Quotient of polynomial ring is a localization of a polynomial ring

Let , where is an algebraically closed field. The goal is to show that some ring obtained by inverting certain elements of is isomorphic to a polynomial ring, again with certain elements inverted. This will show that the quotient field of is isomorphic to the field of rational functions in some (should be 2) variables (which is the ACTUAL goal).

My first thought was to consider the ring , where the relationship can be rewritten as . Then do a change of variables, say , and then solve for . That relation should then let me define a map to some localization of .

I could be doing it right, but the equations seem to get way too complicated to check whether or not I have an isomorphism. Anyone have an idea?

Re: Quotient of polynomial ring is a localization of a polynomial ring