In this study, we explore an advanced stochastic approach tailored for solving large-scale systems of linear algebraic equations. The core of our research is the application and enhancement of the Monte Carlo method, specifically designed to tackle the computational challenges posed by such extensive systems. Our method builds upon the “Walk on Equations” Monte Carlo model, a technique recently developed by Ivan Dimov, Sylvain Maire, and Jean Michel Sellier. We introduce a novel hybrid algorithm that synergizes this model with traditional methods, aiming to optimize performance and efficiency. A key innovation in our approach is the introduction of the “dominancy number”, a critical parameter that significantly influences the algorithm’s convergence and accuracy. Through rigorous analysis and experimentation, we demonstrate the potential of this hybrid method in improving the computational feasibility of solving large-scale linear algebraic systems.