The fondamental problem of determining the phase equilibria of binary mixtures is discussed in the context of two-component phospholipid bilayer membranes of saturated phospholipids with different acyl-chain lengths. Results are presented from mean-field calculations and Monte Carlo simulations on a statistical mechanical model in which the interaction between lipid acyl chains of different length is formulated in terms of a hydrophobic mismatch. The model permits a series of binary phase diagrams to be determined in terms of a single "universal" interaction parameter. The part of the free energy necessary to derive phase equilibria is determined from the simulations using distribution functions and histogram techniques, and the nature of the phase equilibria is determined by a finite-size scaling analysis which also permits the interfacial tension to be derived. Results are also presented for the enthalpy and the compositional fluctuations. It is shown, in accordance with experiments, that the nonideal mixing of lipid species due to mismatch in the hydrophobic lengths leads to a progressively nonideal mixing behavior as the chain-length difference is increased. Moreover, indications are found that a phase transition in a strict thermodynamic sense may be absent in some of the short-chain one-component lipid bilayers, but a transition can be induced when small amounts of another species are mixed in, leading to a closed phase separation loop with critical points. The physical mechanism of inducing the transition is discussed in terms of the molecular properties of the lipid acyl chains. The results of the numerical model study are expected to have consequences for the interpretation of experimental measurements on lipid bilayer systems in terms of phase diagrams.