Try and think back to your high school mathematics classes.  Most of you will recall learning how to solve simultaneous equations.  Equations (1.1) and (1.2) below are a simple example of one of these problems, it may o be a little over simplistic?

  x +   y = 2                                                                                                   (1.1)

3x + 2y = y                                                                                                   (1.2)

To solve this problem we first re arrange equation (1.1):

x = 2 – y                                                                                                       (1.3)

We then substitute (1.3) into (1.2):

3(2 – y) + 2y = y

6 – 2y            = 0

y                    = 3                                                                                          (1.4)

Finally we substitute (1.4) into (1.3):

x = 2 – 3

x = -1                                                                                                           (1.5)

You may also remember being shown how to think of this problem geometrically.  Stated geometrically, the problem asks you to find those points on the x-y plane that lie on both line (1.6) and line (1.7).  Clearly the only solution to this problem is the point where the two lines intersect, which is the point (-1,3).  See figure (1a).

y = 2 – x                                                                                                       (1.6)

y = -3x                                                                                                         (1.7)

The aim of this article is to generalize on the work done in high school.  That is, we wish to solve simultaneous equations where there are more than two equations in more than two unknowns.