Solving Nonlinear Systems

Essential Question: How can you solve a nonlinear system of equations? Exploration: Solving Nonlinear Systems of Equations Work with a partner. Match each system with its graph. Explain1 your reasoning. Then solve each system using the graph.2   Systems of Equations: (y = x^{2}) (y = x + 2) (y = x^{2} + x - 2) (y = x + 2) (y = x^{2} - 2x - 5) (y = -x + 1) (y = x^{2} + x - 6) (y = -x^{2} - x + 6) (y = x^{2} - 2x + 1) (y = -x^{2} + 2x - 1) (y = x^{2} + 2x + 1) (y = -x^{2} + x + 2) Graphs: A. Two parabolas intersecting at two points. B. A parabola and a line intersecting at two points. C. Two parabolas intersecting at one point. D. A parabola and a line intersecting at two points. E. Two parabolas intersecting at two points. F. Two parabolas intersecting at two points. Solutions: A. 2 solutions B. 2 solutions C. 1 solution D. 2 solutions E. 2 solutions F. 2 solutions Notetaking with Vocabulary Subtitle: For use after Lesson 3.5 Vocabulary Term: system of nonlinear equations Solutions: One line and one parabola: 0, 1, or 2 points Two parabolas: 0, 1, 2, or no solutions Core Concepts Solve Equations by Graphing Step 1: To solve the equation f(x) = g(x), write a system of two equations: y = f(x) and y = g(x) Step 2: Graph the system of equations y = f(x) and y = g(x). The x-value of each solution of the system is a solution of the equation f(x) = g(x). 1   Solve Equations Algebraically (Substitution or Elimination) Look at pre-1.4 notes Example: y = 5 - x y - x^2 = 7 y = 5 - x y - x^2 = 7 (5 - x) - x^2 = 7 ... is substitution y = 5 - x y - x^2 = 7 y + x = 5 y - x^2 = 7 x + x^2 = -2 ... is elimination Instructions: In Exercises 4 and 5, solve the system of nonlinear equations by using the graph. Exercise 4: A graph with two parabolas intersecting at two points. The points of intersection are labeled: (-1,-10) and (2,-10) Handwritten note: "Look at graph" Exercise 5: A graph with two parabolas that do not intersect. Handwritten note: "No Solutions" Instructions: In Exercises 1-3, solve the system by graphing. Check your solution(s). Exercise 1: The system of equations: (y = \frac{1}{2}x^2 - 3) (y = -4 - 2x^2) Handwritten note: "Find vertex then make table to graph" Vertex: (0,-3) Vertex: (0,-4) Exercise 2: The system of equations: (y = (x - 2)^2) (y = \frac{1}{4}x - \frac{1}{2}) Handwritten note: "Vertex: (2,0)" Vertex: (0,-1/2) Exercise 3: The system of equations: (y = -x^2 - 2) (y = 4(x + 1) - 3) Handwritten note: "Vertex: (0,-2)" Vertex: (-1,3) Exercise 6: System of equations: y = x + 4 y = (x + 2)² + 1 Solution steps: (x + 4) = (x + 2)² + 1 x + 4 = x² + 4x + 5 0 = x² - 3x + 1 Quadratic Formula: x = (3 ± √5) / 2 Handwritten note: "This will be nice when you find y" Exercise 7: System of equations: x² + y² = 16 y = -x + 4 Solution steps: x² + (-x + 4)² = 16 x² + x² - 8x + 16 = 16 2x² - 8x = 0 Factor: 2x(x - 4) = 0 Solutions: x = 0, x = 4 Find y: y = -(0) + 4 = 4 y = -(4) + 4 = 0 Double Check: (0)² + (4)² = 16 (4)² + (0)² = 16 Solutions: (0, 4) and (4, 0) Exercise 8: System of equations: 2x² + 10x + 48 = y - 10x -4x² - 16x = y Solution steps: 2x² + 10x + 48 = (-4x² - 16x) - 10x 2x² + 10x + 48 = -4x² - 26x 6x² + 36x + 48 = 0 Factor: 6(x² + 6x + 8) = 0 (x + 2)(x + 4) = 0 Solutions: x = -2, x = -4 Find y: -4(-2)² - 16(-2) = y -4(-4)² - 16(-4) = y Double Check: 2(-2)² + 10(-2) + 48 = (-4(-2)² - 16(-2)) - 10(-2) 2(-4)² + 10(-4) + 48 = (-4(-4)² - 16(-4)) - 10(-4) Solutions: (-2, 16) and (-4, 0)