Systems of Non-Linear Equations

This section covers:

Systems of Non-Linear Equations

(Note that solving trig non-linear equations can be found here).

We learned how to solve linear equations here in the Systems of Linear Equations and Word Problems Section.  Sometimes we need solve systems of non-linear equations, such as those we see in conics.

We can use either Substitution or Elimination, depending on what’s easier.  The main difference is that we’ll usually end up getting two (or more!) answers for a variable (since we may be dealing with quadratics or higher degree polynomials), and we need to plug in answers to get the other variable.  So we’ll typically have multiple sets of answers with non-linear systems.

Here are some examples.  Note that we could use factoring to solve the quadratics, but sometimes we will need to use the Quadratic Formula.

Systems of Non-Linear Equations

You can also use your graphing calculator:

Non-Linear Systems in Calculator

Non-Linear Equations Application Problems

Here are a few Non-Linear Systems application problems.


The difference of two numbers is 3, and the sum of their cubes is 407.  Find the numbers.


Let’s set up a system of non-linear equations:    \(\left\{ \begin{array}{l}x-y=3\\{{x}^{3}}+{{y}^{3}}=407\end{array} \right.\).    Substituting the y from the first equation into the second and solving yields:

Non-Linear Word Problem

So x = 7 works, and to find y, we use y = x – 3.  So when x = 7, y = 4.  So the two numbers are 4 and 7.  They work!

Lacy is speeding in her car, and sees a parked police car on the side of the road right next to her at t = 0 seconds.  She immediately decelerates, but the police car accelerates to catch up with her.   (Assume the two cars are going in the same direction in parallel paths).

The distance that Lacy has traveled in feet after t seconds can be modeled by the equation  \(d\left( t\right)=150+75t-1.2{{t}^{2}}\).  The distance that the police car travels after t seconds can be modeled by the equation   \(d\left( t \right)=4{{t}^{2}}\).

(a)  How long will it take the police car to catch up to Lacy? 

(b)  How many feet has Lacy traveled from the time she saw the police car (time t = 0) until the police car catches up to Lacy?


We need to find the intersection of the two functions, since that is when the distances are the same.   Remember that the graphs are not necessarily the paths of the cars, but rather a model of the how far they go given a certain time in seconds.

Note that since we can’t factor, we need to use the Quadratic Formula  to get the values for t.

We could also solve the non-linear systems using a Graphing Calculator, as shown below.

Application of Non-Linear Systems of Equations

Learn these rules, and practice, practice, practice!

On to Introduction to Vectors  – you are ready!

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