In this paper, we provide a method to solve the Cauchy problem of systems of quasi-linear parabolic equations, such systems can be transformed to the systems of linear parabolic equations with variable coefficients via the hodograph transformations. Our approach to solve the linear systems with variable coefficients is to use their fundamental solutions, which are constructed by using the Lie's symmetry method. In consequence, we can derive explicit solutions to the Cauchy problem of the quasi-linear systems in terms of the solutions of the linear systems and the hodograph transformations relating to the quasi-linear and the linear systems.