Lesson 7: Solving Nonlinear Systems of Equations

Property

A linear-quadratic system of equations consists of one linear equation and one quadratic equation. The general form of such a system is:

Examples

• A system with a parabola and a line:

• A system where the linear equation is not in slope-intercept form:

Explanation

A linear-quadratic system pairs a quadratic equation, which graphs as a parabola, with a linear equation, which graphs as a straight line. The solutions to the system are the points where the parabola and the line intersect. Because a line can cross a parabola in at most two places, there can be zero, one, or two real solutions. These systems model situations where a linear path or rate interacts with a parabolic trajectory.

Property

To solve a linear-quadratic system by graphing:

Step 1. Identify that one equation is linear and one is quadratic.
Step 2. Graph the linear equation (line).
Step 3. Graph the quadratic equation (parabola) on the same coordinate system.
Step 4. Determine whether the graphs intersect.
Step 5. Identify the points of intersection.
Step 6. Check that each ordered pair satisfies both original equations.

Property

To solve a linear-quadratic system by substitution:

Step 1. Identify which equation is linear and which is quadratic.
Step 2. Solve the linear equation for either variable.
Step 3. Substitute the expression from Step 2 into the quadratic equation.
Step 4. Solve the resulting quadratic equation.
Step 5. Substitute each solution from Step 4 into the linear equation to find the other variable.
Step 6. Write each solution as an ordered pair and check it in both original equations.

Property

To solve a linear-quadratic system by elimination:

Step 1. Write both equations in standard form.
Step 2. If needed, multiply one or both equations so that the coefficients of one variable are opposites.
Step 3. Add or subtract the equations to eliminate one variable.
Step 4. Solve the resulting equation for the remaining variable.
Step 5. Substitute each solution back into either original equation to find the other variable.
Step 6. Write the solutions as ordered pairs and check in both original equations.