Models of the equation of state (EOS) type, used for describing fluid phase equilibria, typically assume that the system behavior can be described from binary contributions only. This is not enough for obtaining good predictions for the high-pressure phase equilibria of ternary highly non ideal systems. On the other hand, cubic mixing rules (CMRs) provide both, binary and ternary interaction parameters. Such ternary parameters have the potential of improving the reproduction of ternary experimental data while leaving invariant the description of the three constituent binary subsystems. In this work, we evaluate the possibilities of the CMRs approach for ternary systems by studying the phase equilibria of two of them, i.e., CO2 + n-hexadecane + 1,8-octanediol and CO2 + H2O + 2-propanol. For the system CO2+ n-hexadecane + 1,8-octanediol, we have available both, experimental isothermal binary and ternary phase equilibrium data. For such system, we investigate the influence of the ternary parameters on the size of ternary two-phase and three-phase equilibrium regions, and compare it to that of the experimental data. We also consider predicted values of the ternary parameters, which we need when ternary experimental data are not available. For the system CO2 + H2O + 2-propanol, we examine the effect of the ternary interaction parameters on some calculated ternary univariant lines, ternary invariant points and ternary three-phase equilibria, over wide ranges of conditions. In this study, we use the Peng-Robinson (PR) EOS and Soave-Redlich-Kwong (SRK) EOS, both coupled to CMRs. Our results make evident the flexibility and improvement that can be gained in the description of ternary phase equilibria by resorting to ternary interaction parameters. (C) 2015 Elsevier B.V. All rights reserved.