Answer
The linear system is Inconsistent.
Work Step by Step
$\begin{cases}
2x+3y-z=1\\
x+2y=3\\
x+3y+z=4
\end{cases}$
Multiplying Equation 2 by -1 and adding it to Equation 3, results in new Equation 2.
$\begin{cases}
-x-2y=-3\\
x+3y+z=4\\
-- -- -- -\\
y+z=1
\end{cases}$
Multiplying Equation 3 by -2 and adding it to Equation 1, results in new Equation 3.
$\begin{cases}
2x+3y-z=1\\
-2x-6y-2z=-8\\
-- -- -- --\\
-3y-3z=-7
\end{cases}$
Multiplying new Equation 2 by 3 and adding it to new Equation 3.
$\begin{cases}
3y+3z=3\\
-3y-3z=-7\\
-- -- --\\
0=-4
\end{cases}$
Therefore, the linear system is Inconsistent.