Answer
$w=(\frac{1}{3}, -\frac{8}{3},-\frac{16}{3},3)$.
Work Step by Step
Let $w$ be given by $w=(a,b,c,d)$
\begin{align*}
2u+v-3w &=2(0,0,-8,1)+(1,-8,0,7)-3(a,b,c,d)\\
&=(1-3a,-8-3b,-16-3c,9-3d)\\
&=(0,0,0,0)
\end{align*}
we get the system $$1-3a=0,\quad -8-3b=0,\quad -16-3c=0,\quad 9-3d=0.$$
By solving the above system we get the solution
$$a=\frac{1}{3}, \quad b= -\frac{8}{3}, \quad c=-\frac{16}{3}, \quad d=3.$$
Then, $w=(\frac{1}{3}, -\frac{8}{3},-\frac{16}{3},3)$.