Solving Nonlinear Systems for Intermediate Algebra

By — McGraw-Hill Professional
Updated on Aug 12, 2011

Practice problems for these concepts can be found at:

A nonlinear equation has degree two or more. The graph is not a straight line hence the term nonlinear. A nonlinear system has at least one nonlinear equation. Our concentration here is on solving nonlinear systems. The techniques employed eliminate variables, one by one, until an equation in one variable that we can solve remains. There are two methods used primarily: the substitution method and the addition method.

Substitution Method

The substitution method works very well when one of the equations in the system is linear. We simply solve the linear equation explicitly for one variable and substitute into the other equation. The resulting equation must be an equation we can solve.

See solved problem 7.15.

Addition Method

If both equations in a system are second-degree in both variables, the addition (elimination) method often works well.

See solved problem 7.16.

Graphical Method

A graphing calculator may be employed to solve nonlinear systems also. We graph the equations of the system and use the trace feature to approximate the points of intersection of the graphs, if any. The values thus obtained re the approximate solutions to the system.

See solved problem 7.17.


Systems of inequalities can be solved graphically. The solution set of a system of inequalities is simply the intersection of the solution sets of the individual inequalities. We first graph each inequality separately. Finally, the graphs are combined to display the solution set of the system.

See solved problem 7.18.

Practice problems for these concepts can be found at:

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