Load flow problems in electricity distribution systems (EDS) are generally solved through iterative methods such as Newton-Raphson, Gauss-Seidel, Forward-backward Sweep-based methods, etc. These methods are subject to problems such as poor convergence for EDSs with high R/X ratio branches and huge mathematical/programing burden in some cases. Moreover, in EDS studies such as power loss minimization, energy management, EDS planning/expansion, etc., load flow calculations need to be conducted alongside the required objective function and the related constraints of the study. To solve such models, optimization techniques such as metaheuristic methods, are used in which load flow calculation is conducted in each iteration, bringing extra mathematical burden. To address these limitations, this paper presents a new directly solvable and non-iterative load flow model which is developed using a new bus indexing procedure. A connectivity matrix is introduced to characterize the configuration of EDS and provide a feasible general representation of load flow equations. This enables the proposed modified load flow equations to be mergeable in any type of EDS study as model constraints. Moreover, unlike previous iterative models, it can be solved directly through standard optimization packages in a single shot with no need to further iterative optimization procedures, such as metaheuristic methods. This results in a moderate mathematical and programing burden. The 33-bus system is employed to evaluate the effectiveness of the proposed load flow model. The obtained results are reported and compared with forward-backward sweep-based load flow as a benchmark. Accordingly, it is shown that the proposed model maintains the same outcome without using iterative optimization techniques.