the system of equations is this:
x - 2y + z = 3
2x - 4y + z = 2
x - 2y - 2z = 1
First thing I'd do is write it out as an augmented matrix like so.
$\displaystyle \left[ \begin{array}{ccc|c} 1 & 2 & 1 & 3 \\ 2 & 4 & 1 & 2 \\ 1 & 2 & 2 & 1 \\ \end{array} \right]$
Then try to reduce it to Row Echelon form using the elementary row operations. You should end up with some form of a contradiction if the equations have no solutions. And that contradiction tells you that there are no solutions.
In other words, you won't be able to reduce it in a nice neat Row Echelon form if it has no solutions, and the fact that you can't reduce it is what tells you that it's an inconsistent system.
By the way, I'm not a professional mathematician, so don't trust my answer.
But I think I'm right on this.