A crucial question within the fields of origins of life and metabolic networks is whether or not a self-replicating chemical reaction system is able to persist in the presence of side reactions. Due to the strong nonlinear effects involved in such systems, they are often difficult to study analytically. There are however certain conditions that allow for a wide range of these reaction systems to be well described by a set of linear ordinary differential equations. In this article, we elucidate these conditions and present a method to construct and solve such equations. For those linear self-replicating systems, we quantitatively find that the growth rate of the system is simply proportional to the sum of all the rate constants of the reactions that constitute the system (but is nontrivially determined by the relative values). We also give quantitative descriptions of how strongly side reactions need to be coupled with the system in order to completely disrupt the system.