This paper presents an integrated approach to the numerical simulation of non-linear algebraic and transcendental equations using modified versions of the Bisection and Regula Falsi methods. The study aims to improve the efficiency and accuracy of root-finding algorithms for complex non-linear equations. We introduce modifications to the classical Bisection and Regula Falsi methods and compare their performance against traditional approaches. The research includes a comprehensive analysis of convergence rates, error margins, and computational complexity. Our findings demonstrate that the modified methods offer significant improvements in terms of speed and precision for a wide range of non-linear equations. The paper includes detailed tables presenting comparative results and discusses the implications of these findings for various fields of applied mathematics and engineering.