The computation of the fundamental matrix given a set of point correspondences between two images has been the critical point of research for decades. The fundamental matrix should be of rank two for all the epipolar lines to intersect in a unique epipole. Traditional methods of enforcing the rank two property of the matrix are to parameterize the fundamental matrix during the estimation. This usually results in a system of nonlinear multivariable equations of higher degree and it is hard to solve. This paper presents an effective method to solve the typical nonlinear multivariable equations encountered in the fundamental matrix estimation with rank constraint. The method is based on the classical Lagrange multipliers method. After careful transformations of the problem, we reduce the solution of multivariable nonlinear equations to the solution of some single variable equations.