Linear Algebra can solve for a,b,d, and e if a solution exists. It doesn't do anything regarding x and y.
You can not solve u and v through that at all.

Let me try explaining it a different way:
Suppose the system of equations
au + bv = c
du + ev = f
has a unique solution (u,v) = (m,n). (As is typical with 2 equations and 2 unknowns)
Then, if you were asked to solve the system of equations
a*f(x) + b*g(y) = c
d*f(x) + e*g(y) = f
for x and y.
The previous work allows you to immediately reduce this problem to solving
f(x) = m
g(y) = n