Unless you mean something else you can solve exactly if you know y and z (or to the tolerance of your numerical representaion for e.g. float)
x = -(y+z)
if you don't know y and z then there are infinite possible answers
i'm guessing the question is missing something?
I don't write much Java myself, but it sounds like you want a nonlinear solver. Java Numerics has links to many different numerical Java libraries. I think you could also call C libraries from Java, so you could use GNU Scientific Library. Just search for nonlinear solver/optimization in those pages.
As you stated in a comment, the Bisection algorithm is one way of solving this problem. Problems of this type are generally referred to as root-finding algorithms, of which there are many. Probably the most powerful(Or at least, the most popular) is the Newton-Raphson method, if you know that your function is differentiable. However, you can find pretty good implementations of many different solvers (including a Bisection and Newton solver) in the Apache Commons Math package.
I like the JScience library for such things. This example implements the Durand-Kerner-Weierstrass method to find the roots of a univariate polynomial with complex coefficients.