# But There’s a Square in My System…

First they want you to solve linear equations, then it’s quadratic equations, then systems of linear equations… and now it’s systems of nonlinear equations… WHERE DOES IT END?!?!

I’ve got some sad news for you… it doesn’t end, not really.  But there is a secret to solving systems of nonlinear equations.  If you can solve a system of linear equations using substitution, and if you know how to use the quadratic formula, you’re golden.  Here’s how to combine those two techniques you’ve (hopefully) mastered already.

We know y=x+3, so we can substitute this value for y into the first equation to solve for x.  Like this:

This is a quadratic equation that we can solve using the quadratic formula.  We end up with

(try it yourself if you don’t believe me. geez.)
Then we use these values of x in y=x+3 to get our y values. Like this:

So the solutions to this system of equations are the ordered pairs

You’ll notice that unlike linear systems, there is more than one solution.  Which makes sense, if you think about the graphs of these equations… one is a parabola, and one is a line, so the graphs intersect can at two points… check it:

So next time you have to solve a nonlinear system, don’t freak out.  Just remember that you actually already know how to solve nonlinear systems… substitute and use the quadratic formula.  NBD.