1. Slide 8.5- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Nonlinear Systems of Equations and Inequalities Learn techniques for solving nonlinear systems of equations. Learn a procedure for solving a nonlinear system of inequalities. SECTION 8.5 1 2

3. Slide 8.5- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definitions The statements x + y > 4, 2x + 3y < 7, y ≥ x, and x + y ≤ 9 are examples of linear inequalities in the variables x and y. A solution of an inequality in two variables x and y is an ordered pair (a, b) that results in a true statement when x is replaced by a, and y is replaced by b in the inequality. The set of all solutions of an inequality is called the solution set of the inequality. The graph of an inequality in two variables is the graph of the solution set of the inequality.

4. Slide 8.5- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PROCEDURE FOR GRAPHING A LINEAR INEQUALITY IN TWO VARIABLES Step 1.Replace the inequality symbol by an equals (=) sign. Step 2.Sketch the graph of the corresponding equation in Step 1. Use a dashed line for the boundary if the given inequality sign is, and a solid line if the inequality symbol is ≤ or ≥.

5. Slide 8.5- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PROCEDURE FOR GRAPHING A LINEAR INEQUALITY IN TWO VARIABLES Step 3.The graph in Step 2 will divide the plane into two regions. Select a test point in the plane. Be sure that the test point does not lie on the graph of the equation in Step 1. Step 4.(i) If the coordinates of the test point satisfy the given inequality, then so do all the points of the region that contains

6. Slide 8.5- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PROCEDURE FOR GRAPHING A LINEAR INEQUALITY IN TWO VARIABLES the test point. Shade the region that contains the test point. (ii) If the coordinates of the test point do not satisfy the given inequality, shade the region that does not contain the test point. The shaded region (including the boundary if it is a solid curve) is the graph of the inequality.

7. Slide 8.5- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Using Substitution to Solve a Nonlinear System Solve the system of equations by the substitution method. Solution Step 1Solve for one variable. Express y in terms of x in equation (2). Step 2Substitute. Substitute x 2 +1 for y in equation (1).

8. Slide 8.5- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Using Substitution to Solve a Nonlinear System Solution continued Step 3Solve the equation resulting from step (2).

9. Slide 8.5- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Using Substitution to Solve a Nonlinear System Solution continued Step 4Back substitute. Substitute x = –2 in Equation (3) to obtain the corresponding y-value. Since x = –2 and y = 5, the apparent solution set of the system is {(–2, 5)}.

10. Slide 8.5- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Using Substitution to Solve a Nonlinear System Solution continued Step 5Check. Replace x by –2 and y by 5 in both equations (1) and (2). Confirm the solution with a graph.  ? ?  ? ?

11. Slide 8.5- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Using Substitution to Solve a Nonlinear System Solution continued The graphs of the line 4x + y = –3 and the parabola y = x 2 + 1 confirm that the solution set is {(–2, 5)}.

12. Slide 8.5- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Using Elimination to Solve a Nonlinear System Solve the system of equations by the elimination method. Solution Step 1Adjust the coefficients. Multiply Equation (2) by –1 to eliminate x. Step 2

13. Slide 8.5- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Using Elimination to Solve a Nonlinear System Solution continued Step 3Solve the equation obtained in Step 2. Step 4Back substitute the values in one of the original equations to solve for the other variable.

14. Slide 8.5- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Using Elimination to Solve a Nonlinear System Solution continued (i) Substitute y = –5 in Equation (2) & solve for x. Thus (0, –5) is a solution of the system.

15. Slide 8.5- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Using Elimination to Solve a Nonlinear System Solution continued (ii) Substitute y = 4 in Equation (2) & solve for x. Thus (3, 4) and (–3, 4) are the solutions of the system.

16. Slide 8.5- 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Using Elimination to Solve a Nonlinear System Solution continued Step 5Check (0, –5), (3, 4), and (–3, 4) in the equations x 2 + y 2 = 25 and x 2 – y = 5.  ?  ? ?  ? ?  ?  ? ?  ? ?

17. Slide 8.5- 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Using Elimination to Solve a Nonlinear System Solution continued The graphs of the circle x 2 + y 2 = 25 and the parabola y = x 2 – 5 confirm that the solution set is {(0, –5), (3, 4), (–3, 4)}.

18. Slide 8.5- 18 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PROCEDURE FOR GRAPHING A NONLINEAR INEQUALITY IN TWO VARIABLES Step 1.Replace the inequality symbol by an equals (=) sign. Step 2.Sketch the graph of the corresponding equation in Step 1. Use a dashed curve if the given inequality sign is, and a solid line if the inequality symbol is ≤ or ≥.

19. Slide 8.5- 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PROCEDURE FOR GRAPHING A NONLINEAR INEQUALITY IN TWO VARIABLES Step 3.The graph in Step 2 will divide the plane into two regions. Select a test point in the plane. Be sure that the test point does not lie on the graph of the equation in Step 1. Step 4.(i) If the coordinates of the test point satisfy the given inequality, then so do all the points of the region that contains

20. Slide 8.5- 20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PROCEDURE FOR GRAPHING A NONLINEAR INEQUALITY IN TWO VARIABLES the test point. Shade the region that contains the test point. (ii) If the coordinates of the test point do not satisfy the given inequality, shade the region that does not contain the test point. The shaded region (including the boundary if it is solid) is the graph of the given inequality.

21. Slide 8.5- 21 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solving a Nonlinear System of Inequalities Graph the solution set of the following system of Solution Graph each inequality separately in the same coordinate plane. Since (0, 0) is not a solution of any the corresponding equations, use (0, 0) as a test point for each inequality. inequalities:

22. Slide 8.5- 22 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solving a Nonlinear System of Inequalities Solution continued Step 2 Sketch as a solid curve with vertex (0, 4). Step 3Test (0, 0). 0 ≤ 4 – 0 is a false statement. Step 4 Shade the region. Step 1 y = 4 – x 2

23. Slide 8.5- 23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solving a Nonlinear System of Inequalities Solution continued Step 2 Sketch as a solid line through (0, –3) & (2, 0). Step 3Test (0, 0). 0 ≥ 0 – 3 is a true statement. Step 4 Shade the region. Step 1

24. Slide 8.5- 24 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solving a Nonlinear System of Inequalities Solution continued Step 3Test (0, 0). 0 ≥ 0 – 3 is a true statement. Step 4 Shade the region. Step 1 y = –6x – 3 Step 2 Sketch as a solid line through (0, –3)

25. Slide 8.5- 25 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solving a Nonlinear System of Inequalities Solution continued The region common to all three graphs is the graph of the solution set of the given system of inequalities.