\(\mathbb{S}_1\):
\(\begin{eqnarray*} x - y + z &=& 1\\ x\quad~~~ + z &=& 1 \quad E_2 \to -E_1 + E_2\\ x + y + z &=& 2 \quad E_3 \to -E_1 + E_3 \end{eqnarray*}\)
\(\mathbb{S}_2\):
\(\begin{eqnarray*} x - y + z &=& 1\\ y\quad~~~ &=& 0\\ 2y\quad~~~ &=& 1 \quad E_3 \to -2E_2 + E_3 \end{eqnarray*}\)
\(\mathbb{S}_3\):
\(\begin{eqnarray*} x - y + z &=& 1\\ y\quad~~~ &=& 0\\ 0 &=& 1 \end{eqnarray*}\)
The system is inconsistent because \(0=1\) is false.