We present a computational approach for constructing Sylvester style resultants for sparse systems of bivariate polynomial equations. Necessary and sufficient conditions are derived which guarantee that a multiplying set of monomials generates an exact Sylvester style resultant for three bivariate polynomials with a given planar Newton polygon. These conditions include a set of Diophantine equations that can be solved to generate multiplying sets of monomials and therefore the corresponding Sylvester resultants. We have implemented this method in Mathematica, and the results show that such Sylvester style sparse resultants often exist, and they appear in certain specific patterns. This method of Diophantine equations can also be used together with moving planes and moving quadrics [16, 17] to find the implicit equation of a rational surface. Moving planes and moving quadrics were originally introduced for tensor product surfaces---that is, bivariate polynomial systems whose Newton polygons are rectangles. Now by a method similar to our technique for generating Sylvester style sparse resultants, we can use moving quadrics to generate implicit equations for certain rational parametric surfaces whose Newton polygons are not rectangles.