Liquid-vapor critical point for a thermodynamic mixture is an important parameter for the design of mixture transportation and processing. The classical critical points model for the multicomponent mixture is based on the tangent plane distance in terms of the Helmholtz energy. In this work, we have applied the damped Newton-Raphson scheme to solve the model using the critical temperature, critical specific volume and change in mole fraction of each component as independent variables. According to numerical experiments, the convergence criteria, initial guesses of variables and damped factor formulation were determined. The method was tested on twenty five mixtures containing two to eleven components using the SRK and PR equation of state. Comparisons between the calculated critical parameters and experimental values illustrated the overall accuracy of the solution method is satisfactory. This method can solve all the variables simultaneously using the whole iteration, which is more reliable than the nested two-singe variable loops method proposed by Heidemann and Khalil. Thus it can be used for practical purposes of locating the liquid-vapor critical points of multi-component mixtures.