Lesson 6: Solving Nonlinear Systems of Equations

Property

A system of nonlinear equations is a system where at least one of the equations is not linear.

Just as with systems of linear equations, a solution of a nonlinear system is an ordered pair that makes both equations true. In a nonlinear system, there may be more than one solution. The graphs may be circles, parabolas or hyperbolas and there may be several points of intersection, and so several solutions.

Examples

• A system with a parabola and a circle:

Property

To solve a system of nonlinear equations by graphing:

Step 1. Identify the graph of each equation. Sketch the possible options for intersection.
Step 2. Graph the first equation.
Step 3. Graph the second equation on the same rectangular coordinate system.
Step 4. Determine whether the graphs intersect.
Step 5. Identify the points of intersection.
Step 6. Check that each ordered pair is a solution to both original equations.

Examples

• Solve the system $\begin{cases} y = x^2 \\ y = 4 \end{cases}$. Graphing the parabola $y=x^2$ and the horizontal line $y=4$ shows they intersect at $(2, 4)$ and $(-2, 4)$.