Two-parameter equations of state (EOSs), such as the Peng-Robinson and Redlich-Kwong EOSs, or their variants, are widely used for the simulation of industrial processes. In this work, some algebraic properties of two-parameter cubic and noncubic equations of state and of their translated forms are investigated. For saturated pure compounds, translated and untranslated forms describe the same universal curves for pairs of certain dimensionless variables. In the present work we provide general parameters, not previously given in the literature, representing the universal relations set by the van der Waals (VdW), Redlich-Kwong (RK), and modified Carnahan-Starling-van der Waals (MCSV) equations of state. Direct calculation procedures, which make use of the pure compound universal relations, are proposed for the calculation of properties at infinite dilution, leading to the determination of regions of azeotropy. For mixing rules which do not depend on density, it has been found that the universal relations which are valid for saturated pure compounds are also valid for homogeneous azeotropes of any number of components. This finding can be exploited within any existing algorithm designed to find azeotropic points, in order to get faster and more reliable outputs. On the basis of the identity between pure compound and azeotropic universal relations, noniterative methods of calculation are proposed to establish the existence of simple or multiple azeotropy and of pseudocritical points. Simplified calculation procedures are also proposed to obtain values of binary parameters from experimental azeotropic information. All numerical examples have been done using the Peng-Robinson EOS (PR EOS).