Cramer's Rule

Here's a significantly more useful result from Linear Algebra:

Let A be a non-singular n x n matrix and b an n-dimensional real vector.

Let A_i be A with the ith column replaced with b.

If x is a unique solution to Ax = b, then x_i = |A_i| / |A|

Well, useful in the Q-question world where n < 4. After that, taking the determinants gets to be an awful lot of work and you're back to busting out a numerical package.