My class was able to produce solutions using Substitution on the following System:
$$
\left\{
\begin{array}{c}
x+y+z=0 \\
2x+3y+2z=-1\\
x-y+z=2
\end{array}
\right.
$$
The solution was: x = 1, y = -1, z = 0
However, when I tried solving that system using matrices in the following format, I received a Singular Matrix Error in my calculator.
$$ \left[ \begin{array}{ccc|c} 1&1&1&0\\ 2&3&2&-1\\ 1&-1&1&2 \end{array} \right] $$
I solved the matrix by hand, and it is true. The det = 0.
So, why are there solutions to the system using substitution, but no solutions when using matrices?