We present a method for solving systems of nonlinear equations suitable to problems where convergence of an approximate Newton method is initially slow. The method, nonlinear elimination (NlEm), eliminates the nonlinear equations and appropriate variables deemed to be causing the problem. We give an analysis of the method. The analysis leads to a detailed algorithm which we show reduces automatically to approximate Newton method near the root of the system of equations. We conclude with several examples demonstrating the efficacy of the method.