i got here 5 equations
and 6 variables
how to know who is the free variable
without making a triangle of zeros
on the bottom
From the matrix there are several "natural" candidates: take row 1, for instance: (if you want the first variable to be the free one), or , if you want the fourth variable to be the free one.
But this time is better to put as function of since then row 2 gives you as function of the first one, and row 4 gives as function of , so as function of the first one, and then 3 row gives you the fifth and...etc.
In fact, and after checking quickly the matrix, the above seems to be the best option, and by far. But it is always better to bring your matrix to echelon form