Dixon's resultant method is an efficient way of simultaneously eliminating several variables from a system of nonlinear polynomial equations at a time. However, the method only works for systems of n + 1 generic n-degree polynomials in n variables and does not work for most algebraic and geometric problems. In this paper, by using techniques from pseudoinverse theory and linear transformations, the author extends Dixon's resultant method to an arbitrary system of n + 1 nontrivial polynomials in n variables where the Dixon matrix can be singular or even nonsquare. The extended method does not require any precondition — this is the main contribution of the paper. The extended method can be used efficiently as an elimination method in geometric reasoning, computer aided geometric design (CAGD) and solid modeling. Several examples show that the new method works well also in situations where other methods (of the same subject) may fail to give a correct answer.