The computational differences between two methods (monomial method and S-system method), which are actually mathematically the same and are both equivalent to Newton’s method operating on a transformed system of equations are described. It is shown that it is unclear at this time as to whether using the smaller but relatively dense monomial method linear system is more efficient than using the considerably larger, sparse linear system created using the S-system method. This paper further investigates the effectiveness of a combinatorial strategy for locating many, and sometime all, solutions to a system of nonlinear equations when applied to either of the two methods.