Actually, I was working on a physics problem, but the question which I had was a kind of math question. So, I encountered the following problem. While solving the Schrodinger equation for some potential and using the boundary conditions for the solutions such that I had 2 equations which 4 unknowns $A,B,k$ and $q$- $$A+B=Ae^{i(k-q)L}+Be^{-i(k+q)L}$$ $$ik(A-B)-ik(Ae^{i(k-q)L}-Be^{-i(k+q)L})=\frac{g_0}{L}(A+B)$$ where $g_0$ and $L$ are constants.
Now, is there any way to solve these equations for some family of solutions? Also, why would that method work? Any sort of help is appreciated.
One way is to solve the first for $A$ to get $$ A = B \frac{e^{-i(k+q)L}-1}{1-e^{-i(k-q)L}} $$ and plug that into the second one...