Page "Gaussian elimination" Paragraph 27

Some systems cannot be reduced to triangular form, yet still have at least one valid solution: for example, if y had not occurred in and after the first step above, the algorithm would have been unable to reduce the system to triangular form.

However, it would still have reduced the system to echelon form.

In this case, the system does not have a unique solution, as it contains at least one free variable.

The solution set can then be expressed parametrically ( that is, in terms of the free variables, so that if values for the free variables are chosen, a solution will be generated ).