The restricted four-body problem is used to derive linearized equations of relative motion that take into account the perturbing effects of a larger, secondary gravitational source. The result is a system of coupled, first-order, linear differential equations that has a complete analytical solution. In this paper, we numerically integrate these linearized equations and compare the results to the outcomes of the restricted four-body problem and the well known Hill-Clohessy-Wiltshire equations. Various cases pertaining to two scenarios are analyzed: the relative motion of a chase spacecraft with respect to a target satellite orbiting an asteroid while both are perturbed by the Sun, and the relative motion of a chase spacecraft with respect to a target satellite orbiting the Moon while both are perturbed by the Earth. The results demonstrate that the Benavides-Spencer formulation is far more accurate than the results given by the Hill-Clohessy-Wiltshire equations when compared to the real-life outcomes returned by the numerical integration of the restricted four-body problem. Future work will unveil the complete analytical solution of the Benavides-Spencer formulation both as an initial value problem and a boundary value problem.