###### Item 3

Consider the matrix

\begin{equation*}
\left[
\begin{array}{rrr|r}
1 \amp 0 \amp 2 \amp 0 \\
0 \amp 1 \amp -1 \amp 0 \\
0 \amp 0 \amp 0 \amp 1 \\
\end{array}
\right].
\end{equation*}

Beginning with the last equation, we see that

\begin{equation*}
0x +0y+0z = 0 = 1\text{,}
\end{equation*}

which is not true for any \((x,y,z)\text{.}\) There is no solution to this particular equation and therefore no solution to the system of equations.