Dislocation motion through a random alloy is impeded by its interactions with the compositional fluctuations intrinsic to the alloy, leading to strengthening. A recent theory predicts the strengthening as a function of the solute-dislocation interaction energies and composition. First-principles calculations of solute/dislocation interaction energies are computationally expensive, motivating simplified models. An elasticity model for the interaction reduces to the pressure field of the dislocation multiplied by the solute misfit volume. Here, the elasticity model is formulated and evaluated for cubic anisotropy in fcc metals, and compared to a previous isotropic model. The prediction using the isotropic model with Voigt-averaged elastic constants is shown to represent the full anisotropic results within a few percent, and so is the recommended approach for studying anisotropic alloys. Application of the elasticity model using accessible experimentally-measured properties and/or first-principles-computed properties is then discussed so as to guide use of the model for estimating strengths of existing and newly proposed alloys.