In Lennard-Jones (LJ) potential argon system investigated by molecular dynamics simulation, the different cutoff distances are frequently applied to calculate the interactive force between atoms, and some reports have given out the influence of the cutoff distances on the simulation systems. More and more calculations suggest to use 4.5? or even larger truncation distances (? is the diameter of argon atom) to obtain the more accurate thermodynamic properties of the systems. In this work, a simple method was proposed to solve the problem of superheating encountered in the calculation of melting point by direct heating, where an independent track and ensemble at each temperature point are run. And then, the effect of different cutoff distances on the phase diagrams of the melting and boiling points of argon system in the NPT ensemble was studied. The melting point was in good agreement with the experimental and theoretically calculated values when the cutoff distance of 2.5? was used. However, the deviation from the experimental melting point became more evident when the larger cutoff distances were employed. In order to find out the underlying mechanism behind the deviation, the radial distribution functions and velocity autocorrelation function at the melting points and different thermodynamic states of the liquid argon with different cutoff distances were analyzed. It was found that the same thermodynamic properties can be obtained at the corresponding thermodynamic state points under different truncation distances. The mapping between the different thermodynamic state points was understandable due to the different thermodynamic states at the same temperatures under the varied truncation distances, and was beneficial to significantly reduce the computational workload at the smaller cutoff distance. This work proposed an exploratory way for the selection of the cutoff distance in the simulation of liquid argon, where the truncation distance of 2.5? can meet the requirements of computational accuracy and performance in the simulations.