Itemc

After applying two replacement and one scaling operation, we find

\begin{equation*} \begin{alignedat}{4} -x \amp {}-{} \amp 2y \amp {}+{} \amp 2z \amp {}={} \amp -1 \\ \amp \amp \amp \amp z \amp {}={} \amp 1 \\ \amp \amp \amp \amp 2z \amp {}={} \amp 1 \\ \end{alignedat} \end{equation*}

Another replacement operation leads to the system

\begin{equation*} \begin{alignedat}{4} -x \amp {}-{} \amp 2y \amp {}+{} \amp 2z \amp {}={} \amp -1 \\ \amp \amp \amp \amp z \amp {}={} \amp 1 \\ \amp \amp \amp \amp 0 \amp {}={} \amp 1 \\ \end{alignedat} \end{equation*}

Since the third equation has no solutions, the original system can have no solutions as well.

in-context