# 3.9 – Nonlinear Systems of Equations

## Objectives

• Define nonlinear system of equations.
• Determine the number of solutions that a system of nonlinear equations has when given a graph of the system.
• List the steps for solving systems of nonlinear equations using the substitution method.
• Solve systems of nonlinear equations.
• Use algebra to isolate a variable in a nonlinear equation.

## Key Terms

• The solutions to the quadratic equation $ax^2+bx+c=0$ are given by the formula $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$.
• Isolate – To separate from others.
• To get the variable alone on one side of the equation or inequality.
• Nonlinear System of Equations – A group of nonlinear equations that have the same variables and are used together to solve a problem.
• The equations may be some combination of linear equations and nonlinear equations (all may be nonlinear, or only one may be nonlinear).
• System of Linear Equations – A group of linear equations that have the same variables and are used together to solve a problem.
• A linear equation can be written in the form y = mx + b.
• The graph of a linear equation is a straight line.

## Notes

Solutions of Nonlinear Systems of Equations
• The first step in solving a system of nonlinear equations by substitution is often to isolate a variable in one of the equations.
• To solve a system of equations in which there is no linear equation to start with, you can sometimes begin by isolating and substituting a variable that is squared in both equations.
• Some nonlinear equations represent real-world situations.
• In the real world, many x-values will start at zero and go up.
• So, if you get a negative x-value as a solution for a frog jumping forward, a human being shot out of a cannon, or you throwing a ball forward, there won’t be any negative x-values.
• Graphs for nonlinear systems of equations:

• Example 1:  To solve the system of equations below, Kira isolated the variable y in the first equation and then substituted it into the second equation. What was the resulting equation?

$\left\{\begin{array}{l}3y=12x\\x^2+y^2=81\\\end{array}\right\}$

Step 1: Isolate y in the 1st equation (divide by 3 on both sides).

$y=4x$

Step 2: Substitute the y-value of “4x” into the 2nd equation.

$x^2+(4x)^2=81$

Step 3: Square the 4x term.

$x^2+16x^2=81$

• Example 2: How many solutions are there to the following system of equations?