Answer
$x=\left( \frac{5}{2},-6, 0\right)$.
Work Step by Step
Assume that $x=(a,b,c)$ and $5 {u}-2 {x}=3 {v}+{w}$, then we have
\begin{align*}
&\Longrightarrow 5 (1,-1,2) -2 (a,b,c)=3(0,2,3)+(0,1,1)\\
&\Longrightarrow (5,-5,10) + (-2a,-2b,-2c)=(0,6,9)+(0,1,1)\\
&\Longrightarrow (5,-5,10) + (-2a,-2b,-2c)=(0,7,10)\\
&\Longrightarrow (-2a,-2b,-2c)=(0,7,10)-(5,-5,10)\\
&\Longrightarrow (-2a,-2b,-2c)=(-5,12,0).
\end{align*}
Hence, $a=\frac{5}{2}$, $b=-6$, $c=0$ and $x=\left( \frac{5}{2},-6, 0\right)$.
