The service rate region (SRR) has emerged as a critical performance metric for distributed systems that store data redundantly. It measures the system's ability to serve multiple users concurrently. Mathematically, the SRR is a polytope in R^k where each dimension corresponds to the service request rate of one of the k data objects. This paper focuses on systems employing a class of Maximum Distance Separable (MDS) codes. For each code in the class, we characterize the k axes intercept points of its SRR, and the smallest standard simplex that includes the SRR. We use these results to show that the SRR grows with the increasing number of systematic columns in the generator matrices. We establish a graph-theoretic framework associating this SRR problem with fractional matchings in quasi-uniform hypergraphs. Identifying the SRR polytope is equivalent to determining a particular image of the fractional-matching polytope. We introduce a notion of Greedy Matching and show that it is sufficient to focus on these matchings to characterize the SRR rather than the entire matching polytope. With these tools, we determine the SRR of a large subset of the considered class of codes. Our results generalize previous characterizations of systematic and non-systematic MDS-coded systems, offering a unified framework for analyzing service rate regions of codes.