Answer
$x=\left( -\frac{2}{3},\frac{4}{3}, -\frac{1}{3}\right)$.
Work Step by Step
Assume that $x=(a,b,c)$ and $3 x+2 u-v+2 w=0$, then we have
\begin{align*}
&\Longrightarrow 3 (a,b,c)+2(1,-2,1)- (0,2,3)+2(0,1,1)=(0,0,0)\\
&\Longrightarrow 3 (a,b,c)+(2,-4,2)+ (0,-2,-3)+(0,2,2)=(0,0,0)\\
&\Longrightarrow (3a,3b,3c)+(2,-4,1)=(0,0,0)\\
&\Longrightarrow (3a,3b,3c)=(0,0,0)-(2,-4,1)\\
&\Longrightarrow (3a,3b,3c)=(-2,4,-1).
\end{align*}
Hence, $a=-\frac{2}{3}$, $b=\frac{4}{3}$, $c=-\frac{1}{3}$ and $x=\left( -\frac{2}{3},\frac{4}{3}, -\frac{1}{3}\right)$.
