Hmm. I may have started you down a path that's more work. Try eliminating the trig functions. How, might you ask? By solving each equation for the trig function, squaring everything in sight, adding one to the other to get 1. So, for example,

$\displaystyle \cos(\theta)=\frac{1}{800}\left(S\cos(80^{\circ})-70\cos(100^{\circ})\right)$

$\displaystyle \sin(\theta)=\dots$

Then, set $\displaystyle \cos^{2}(\theta)+\sin^{2}(\theta)=1$,

and solve for S. I think that might be easier.