Lesson 11.5: Solve Systems of Nonlinear Equations

New Concept

A system of nonlinear equations has at least one equation that isn't a line, like a circle or parabola. We'll find intersection points (solutions) using graphing, substitution, and elimination, then apply these skills to solve real-world problems.

What’s next

First, you’ll learn to solve these systems by graphing. Then, you'll tackle practice problems using the substitution and elimination methods to master each technique.

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 $(-2, 4)$.

Property

To solve a system of nonlinear equations by substitution:

Step 1. Identify the graph of each equation and sketch possible intersections.
Step 2. Solve one of the equations for either variable.
Step 3. Substitute the expression from Step 2 into the other equation.
Step 4. Solve the resulting equation.
Step 5. Substitute each solution from Step 4 into one of the original equations to find the other variable.
Step 6. Write each solution as an ordered pair and check it in both original equations.

Examples

• To solve $\begin{cases} x^2 + y^2 = 13 \\ y = x-1 \end{cases}$, substitute $(x-1)$ for $y$ in the first equation: $x^2 + (x-1)^2 = 13$. This simplifies to $2x^2 - 2x - 12 = 0$, giving solutions $(3, 2)$ and $(-2, -3)$.

Property

To solve a system of equations by elimination:

Step 1. Identify the graph of each equation and sketch possible intersections.
Step 2. Write both equations in standard form.
Step 3. Multiply one or both equations so that the coefficients of one variable are opposites.
Step 4. Add the equations to eliminate one variable.
Step 5. Solve for the remaining variable.
Step 6. Substitute each solution into an original equation to solve for the other variable and write the solution as an ordered pair.

Examples

• In the system $\begin{cases} x^2 + y^2 = 20 \\ x^2 - y^2 = 12 \end{cases}$, adding the equations eliminates $y^2$, giving $2x^2 = 32$. The solutions are $(4, 2)$, $(4, -2)$, $(-4, 2)$, and $(-4, -2)$.

Property

To solve applications with a system of nonlinear equations:

Step 1. Identify what you are looking for.
Step 2. Define the variables for the unknown quantities.
Step 3. Translate the given information into a system of equations. For geometric problems, use formulas for area, perimeter, or the Pythagorean theorem. For number problems, translate the sentences directly into equations.
Step 4. Solve the system using substitution or elimination.

Examples

• The sum of two numbers is 12 and their product is 35. Find the numbers. The system is $\begin{cases} x+y=12 \\ xy=35 \end{cases}$. Solving gives the numbers 5 and 7.