The presentation of the simplex method, which solves linear programming (LP) problems, is not universal. In the U.S.A. instructors on the West coast enjoy solving the minimization problems, while in the East the maximization version is preferred. Even within each of these groups you will find differences in presenting the simplex rules. Consider an LP problem in which some constraints are in ( = ) or ( 2 ) forms with the right- hand side (R.H.S.) non-negative. For this type of problem the usual simplex algorithm (big M-method) requires additional variables (artificial variables) and introducing penalty terms in the objective function. Enough experience has been gained showing that some students, particularly non-mathematical majors, have difficulty in understanding the intuitive notion of this algorithm. This is the reason and the key factor in motivating some authors in trying to develop algorithms which do not involve any artificial variables and penalty terms. Recently, several new algorithms which generally avoid the use of artificial variables, for the sake of simplicity, appeared in some textbooks (see, for example, Refs 1, p. 302; 2, p. 2531 for such algorithms). Unfortunately, these algorithms are not for the general purpose of solving all types of LP problems and somehow mislead students. For example, in following all the steps given in these types of algorithms for the following problem: one never gets to the final tableau, in other words the algorithm never terminates. In Section 2 a new algorithm is presented which efficiently incorporates the regular and dual simplex algorithms. The strategy adapted for the new algorithm is summarized by the following two phases: Phase I. Phase II. Push toward a neighboring vertex of the optimal solution while trying to maintain feasibility. If pushed too far in Phase I, pull back toward the optimal vertex (if any).