The search for a more efficient and robust numerical method for solving problems have become an interesting area for many researchers as most problems resulting into nonlinear system of equations would require a very good numerical method for its computation. The introduction of the Broyden method has served as the foundation to developing several others, which are referred to as Broyden – like methods by some authors. These methods, in most cases, have proven to be superior to the original classical Broyden method in terms of the number of iterations needed and the CPU time required to reach a solution. This research sought to develop new Broyden – like methods using weighted combinations of quadrature rules (i.e., Simpson -1/3 and Simpson -3/8 rules against Midpoint, Trapezoidal, and Simpson quadrature rules). The weighted combination of the quadrature rules in the development of the new methods led to the discovery of several new methods. Some of which have proven to be more efficient and robust when compared with some existing methods. A comparison of these newly developed methods with the classical Broyden method together with some existing improved Broyden method revealed that, one of the newly developed methods namely, Midpoint–Simpson-3/8 (MS–3/8) method, outperformed all the others, with the MS–3/8 method giving the best of numerical results in all the benchmark problems considered in the study.