AbstractBased on the root–coefficient relations for a cubic function, quadratic functions are constructed that strictly relate the saturated volumes of liquid and vapor phases and the third solution from a cubic equation of state (EoS). The vapor–liquid equilibrium (VLE) calculation with a cubic EoS is thus reduced to solving a single nonlinear equation. In light of a recent finding that the “unphysical” third solution, namely the Maxwell crossover or the M-line, plays a central role as the dividing interface in the density gradient theory, here we show that it can also be used to derive explicit approximations for a VLE problem. The van der Waals EoS and the Soave–Redlich–Kwong (SRK) EoS are discussed as examples. The method proposed in this work simplifies the calculations of the traditional VLE problem with a cubic EoS. With one-time-only effort for a given system, simple explicit approximations can be obtained to avoid the repetitively iterative computations for a VLE problem. Finally, the relationship between the Widom line in the supercritical region and the M-line is briefly discussed with the SRK EoS. Graphical Abstract