In a first order monomolecular reversible reaction system, the time evolution of molar concentration of the reacting species in a batch reactor is governed by linear ordinary differential equations. In this work, a flow graph theory approach is considered to derive the analytical solution for the kinetic equations of two and three species reacting systems. The flow graph is based on the image of reaction stoichiometry and utilizes Cramer's method of determinants to find an analytical solution for the chemically reacting system. The exact solutions derived for the reversible reaction systems through the flow graph theory approach are consistent with the reported analytical solutions obtained through Laplace transforms.