In this work, we establish the $R$-linear convergence rate of the inertial extragradient method for solving strongly pseudo-monotone equilibrium problems with a new self adaptive step-size. The linear convergence rate of the proposed methods is obtained without the prior knowledge of the Lipschitz-type constants of the bifunction. We also discuss the application of the obtained results to variational inequality problems involving strongly pseudomonotone and Lipschitz continuous mapping.