Answer
$w=(-1, \frac{5}{3},6, \frac{2}{3})$.
Work Step by Step
Let $w$ be given by $w=(a,b,c,d)$
\begin{align*} 2u+v-3w &=2(0,2,7,5)+(-3,1,4,-8)-3(a,b,c,d)\\ &=(-3-3a,5-3b,18-3c,2-3d)\\ &=(0,0,0,0) \end{align*}
we get the system
$$-3-3a=0,\quad 5-3b=0,\quad 18-3c=0,\quad 2-3d=0.$$ By solving the above system we get the solution
$$a=-1, \quad b= \frac{5}{3}, \quad c=6, \quad d=\frac{3}{2}.$$ Then, $w=(-1, \frac{5}{3},6, \frac{2}{3})$.
