This concise but detailed and thorough treatment discusses the rudiments of the well-known simplex method for solving optimization problems in linear programming. Geared toward undergraduate students, the approach offers sufficient material for readers without a strong background in linear algebra. Many different kinds of problems further enrich the presentation. The text begins with examinations of the allocation problem, matrix notation for dual problems, feasibility, and theorems on duality and existence. Subsequent chapters address convex sets and boundedness, the prepared problem and boundedness and consistency, optimal points and motivation of the simplex method, and the simplex method and tableaux. The treatment concludes with explorations of the effectiveness of the simplex method and the solution of the dual problem. Two helpful Appendixes offer supplementary material.