Quasi-steady state (QSS) reduction is a commonly used method to lower the dimension of a differential equation model of a chemical reaction network. From a mathematical perspective, QSS reduction is generally interpreted as a special type of singular perturbation reduction, but in many instances the correspondence is not worked out rigorously, and the QSS reduction may yield incorrect results. The present paper contains a thorough discussion of QSS reduction and its relation to singular perturbation reduction for the special, but important, case when the right-hand side of the differential equation is linear in the variables to be eliminated (but the differential equation model might otherwise be nonlinear). For this class we give necessary and sufficient conditions for a singular perturbation reduction (in the sense of Tikhonov and Fenichel) to exist, and to agree with QSS reduction. We then apply the general results to chemical reaction networks with noninteracting species, generalizing earlier results and methods for steady states to QSS scenarios. We provide easy-to-check graphical conditions to select parameter values for which the singular perturbation reduction applies, and additionally, we identify when the singular perturbation reduction agrees with the QSS reduction. Finally we consider a number of examples.