\(S_1\):
\(\begin{eqnarray*} x + 2 y - ~~z &=& ~~~0 &\\ 2 x + ~~y + ~~z &=& ~~~0 &\quad E_2\to -2 E_1 + E_2\\ - 3 x + ~~y + 2 z &=& -6 &\quad E_3\to ~~~3 E_1 + E_3 \end{eqnarray*}\)
The two solution-preserving operations on equations \(2\) and \(3\) result in the equivalent (in the sense that it has the same solution) system:
\(S_2\):
\(\begin{eqnarray*} x + 2 y - ~~z = ~~~0\\ ~~-3 y + ~3 z = ~~~0\\ 7 y - ~~~z = - 6 \end{eqnarray*}\)