The problem you are asking about is called elimination theory ( http://en.wikipedia.org/wiki/Elimination_theory), http://en.wikipedia.org/wiki/Elimination_theory ), which shows how to eliminate unknown variables x, y, ... from sets of polynomial equations. One of the major tools for doing this explicitly is the resultant ( http://en.wikipedia.org/wiki/Resultant) http://en.wikipedia.org/wiki/Resultant ) of 2 polynomials, that gives polynomial conditions on their coefficients for them to have a common root.
Carrying out these calculations is too messy to do by hand except for small examples, though some computer algebra systems are now quite good at it.
