Solving Non-Linear Systems of Equations

Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Many systems of equations are non-linear, and knowing how to solve these non-linear systems is important. In this lesson, learn how to recognize and solve non-linear systems of equations.

What is a Non-Linear System of Equations?

How do you know if you have a system of equations? How many equations do you need to make a system? What do the equations need to look like? What makes an equation non-linear? Before you can learn how to solve a non-linear system of equations, you first need to know how to recognize one.

A system of equations is made up of two or more independent equations. The equations can be linear or non-linear. A non-linear equation contains one term that has an exponent, like x2 or y2 (it might look like 4x2 + 3y = 12), while a linear equation does not (something like 5x - 7y = 10). So, a non-linear system of equations has two or more equations with at least one term being raised to a power (having an exponent of two or greater) and one of the equations has a product of variables.

Which of the following do you think are systems of equations?

systems of equations

The first is definitely not a system of equations because it is all alone! Remember that you need at least two equations to have a system. The others are systems of equations, but did you notice that only one is a non-linear system of equations? That's right! Only the last one is non-linear, because it contains x2.

How to Solve Non-Linear Systems of Equations

Now that you know how to identify a non-linear system of equations, let's look at how to solve it. We'll begin with a simple example.

Solve the following system of equations:

x2 = 2y + 6

x2 = 3y + 7

What exactly does it mean to solve a system of equations? The solution is the point (or points, there can be more than one) where the graphs of the two equations intersect. Graphing the two equations above will show you that there are two points of intersection for this system of equations, so there should be two solutions.

graph of both equations

You can estimate the solution from a graph like this, but it is also helpful to be able to find the solution analytically. Since x2 = 2y + 6 and x2 = 3y + 7, then 2y + 6 must also be equal to 3y+ 7.

2y + 6 = 3y + 7

We have just eliminated the x2 terns and now we have one, easy to solve equation. To solve this equation, rearrange the terms to get y by itself. If you need a review of solving linear equations like this, be sure to check out one of our other algebra lessons!

2y - 3y = 7 - 6

-y = 1

y = -1

Wait! The problem isn't over yet. We know what y is, but what about x? To find x, substitute the value we just found for y into either of the original two equations.

x2 = 3 (-1) + 7

x2 = 4

x = +2 or x = -2

So, the two points of intersection are (2,-1) and (-2,-1) and these are the solutions to the system of equations.

More Challenging Problems

The solutions to the previous problem were both whole numbers, but that is not always the case. Let's look at a problem that is a little different.

Solve the following system of equations:

y = x2 + 5x - 7

y = 4x+3

Just like in the first problem, the solutions to this system are the points where the two equations overlap on a graph.

non linear system graph #2

Just from looking at the graph, you can see that the solutions are somewhere around (2.5,13) and (-3.5,-12), but we can be more exact than that. To find the exact solutions, set the two equations equal to each other (we can do this since they both are equal to the same thing, y).

x2 + 5x - 7 = 4x+3

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