Dixon's method for computing multivariate resultants by simultaneously eliminating many variables is reviewed. The method is found to be quite restrictive because often the Dixon matrix is singular, and the Dixon resultant vanishes identically yielding no information about solutions for many algebraic and geometry problems. We extend Dixon's method for the case when the Dixon matrix is singular, but satisfies a condition. An efficient algorithm is developed based on the proposed extension for extracting conditions for the existence of affine solutions of a finite set of polynomials.Using this algorithm, numerous geometric and algebraic identities are derived for examples which appear intractable with other techniques of triangulation such as the successive resultant method, the Groebner basis method, Macaulay resultants and Characteristic set method. Experimental results suggest that the resultant of a set of polynomials which are symmetric in the variables is relatively easier to compute using the extended Dixon's method.