A new decomposition method for solution of multi-component advection-dispersion-reaction equations coupled by first-order reactions is developed. This new method overcomes some of the limitations that were implicit in previously published algorithms. The approach is based on the Singular Value Decomposition (SVD) method. We derived analytical solution formulas, which can also be applied to systems with identical reaction rates. The proposed approach is flexible for solving one-, two- or three-dimensional advection-dispersion-reaction networks. The methodology is demonstrated on the reductive biodegradation of chlorinated solvents.