This thesis discusses the basic problems of solving a linear programming problem. A definition of the linear programming problem is stated.Basic linear algebra methods are necessary to solve a linear programming (LP) problem. A short synopsis of the necessary methods used are presented, including examples.The simplex algorithm is introduced and an example is used to demonstrate its applicability. A computer program is presented which utilizes the basic simplex algorithm and the extended tableau form.The problem of degeneracy as it affects the simplex method is explained. Research of more efficient programs are always in progress. The power of the simplex mehtod is evident in the business applications area.