How to Solve Nonlinear Systems

In a nonlinear system, at least one equation will have a graph that isn’t a straight line. You can always write a linear equation in the form Ax + By = C (where A, B, and C are real numbers); a nonlinear system is represented by any other form. Examples of nonlinear equations include, but are not limited to, any conic section, polynomial, rational function, exponential, or logarithm. The nonlinear systems you’ll see in pre-calc will have two equations with two variables, as the three-dimensional systems are extremely difficult to solve. Because you’re really working with a system with two equations and two variables (even though one or both equations are nonlinear), you have the same two methods at your disposal: substitution and elimination.

The method of solving nonlinear systems is different from that of linear systems in that these systems are much more complicated and therefore require much more work. Usually, substitution is your best bet. Unless the variable you want to eliminate is raised to the same power in both equations, elimination won’t get you anywhere.

When one system equation is nonlinear

If one equation in a system is nonlinear, your first thought before solving should be, “Bingo! Substitution method!” (or something to that effect). In this situation, you can solve for one variable in the linear equation and substitute this expression into the nonlinear equation, because solving for a variable in a linear equation is a piece of cake! And any time you can solve for one variable easily, you can substitute that expression into the other equation to solve for the other one.

For example, follow these steps to solve the system

1. Solve the linear equation for one variable.

   In the example system, the top equation is linear. If you solve for x, you get x = 3 + 4y.

2. Substitute the value of the variable into the nonlinear equation.

   When you plug 3 + 4y into the second equation for x, you get (3 + 4y)y = 6.

3. Solve the nonlinear equation for the variable.

   When you distribute the y, you get

   

   Because this is a quadratic equation, you must get 0 on one side, so subtract the 6 from both sides to get

   

   You have to use the quadratic formula to solve this equation for y:

   

   When you square root something, you get a positive and a negative answer, which means you have two different answers in this situation.

4. Substitute the solution(s) into either equation to solve for the other variable.

   Because you find two solutions for y, you have to substitute them both to get two different coordinate pairs. Here’s what happens when you do:

   

   This gives you the solutions to the system:

   

   These solutions represent the intersection of the line x – 4y = 3 and the rational function xy = 6.

When both system equations are nonlinear

If both of the equations in a system are nonlinear, well, you just have to get more creative to find the solutions. Unless one variable is raised to the same power in both equations, elimination is out of the question. Solving for one of the variables in either equation will not necessarily be easy, but it can usually be done. Then, plug this expression into the other equation and solve for the other variable just as you did before. Unlike linear systems, there may be many operations involved in the simplification or solving of these equations. Just remember to keep your order of operations in mind at each step of the way.

When both equations in a system are conic sections, you’ll never find more than four solutions (unless the two equations describe the same conic section, in which case the system has an infinite number of solutions — and therefore is a dependent system). This is because conic sections are all very smooth curves with no sharp corners or crazy bends, so there is no way for two different conic sections to intersect more than four times.

For example, suppose a problem asks you to solve the following system:

Doesn’t that just make your skin crawl? Don’t break out the calamine lotion just yet, though. Follow these steps to find the solutions:

1. Solve for x² or y² in one of the given equations.

   The second equation is attractive because all you have to do is add 9 to both sides to get

   

2. Substitute the value from Step 1 into the other equation.

   You now have

   

   A-ha! This is a quadratic equation.

3. Solve the quadratic equation.

   Subtract 9 from both sides to get

   

   Remember that you’re not allowed, ever, to divide by a variable.

   You must factor out the greatest common factor (GCF) instead to get y(1 + y) = 0. Use the zero product property to solve for y = 0 and y = –1.

4. Substitute the value(s) from Step 3 into either equation to solve for the other variable.

   In this example, you use the equation solved for in Step 1.

   

Be sure to keep track of which solution goes with which variable, because you have to express these solutions as points on a coordinate pair. Your answers are

This solution set represents the intersections of the circle and the parabola given by the equations in the system.