Characterizing the exact repair storage-vs-repair bandwidth tradeoff for distributed storage systems remains an open problem for more than four storage nodes. Motivated by the prevalence and practical applicability of linear codes, the exact repair problem when restricted to linear codes is considered. The main result of this paper is a new approach to develop bounds for exact repair distributed storage systems with linear codes (LDSS). Using this approach, the exact repair region for the (k + 1, k, k) LDSS is completely characterized. The new approach utilizes the properties of linear codes together with the exact repair constraints. These constraints are formally captured through an optimization problem with a recursive structure, and its solution finally yields the new bounds for the LDSS. These bounds together with recent code constructions characterize the exact repair region for (k + 1, k, k) LDSS.