Monte Carlo computer simulations have been performed on a three-dimensional simple-cubic lattice for mixtures of linear and branched chains with monomeric solvent molecules. Mixtures with chains of length 10 and of length 40 have been considered for both linear and branched architectures at a variety of polymer volume fractions. For the internal energy of mixing of athermal chains with solvent, the lattice cluster theory (LCT) is found to give predictions in reasonably good agreement with the results of the simulations, especially for short chains. At lower temperatures, the theory is not as satisfactory. However, it does prove to be more accurate for branched polymer with solvent than for linear polymer with solvent. The origins of the changes in the internal energy of mixing are understood in terms of additional internal contacts made possible by the presence of the branches. The effects of branching on the radii of gyration are also presented.