Answer
$x=\left( -\frac{2}{3},\frac{5}{3},\frac{1}{3}\right)$.
Work Step by Step
Assume that $x=(a,b,c)$ and $2 {u}+3 {x}=2 {v}-{w}$, then we have
\begin{align*}
&\Longrightarrow 2 (1,-1,2) +3 (a,b,c)=2(0,2,3)-(0,1,1)\\
&\Longrightarrow (2,-2,4) + (3a,3b,3c)=(0,4,6)+(0,-1,-1)\\
&\Longrightarrow (2,-2,4) + (3a,3b,3c)=(0,3,5)\\
&\Longrightarrow (3a,3b,3c)=(0,3,5)-(2,-2,4)\\
&\Longrightarrow (3a,3b,3c)=(-2,5,1).
\end{align*}
Hence, $a=-\frac{2}{3}$, $b=\frac{5}{3}$, $c=\frac{1}{3}$ and $x=\left( -\frac{2}{3},\frac{5}{3},\frac{1}{3}\right)$.
