Answer
$x=1$
$y=0$
$z=1$
Work Step by Step
$ \begin{bmatrix}
1 & 1 & 1 & 2\\
2 & -3 & 2 & 4\\
4 & 1 & -3 & 1
\end{bmatrix} $
Add -2 times the 1st row to the 2nd row to produce a new 2nd row.
Add -4 times the 1st row to the 3rd row to produce a new 3rd row.
$ \begin{bmatrix}
1 & 1 & 1 & 2\\
0 & -5 & 0 & 0\\
0 & -3 & -7 & -7
\end{bmatrix} $
From the second row we can express $-5y=0 \rightarrow y=0$
Use back-substitution to find the solution.
$y=0$
$-3y-7z=-7 \rightarrow z=1$
$x+y+z=2 \rightarrow x=1$