A fundamental question in the study of stochastic biochemical reaction networks is what values of mean and variance of the species present in the network are obtainable by perturbing the system with an external input. Here, we propose a computationally efficient technique to answer this question, for networks involving zero and first order reactions. Specifically, we adopt the hyperplane method to compute inner and outer approximations of the reachable set of the linear system describing the moments evolution. A remarkable feature of this approach is that it allows one to easily compute projections of the reachable set for pairs of species of interest, without requiring the computation of the full reachable set, which can be prohibitive for large networks. To illustrate the benefits of this method we consider a standard controlled gene expression model involving two species: the mRNA and the corresponding protein. We verify that the proposed approach leads to estimates of the reachable set, for the protein mean and variance, that are more accurate than those available in the literature and that are consistent with experimental data.