Heckscher-Ohlin with different technologies
As the companies doing the production will minimize costs in equilibrium
$$
\begin{align*}
|MRTS_c(K_c,L_c)| &amp; = r/w \\
\\
|MRTS_w(K_w,L_w)| &amp; = r/w
\end{align*}
$$
will hold. (Or you can also use the general form of your equations (1),(2),(5) and (6) to get this.) Given your particular production functions the marginal rates of technical substitution are
$$
\begin{align*}
|MRTS_c(K_c,L_c)| &amp; = \frac{L_c}{K_c} \\
\\
|MRTS_w(K_w,L_w)| &amp; = \frac{L_w}{K_w}.
\end{align*}
$$
Taken together, this means
$$
\frac{L_c}{K_c} = r/w = \frac{L_w}{K_w}.
$$
It follows that
$$
L_c + L_w = (K_c + K_w) \cdot r/w.
$$
The total amount of Labor and Capital are given in your exercise, and these will determine the factor price ratios as well as the $L_c/K_c = L_w/K_w$ ratio. Once you plug those in, your equation system will become linear.

Notice that this is true both in autarky and in free trade equilibrium, we have not used demand functions so far at all. Setting the appropriate demand equal to the appropriate production will give you the remaining equations you need to determine the equilibrium price ratio.