For looking at the coefficients, you probably want to use
My notation might be off here. But collect distributed writes it as a sum of coefficients times monomials in y,z. Then coeffs will give you the coefficients, and store the corresponding monomials in an optional third argument.
Your 3 equations probably have no common solution for all paramaters. Any 2 of the equations should have a common solution (probably 4 of them). Probably solve will find them. You could solve on say the first two equations and the last two equations. This should give 4 solutions (each) in terms of your paramaters. You could then equate your parameters.