Too many equations. Can I eliminate one this way?

Hello all,

I have a big long system of equations I am trying to solve and I have more equations than unknowns... let me tell you how I got there. Lets say I am trying to solve for x, y, w/x and z/y, and right now I have 4 equations (we are fine right now, but x and y don't mean much in the context I am working in). Two of these four equations are...

y = a x + b z

x = c y + d w

I divide both sides of the top equation by y and both sides of the bottom equation by x. Now I have

1 = a (x/y) + b (z/y)

1 = c (y/x) + d (w/x)

Okay so now I am just trying to solve for x/y w/x and z/y (but I have four equations). There is no reason why I cant just set a (x/y) + b (z/y) equal to c (y/x) + d (w/x) to eliminate this extra equation right?

Cheers!

Nick