This work presents analysis of the simplex method, a fundamental algorithm in the field of linear programming. The introductory part delves into the historical development of the simplex method, emphasizing its significance and contribution to the evolution of this mathematical discipline. A detailed explanation of the mathematical framework of linear optimization problems follows, including definitions of objective functions, constraints, and the structure of linear programs themselves. Particular attention is given to the concept of convexity and its crucial role in the application of the simplex method. Convex sets and convex functions are defined, and the convexity of linear functions is mathematically proven. The work gradually unveils the mechanisms of the simplex method, starting from the graphical method, moving through key elements like the simplex tableau and the pivoting technique, and arriving at the solution of minimization problems. Beyond the basic principles, advanced topics are addressed, such as the Charnes M method, which extends the application of the simplex method to problems with an infeasible initial solution, and sensitivity analysis, which allows for the evaluation of the impact of changes in input parameters on the optimal solution. Finally, the practical relevance of the simplex method is illustrated through its application to a concrete engineering problem, where the solution is obtained using Excel Solver.