1. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Objectives Solve a system of equations containing first- or second-degree equations in two variables. Identify a conic section from its equation. 9.6 Solving Nonlinear Systems

2. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Glossary Terms classifying a conic section system of nonlinear equations 9.6 Solving Nonlinear Systems

3. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Rules and Properties Solutions of a System of Nonlinear Equations 9.6 Solving Nonlinear Systems Independent systems of two conic sections can have 0, 1, 2, 3, or 4 solutions. 0 real solutions

4. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Rules and Properties Solutions of a System of Nonlinear Equations 9.6 Solving Nonlinear Systems 1 real solution Independent systems of two conic sections can have 0, 1, 2, 3, or 4 solutions.

5. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Rules and Properties Solutions of a System of Nonlinear Equations 9.6 Solving Nonlinear Systems 2 real solutions Independent systems of two conic sections can have 0, 1, 2, 3, or 4 solutions.

6. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Rules and Properties Solutions of a System of Nonlinear Equations 9.6 Solving Nonlinear Systems 3 real solutions Independent systems of two conic sections can have 0, 1, 2, 3, or 4 solutions.

7. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Rules and Properties Solutions of a System of Nonlinear Equations 9.6 Solving Nonlinear Systems 4 real solutions Independent systems of two conic sections can have 0, 1, 2, 3, or 4 solutions.

8. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Rules and Properties Classifying a Conic Section 9.6 Solving Nonlinear Systems Ax 2 + Cy 2 + Dx + Ey + F = 0 ellipse (or circle)AC > 0 circle A = C, A  0, C  0 parabolaAC = 0 hyperbolaAC < 0 Type of conicCoefficients

9. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Key Skills Solve a system of nonlinear equations. 9.6 Solving Nonlinear Systems Systems of nonlinear equations can be solved by substitution, elimination, and graphing. Use each method as you would for systems of linear equations.

10. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Key Skills Solve a system of nonlinear equations. 9.6 Solving Nonlinear Systems x 2 + y 2 = 1 y = x 2 Use the graphing method to solve the system: x y Solve equations for y. y = ±  1 – x 2 y = x 2 Graph each function.

11. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Key Skills Solve a system of nonlinear equations. 9.6 Solving Nonlinear Systems x y y = ±  1 – x 2 y = x 2 The solutions are the points of intersection for the graphs of the functions in the system. Use the graphing method to solve the system: The approximate solutions are (–0.786, 0.618) and (0.786, 0.618).

12. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Key Skills Identify a conic section from its equation. x 2 + y 2 – 6x – 10y + 32 = 0 9.6 Solving Nonlinear Systems represents:a circle becauseA = C

13. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Key Skills Identify a conic section from its equation. 2x 2 + y 2 – 12x – 10y + 41 = 0 9.6 Solving Nonlinear Systems represents:an ellipse because AC > 0 and A  C

14. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Key Skills Identify a conic section from its equation. x 2 – 6x – 4y + 29 = 0 9.6 Solving Nonlinear Systems represents:a parabola becauseAC = 0

15. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Key Skills Identify a conic section from its equation. 2x 2 – y 2 – 12x + 10y – 9 = 0 9.6 Solving Nonlinear Systems represents:a hyperbola becauseAC < 0 TOC