One of the basic tasks of numerical analysis is that of solving equations. The first step in solving such equations to is subtract one side completely, leaving an equation of the form

The resulting equation can be one-dimensional or it can be a system of multi-dimensional equations. One-dimensional problems are much, much easier to solve than multi-dimensional ones. The reason for this is that in a one-dimensional equation, you can bracket values. That is to say, you can find an interval in which you are sure there is a root. One you have found a region in which you are sure there is a root, you can then use various methods of homing in on the actual root.

In the case of multi-dimensional problems, you can't be sure that a root exists, unless and until you actual find it. But all is not lost, the implicit function theorem, states that if you have "n" equations in "n" unknowns, you will usually have a root. The "usually" is important in that you will sometimes come across systems of equations which have as their solution a curve, or perhaps have no solution at all.

REFERENCE: Chapter 9. Numerical Recipes in C by Press, DaVaney, Teukolsky, and Vetterling.