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.
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.