Answer
The system is inconsistent.
Work Step by Step
$\left\{\begin{array}{lll}
x-y-z & =1 & \\
2x+3y+z & =2 & /R_{2}=r_{2}+r_{1}\\
3x+2y & =0 &
\end{array}\right.$
... Eliminate z in eq. 2 and 3
$\left\{\begin{array}{lllll}
x & -y & -z & =1 & \\
3x & +2y & & =3 & \\
3x & +2y & & =0 & /R_{3}=r_{3}-r_{2}
\end{array}\right.$
... Eliminate y in equation 3
$\left\{\begin{array}{lllll}
x & -y & -z & =2 & \\
3x & +2y & & =4 & \\
& & 0 & =3 &
\end{array}\right.$
The third equation is impossible - there are no solutions, so
the system is inconsistent.