An iterative numerical method is used to study the quasisteady-state kinetics of systems with competitive first- and second-order reactions. We investigate the nonlinear dissociation-recombination kinetics for five Ar-O2 mixtures at total initial pressures of 0.1 and 1 atm as well as pure O2 at initial total pressures of 0.1, 1, and 10 atm. Three different data sets for the state-to-state energy transfer rate constants and the state-selected dissociation-recombination rate constants were employed. We find that in all cases where V-V energy transfer was neglected, the phenomenological rate law is obeyed with overall phenomenological rate coefficients that are linear functions of compositions, i.e., the linear mixture law holds. However, if V-V energy transfer is included, the linear mixture law breaks down although the deviations are found to be small relative to the experimental uncertainties. The total initial pressure has no effect on the validity of the linear mixture formula or on the role of V-V energy transfer. The linear mixture formula is also verified analytically by singular perturbation methods when the pseudo-first-order, time-dependent transition matrix can be factored into a time-dependent scalar multiplied by a time-independent matrix. In this case, the nonequilibrium internal-state distribution can be characterized by an invariant vector d in the quasisteady state, and the nonequilibrium state populations relax to equilibrium along straight-line paths. In general, however, straight-line paths and invariant vectors do not exist in the quasisteady state of competitive reaction systems. Nevertheless, for various Ar-O2 mixtures with similar ratios of [O]/[O2] and magnitudes of the nonequilibrium effect kd/kd e, we find that d is roughly independent of XAr, and we can exploit this invariance to predict the quasisteady nonequilibrium internal-state distributions and rate coefficients.