jmc

We first find the $y$-coordinate of the intersection point by substituting $x=-8.4$ into the second equation. This gives us $0.5(-8.4)+y=14$, so $y=14-(0.5)(-8.4)=14-(-4.2)=14+4.2=18.2$. Substituting $x=-8.4$ and $y=18.2$ into the first equation gives $$k=-2x+y=-2(-8.4)+18.2=16.8+18.2=\boxed{35}.$$

A faster way to solve this problem is to eliminate $y$ by subtracting the first equation from the second. This gives us $0.5x-(-2x)=14-k$, so $2.5x=14-k$. When $x=-8.4$, this gives us $14-k=2.5(-8.4)=-21$, and solving this equation gives $k=\boxed{35}$.