An explicit analytical solution is derived and illustrated for multidimensional advective-dispersive-reactive transport in systems with three distinct velocities coupled by first-order interactions or reactions. The solution is an exponentially scaled space-time convolution of an advective-dispersive-reactive kernel with a coupled advective transport kernel. Explicit forms are provided for both kernels. The solution process for the coupled advective kernel uses one-sided two-dimensional Laplace transforms and introduces two constraints into the transport problem which are to be relaxed in future work. The behavior of the solution is illustrated with a third-order approximation of single species transport in a heterogeneous aquifer. Results indicate that the early time behavior of the third-order system is that of noninteracting advective-dispersive-reactive transport. The large time mean behavior is that of a single advective-dispersive-reactive transport equation with constant effective parameters. Results further demonstrate that the third-order approximation provides accurate predictions of the mean and uncertainty of concentration distributions calculated from detailed two-dimensional transport simulations at all times.