uby said:
any two pieces of information should be solvable so long as their combinations cannot yield all three unknowns
Actually, that's not true. To take a simpler example, consider the equation
x = y - 1
In this system, it's impossible to solve for the ratio x/y - or rather, you can solve for it, but not in terms of constants only, because the ratio depends on which (x,y) point you choose. You can see this by taking a few sample points: at (3,4) the ratio is 0.75, at (9,10) it's 0.9, and so on.
What the concept of degrees of freedom (DOF) really tells you is that if you have an underdetermined system, the dimensionality of the solution space is equal to the number of variables minus the number of constraints. (At least for "normal" e.g. algebraic systems) In your example, you have 3 variables and 2 unknowns, thus the space of solutions to your system will be a 1-dimensional object, namely a line or curve. If you created a 3D plot of (x,y,z) points that satisfy your equations, you should be able to see that.
Now, if you think about it, hopefully you can see that in order for you to solve for the ratios you're looking for (in terms of A,B,K1,K2 only), the values of those ratios would have to be constant along the entire curve. But they're not. Again, if you had a 3D plot of the solution space, you should be able to calculate x/y and z/(x+y) at various points in that space and you would find an assortment of different values.
Generally speaking, you can't pick any arbitrary function of x, y, and z (such as one of your ratios) and expect it to be constant throughout the solution space (the 1D curve). In fact, most functions of x,y,z will
not be constant throughout the solution space. Finding a function that
is constant is actually an interesting problem of its own. This sort of problem is quite relevant in physics, and it sometimes takes some rather advanced tools to solve it, e.g.
Noether's Theorem.
In principle, I think you could use Noether's Theorem to find quantities that are constant throughout the solution space of your system, although it may not be necessary. One thing you could try is to just combine the two equations by solving one equation for one variable and substituting into the other. In the resulting equation, make sure that the left side contains only constants, and then whatever appears on the right side should give you a quantity that will be constant throughout your solution space. It may be more complicated than a simple ratio, though.