I have this problem $\begin{array}{rcccccl} 3x &-&y &+& 2z &=& 2\\ 2x &+& y &+& z &=& -1\\ x &+& 3y & & &=&-1 \end{array}$ This gives me the matrix: $\left(\begin{array}{rrr|r} 3 & -1 & 2 & 2\\ 2 & 1 & 1 & -1\\ 1 & 3 & 0 & -1 \end{array}\right)$
I remember something about if you can show that a system in inconsistent, you know it's not solvable.
I set Row1->Row1 - 2*Row2 + Row3
The result is: $\left(\begin{array}{rrr|r} 0 & 0 & 0 & 3\\ 2 & 1 & 1 & -1\\ 1 & 3 & 0 & -1 \end{array}\right)$
So 0 = 3 which is inconsistent.
Two things, did I do this right and is my assumption that all linear systems that can be shown as inconsistent are unsolvable?
Are there less painful ways to show it's not solvable? It seems like this could go on for a while with trial and error to either find an inconsistency or an answer for x y and z.