Chapter 12 12-7 Nonlinear Systems

Objectives Solve systems of equations in two variables that contain at least one second-degree equation.

What is a nonlinear system? A nonlinear system of equations is a system in which at least one of the equations is not linear. You have been studying one class of nonlinear equations, the conic sections. The solution set of a system of equations is the set of points that make all of the equations in the system true, or where the graphs intersect. For systems of nonlinear equations, you must be aware of the number of possible solutions.

Solutions You can use your graphing calculator to find solutions to systems of nonlinear equations and to check algebraic solutions.

Example 1: Solving a Nonlinear System by Graphing Solve by Graphing x2 + y2 = 25 4x2 + 9y2 = 145 The graph of the first equation is a circle, and the graph of the second equation is an ellipse, so there may be as many as four points of intersection.

Solution Step 1 Solve each equation for y. Step 2 Graph the system on your calculator, and use the intersect feature to find the solution set. The points of intersection are (–4, –3), (–4, 3), (4, –3), (4, 3).

Check It Out! Example 1 Solve by graphing 3x + y = 4.5 y = 1/2(x – 3)2 The graph of the first equation is a straight line, and the graph of the second equation is a parabola, so there may be as many as two points of intersection.

Substitution Method The substitution method for solving linear systems can also be used to solve nonlinear systems algebraically.

Example 2: Solving a Nonlinear System by Substitution Solve by substitution x2 + y2 = 100 y = x2 – 26 1 2 The solution set of the system is {(6, –8) (–6, –8), (8, 6), (–8, 6)}.

Check it out!!!! The solution set of the system is {(3, –4), (–4, 3)}. Solve the system of equations by using the substitution method. x + y = –1 x2 + y2 = 25 The solution set of the system is {(3, –4), (–4, 3)}.

Check it out!! Solve the system of equations by using the substitution method. The solution set of the system is {(3, –4), (–3, –4), (0, 5)}. x2 + y2 = 25 y – 5 = –x2

Elimination method The elimination method can also be used to solve systems of nonlinear equations.

Example Solve by elimination 4x2 + 25y2 = 41 36x2 + 25y2 = 169 The solution set of the system is {(–2, –1), (–2, 1), (2, –1), (2, 1)}.

Check it out!! Solve by elimination 25x2 + 9y2 = 225 25x2 – 16y2 = 400 There is no real solution of the system.

Application Suppose that the paths of two boats are modeled by 36x2 + 25y2 = 900 and y = 0.25x2 – 6. How many possible collision points are there?

Videos

Student Guided Practice Do odd problems from 2-11 in your book page 866

Homework Do even problems from 15-25 in your book page 866

Closure Today we learned about nonlinear systems Next class we are going to see chapter 6