Multiplying the four equations by $12,6,4,3$ respectively, we get $$\begin{aligned} 24 x+12 y+12 z+12 w & =12 \\ 6 x+18 y+6 z+6 w & =12 \\ 4 x+4 y+16 z+4 w & =12 \\ 3 x+3 y+3 z+15 w & =75 \end{aligned}$$ Adding these yields $37 x+37 y+37 z+37 w=111$, or $x+y+z+w=3$. Subtract this from the fourth given equation to obtain $4 w=22$, or $w=11 / 2$.