Answer
The system is inconsistent.
Work Step by Step
$\left\{\begin{array}{lllll}
2x & -2y & +3z & =6 & \\
4x & -3y & +2z & =0 & /R_{2}=r_{2}-2r_{1}\\
-2x & +3y & -7z & =1 & /R_{3}=r_{3}+r_{1}
\end{array}\right.$
... Eliminate $x$ in eq. 2 and 3
$\left\{\begin{array}{lllll}
2x & -2y & +3z & =6 & \\
& y & -4z & =-12 & \\
& y & -4z & =7 & /R_{3}=r_{3}-r_{2}
\end{array}\right.$
... Eliminate y in equation 3
$\left\{\begin{array}{lllll}
2x & -2y & +3z & =6 & \\
& y & -4z & =-12 & \\
& & 0 & =19 &
\end{array}\right.$
The third equation is impossible - there are no solutions.
The system is inconsistent.