Numerous applications in operations research and computer science require a combination of prediction and optimization -- use historical data to predict the parameters of an optimization problem, and solve the optimization problem to output a decision. Addressing these two challenges independently results in the \emph{predict-then-optimize} problem. This approach can result in discrepancies between the prediction error, minimized during training, and the ultimate objective of minimizing the decision error. Consequently, recent work has focused on the \emph{predict and optimize} (PO) framework which focuses on training an end-to-end model from the data to the decisions. We focus on linear programs (LPs) within the PO framework where the main challenge is handling the non-differentiability of LPs. For a linear prediction model, we present a novel reduction from PO to a convex feasibility problem. This reduction enables us to use alternating projections onto convex sets for solving the PO problem, resulting in a computationally-efficient and theoretically principled algorithm. Finally, we validate the effectiveness of our approach on synthetic shortest path and fractional knapsack problems, demonstrating improved performance compared to the prior work.