Carlton Lemke presents a constructive proof for the existence of solutions to the system of equations \( Mz - w = q \), \( z \geq 0 \), \( w \geq 0 \), and \( z^Tw = 0 \). This system encompasses both quadratic programming problems and the problem of finding equilibrium points in bimatrix games. The proof does not require reducing a functional but instead generates an adjacent extreme-point path leading to a solution, assuming non-degeneracy. The technique combines the concepts of non-degeneracy and extreme-point paths, generating a path that terminates in an equilibrium point. Lemke also discusses the computational aspects and compares his method with those of Dantzig and Cottle, extending the results to various types of problems, including linear programming. The paper provides a general algorithm for solving these problems and extends the class of problems for which an adjacent extreme-point path scheme can lead to a solution.Carlton Lemke presents a constructive proof for the existence of solutions to the system of equations \( Mz - w = q \), \( z \geq 0 \), \( w \geq 0 \), and \( z^Tw = 0 \). This system encompasses both quadratic programming problems and the problem of finding equilibrium points in bimatrix games. The proof does not require reducing a functional but instead generates an adjacent extreme-point path leading to a solution, assuming non-degeneracy. The technique combines the concepts of non-degeneracy and extreme-point paths, generating a path that terminates in an equilibrium point. Lemke also discusses the computational aspects and compares his method with those of Dantzig and Cottle, extending the results to various types of problems, including linear programming. The paper provides a general algorithm for solving these problems and extends the class of problems for which an adjacent extreme-point path scheme can lead to a solution.