The Gauss-Jordan elimination procedure is a systematic method for solving systems of linear equations. It works one variable at a time, eliminating it in all rows but one, and then moves on to the next variable. We illustrate the procedure on three examples.

**Another example:**

First we eliminate from equations 2 and 3.

Then we eliminate from equations 1 and 3.

Equation 3 shows that the linear system has *no solution*.

**A third example:**

Doing the same as above, we end up with

Now equation 3 is an obvious equality. It can be discarded to obtain

The situation where we can express some of the variables (here
and ) in terms of the remaining variables (here ) is
important. These variables are said to be *basic* and
*nonbasic* respectively.
Any choice of the nonbasic variable yields a solution of
the linear system. Therefore the system has infinitely many
solutions.

Mon Aug 24 16:30:59 EDT 1998