$\displaystyle

\begin{pmatrix}

3 &0 &0 &-1 &0 &0 \\

2& 0& 0 & 0 & 0 &-4 \\

8&2 &0 &-3 &-1 &0 \\

0& 1& 0& -1& 0& 0\\

0& 0 &1 &0 &-1 &0

\end{pmatrix}

$

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

?