Iterative differential approximation methods that rely upon backpropagation have enabled the optimization of neural networks; however, at present, they remain computationally expensive, especially when training models at scale. In this paper, we propose a computationally efficient alternative for optimizing neural networks that can both reduce the costs of scaling neural networks and provide high-efficiency optimizations for low-resource applications. We derive an explicit solution to a simple feed-forward language model (LM) by mathematically analyzing its gradients. This solution generalizes from single-layer LMs to the class of all single-layer feed-forward softmax-activated neural models trained on positive-valued features, as is demonstrated by our extension of this solution application to MNIST digit classification. For both LM and digit classifiers, we find computationally that explicit solutions perform near-optimality in experiments showing that 1) iterative optimization only marginally improves the explicit solution parameters and 2) randomly initialized parameters iteratively optimize towards the explicit solution. We also preliminarily apply the explicit solution locally by layer in multi-layer networks and discuss how the solution's computational savings increase with model complexity -- for both single- and mult-layer applications of the explicit solution, we emphasize that the optima achieved cannot be reached by backpropagation alone, i.e., better optima appear discoverable only after explicit solutions are applied. Finally, we discuss the solution's computational savings alongside its impact on model interpretability and suggest future directions for the derivation of explicit solutions to complex- and multi-layer architectures.