This paper introduces an optimal methodology for applying Gaussian elimination to fuzzy systems of linear equations, emphasizing the synergy between classical numerical techniques and fuzzy logic. We commence with essential definitions related to linear equations and matrices, advancing to the formulation of augmented matrices and the execution of Gaussian elimination to attain reduced row echelon form. The discourse encompasses the notions of leading and free variables, row equivalence, and the rank theorem, thereby establishing a thorough framework for comprehending linear systems. We address the distinctive challenges that fuzzy numbers present in linear equations, proposing a polynomial parametric representation for fuzzy systems. Through comprehensive examples, we illustrate the efficacy of Gaussian elimination in resolving these systems, detailing the process of obtaining solutions for both leading and free variables. The outcomes highlight the complex interconnections between parameters and solutions, underscoring the method's versatility in optimization scenarios. Our results emphasize the significance of Gaussian elimination not only within conventional linear algebra but also in the context of fuzzy systems, thereby opening avenues for further exploration in numerical analysis and its applications in uncertain environments.