The calculable R-matrix method has been shown to be an efficient method for describing scattering states from nuclear interactions. The method has been applied with success to calculate solutions to the Schrodinger equation in two body, single channel scattering reactions. The purpose of this study is to extend the R-matrix method to calculate solutions to a non-local, coupled channels reaction. Such a method will calculate wavefunctions using the local and non-local interactions within each channel, and also include coupling potentials between the channels. We use the Woods-Saxon and Perey-Buck potential models for the local and non-local potentials. The Lagrange-mesh method is chosen to compute solutions to the scattering problem. The R-matrix method can be successfully applied to solve for the wavefunction of the projectile in each channel, although calculation of the solution is more computationally intensive than a single channel reaction. The R-matrix method is an effective approach to calculating solutions for coupled channel reactions. While this study focused on a two coupled channels case, in theory the method could be applied to more channels, at the cost of more computation time with each additional channel.