This paper discusses modifications to Newton's method for solving nonlinear simultaneous equations, aiming to reduce the number of function evaluations required. The author presents a class of methods based on quasi-Newton updates, which approximate the Jacobian matrix using corrections derived from function evaluations. These methods are designed to be more efficient than Newton's method, especially when the Jacobian matrix is difficult to compute. The paper includes numerical experiments to evaluate the performance of these methods, comparing their convergence rates and computational efficiency. The results suggest that the full-step norm-reducing variant of Method 1 and the basic method are generally superior, with the former showing better performance for mildly nonlinear problems and the latter being more robust when a good initial estimate is not available. The paper concludes with a discussion of the advantages and limitations of the proposed methods and their potential applications in practical problems.This paper discusses modifications to Newton's method for solving nonlinear simultaneous equations, aiming to reduce the number of function evaluations required. The author presents a class of methods based on quasi-Newton updates, which approximate the Jacobian matrix using corrections derived from function evaluations. These methods are designed to be more efficient than Newton's method, especially when the Jacobian matrix is difficult to compute. The paper includes numerical experiments to evaluate the performance of these methods, comparing their convergence rates and computational efficiency. The results suggest that the full-step norm-reducing variant of Method 1 and the basic method are generally superior, with the former showing better performance for mildly nonlinear problems and the latter being more robust when a good initial estimate is not available. The paper concludes with a discussion of the advantages and limitations of the proposed methods and their potential applications in practical problems.