An interesting connection is found between the coupled-reaction-channel (CRC) approach and the triad of Lippmann-Schwinger (LS) equations for the three-particle scattering state. It turns out that the CRC decomposition ansatz can be used as a vehicle to couple the three uncoupled LS equations. The possibility of using CRC decomposition as a means of coupling the three LS equations does not seem to have been noticed before. This finding allows one to introduce a set of coupled equations for CRC rearrangement operators in which two-particle resolvents appear naturally as the propagator term. The formal operator equations are then reduced to effective two-body form by employing two different separable expansions of the two-particle resolvent. The two versions of the ensuing CRC effective two-body equations are applied to a benchmark model of n+d scattering. Although the two CRC versions differ considerably in the way breakup channel manifests itself in the effective two-body equations, both sets of results for elastic and breakup amplitudes of the benchmark model are almost the same and are in good agreement with the results available in the literature. This study further corroborates that the CRC approach is an alternative versatile tool for three-particle dynamics and is on par with approaches based on Faddeev theory.