In algebra, we often have to solve simultaneous differential equations. If the equations are linear and independent, there is a unique solution when the number of equations equals the number of variables. If there are only two variables and two equations, it's easy, but as the number of equations and unknowns increases, the problem becomes more difficult. Imagine, for example trying to solve 100 equations in 100 unknowns. That's an unthinkably difficult manual exercise, but a good computer program can do it in less than a second.

Suppose you want the solution to this pair of equations:

2 * x + y = 1
4 * y = 12

In the first equation, the coefficient of x is 2 and the coefficient of y is 1. In the second equation, the coefficient of x is 0, and the coefficient of y is 4. The right-side values are 1 and 12, respectively. It's easy to solve this pair of equations by hand. The second equation says y = 12/4 = 3, and substituting this back into the first equation gives x = (1-3)/2 = -1.