Solving two or three polynomials in two or three variables

An algorithm is described that allows two polynomials in two variables to be reduced to a single polynomial in one variable, and then back solved to get all sets of solutions. The algorithm works faster than most other algorithms within its range of utility (providing sets of solutions as large as 70), and can be extended to cover some sets of 3 polynomials in 3 variables (those where one of the polynomials has only two of the variables). While subject to the known stability problems of polynomial root-finding this algorithm can also be extended to provide for reducing to the single variable polynomial with symbols, thus permitting proof of results, and potential removal of extraneous roots. Such manipulation can then materially shorten the numerical process if the problem is to be applied to a number of different cases.