This paper presents an alternative approach of teaching systems of two linear equations to the students in Tanzanian ordinary secondary schools. I conducted a desk-based research on methods that have been used by in-service mathematics teachers for teaching systems of two linear equations in ordinary secondary schools. It was found that five common methods (substitution, elimination, graphical, inverse matrix, Cramer's rule) have been used for teaching a system of two linear equations and all methods yield the same answer regardless of having different ways of approaching the system. I realized that a crossing method (alternative approach) is not found in the literature and yet not used by teachers for teaching students a system of two linear equations. But this crossing method yields similar answers with that resulted when using the five common methods. I present this alternative method in this paper by comparing with the answers obtained using five methods while focusing on two systems of two linear equations. This new approach has implications in teaching, learning and solving systems of two linear equations in ordinary secondary schools, including mathematics teachers and educators can use this method for teaching students in solving systems of two linear equations.