Controllability and stabilization of coupled reaction-diffusion systems has been widely studied in the last decades. In recent years, coupling reaction-diffusion systems to elliptic equations has been dragging attention. These systems arise as a simplified model for coupled reaction-diffusion systems when at least one diffusion coefficient is high. In this article, we study the eigenvalues of a coupled reaction-diffusion system with one high diffusion. As a consequence, when this system is unstable we are able to build a Backstepping-based controller to stabilize it. This is achieved due to the singular perturbation method (SPM), which provides a methodology to study the behaviour of the eigenvalues with respect to the singular perturbation parameter. Additionally, the SPM helps in building controller based on the reduced models, which are used to stabilize the coupled system when one diffusion is high enough.