The ability of cubic equations of state to predict the phase behavior of the Lennard-Jones pure fluid and asymmetric binary mixtures obeying the Lorentz-Berthelot combining rules is examined. It is shown that the commonly used Soave-Redlich-Kwong and Peng-Robinson equations of state give good vapor pressures but incorrect saturated liquid densities for the pure fluid. A modification of the Peng-Robinson equation of state that uses a volume translation term to describe accurately the saturated liquid densities is proposed. For the mixture calculations, we use the van der Waals one-fluid conformal solution mixing rules. All the cubic equations of state tested give excellent pressure-composition phase diagrams for the binary mixtures. Predictions of pressure-density phase envelopes are less satisfactory, but equivalent to those of other more complex equations of state previously developed for the Lennard-Jones fluid. Our results suggest that cubic equations of state can be used to model mixtures of significantly different compounds provided that appropriate combining and mixing rules are used.