The standard way of solving a system of non-linear equations is via Newton's method. For this to work you need the corresponding derivatives (i.e. the Jacobian matrix) or at least their approximations. Depending on your situation, you might be able to calculate the Jacobian explicitely. Otherwise, you can try using finite differences to approximate it. Once you got the Jacobian, you still need to either invert it, or to solve a system of linear equations. There are various algorithms for both of these tasks.

So, if you're system of equations is ``well-behaved'', it is absolutely possible. None of the steps is very complex if you're not striving for the highest possible performance and accuracy. However, getting it right still requires some work. Since this task is fairly common, there are several available implementations, see this website. Note that you can use any Java library in Processing (not only those written explicitely for Processing).