1. 1

2. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Systems of Linear Equations Using Cramer’s Rule 6 1.Evaluate determinants of 2  2 matrices. 2.Evaluate determinants of 3  3 matrices. 3.Solve systems of equations using Cramer’s Rule.

3. Slide 9- 3 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Square matrix: A matrix that has the same number of rows and columns. Every square matrix has a determinant. Determinant of a 2  2 Matrix If then det(A) =

4. Slide 9- 4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the determinant of the following matrix. Solution

5. Slide 9- 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Minor: The determinant of the remaining matrix when the row and column in which the element is located are ignored.

6. Slide 9- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the minor of  4 in Solution To find the minor of  4, we ignore its row and column (shown in blue) and evaluate the determinant of the remaining matrix (shown in red).

7. Slide 9- 7 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Determinant of a 3  3 Matrix If

8. Slide 9- 8 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Cramer’s Rule The solution to the system of linear equations

9. Slide 9- 9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The solution to the system of linear equations

10. Slide 9- 10 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution Example Use Cramer’s Rule to solve First, we find D, D x, and D y. The solution is (1/5, 4/5).

11. Slide 9- 11 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution Example Use Cramer’s Rule to solve We need to find D, D x, D y, and D z.

12. Slide 9- 12 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued

13. Slide 9- 13 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued The solution is (3, 1, –1). The check is left to the student.

14. Slide 9- 14 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the determinant. a)  75 b)  72 c)  27 d) 72

15. Slide 9- 15 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the determinant. a)  75 b)  72 c)  27 d) 72

16. Slide 9- 16 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve using Cramer’s Rule. a) (  1,  4) b) (1,  3) c) (  2,  3) d) No solution

17. Slide 9- 17 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve using Cramer’s Rule. a) (  1,  4) b) (1,  3) c) (  2,  3) d) No solution

18. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Systems of Linear Inequalities 7 1.Graph the solution set of a system of linear inequalities. 2.Solve applications involving a system of linear inequalities.

19. Slide 9- 19 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Graph the solution set of a system of linear inequalities.

20. Slide 9- 20 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving a System of Linear Inequalities To solve a system of linear inequalities, graph all of the inequalities on the same grid. The solution set for the system contains all ordered pairs in the region where the inequalities’ solution sets overlap along with ordered pairs on the portion of any solid line that touches the region of overlap.

21. Slide 9- 21 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Graph the solution set for the system of inequalities. Solution Graph the inequalities on the same grid. Both lines will be dashed. Solution

22. Slide 9- 22 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Graph the solution set for the system of inequalities. Solution Graph the inequalities on the same grid. Solution

23. Slide 9- 23 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example—Inconsistent System Graph the solution set for the system of inequalities. Solution Graph the inequalities on the same grid. The slopes are equal, so the lines are parallel. Since the shaded regions do not overlap there is no solution for the system.

24. Slide 9- 24 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 2 Solve applications involving a system of linear inequalities.

25. Slide 9- 25 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Mr. Reynolds is landscaping his yard. He is looking at some trees and bushes. He would like to purchase at least 3 plants. The trees cost $40 and the bushes cost $20. He cannot spend more than $300 for the plants. Write a system of inequalities that describes what Mr. Reynolds could purchase, then solve the system by graphing. Understand We must translate to a system of inequalities, and then solve the system.

26. Slide 9- 26 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Plan and Execute Let x represent the trees and y represent the bushes. Relationship 1: Mr. Reynolds would like to purchase at least 3 plants. x + y  3 Relationship 2: Mr. Reynolds cannot spend more than $300. 40x + 20y  300

27. Slide 9- 27 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Answer Since Mr. Reynolds cannot purchase negative plants, the solution set is confined to Quadrant 1. Any ordered pair in the overlapping region is a solution. Assuming that only whole trees and bushes can be purchased, only whole numbers would be in the solution set. For example: (4, 2); (3, 4)

28. Slide 9- 28 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graph. a)b) c)d)

29. Slide 9- 29 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graph. a)b) c)d)