrix

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 24 Additive Inverse

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

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 26 Example Matrix Multiplication

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 27 Identity Matrix

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 28 Inverse of a Square Matrix

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 29 Inverse of a 2 × 2 Matrix

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 30 Minors and Cofactors of an n × n Matrix

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 31 Determinant of a Square Matrix

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 32 Transpose of a Matrix

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 33 Example Using the Transpose of a Matrix

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 34 Inverses of n × n Matrices An n × n matrix A has an inverse if and only if det A ≠ 0.

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 35 Example Finding Inverse Matrices

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 36 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

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 37 Reflecting Points About a Coordinate Axis

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 38 Example Using Matrix Multiplication