In general, systems of simultaneous equations are extremely hard to solve. A common technique is the **substitution method**: try to solve one of the equations for one of the variables and substitute the result into the other equations, thereby reducing the number of equations and the number of variables by 1. Continue until you reach a single equation with a single variable, which (hopefully) can be solved; back substitution then yields the values for the other variables.

In the above example, we first solve the second equation for *x*:

Systems of simultaneous *linear* equations are studied in linear algebra and can always be solved; one uses Gauss-Jordan elimination or the Cholesky decomposition. To solve general systems numerically on a computer, the *n*-dimensional Newton's method may be used.
Algebraic geometry is essentially the theory of simultaneous polynomial equations. The question of effective computation with such equations belongs to elimination theory.

Simultaneous equation models are a form of statistical model in the form of a set of linear simultaneous equations. They are often used in econometrics.