Item3

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.

in-context