Algebraic algorithms are powerful methods in solving the selective harmonic elimination (SHE) problem, which can find all exact solutions without the requirements of choosing initial values. However, the huge computational burden and long solving time limit the solving capability of algebraic algorithms. This article presents a novel Newton's identifies-based method to simplify the SHE equations including the order reduction and the variable elimination, thereby reducing the computational burden and the solving time of algebraic algorithms or in other words improving the solving capability of the algebraic algorithms. Compared with existing simplification methods, the proposed method significantly improves the efficiency of solving SHE equations. With the proposed method, the degree of reduction is no longer the bottleneck of solving the SHE equations by using algebraic algorithms. By using the proposed method, the SHE equations with ten switching angles are completely solved with the algebraic algorithm for the first time. The simulation and experimental results indicate that the proposed method is effective and correct.