The relevance of the problem under consideration is growing with the expansion of the element base of modern semiconductor electronic devices and the improvement of manufacturing technologies for modern semiconductor electronic devices. The growing number of elements of the designed electronic devices has led to the need to develop new algorithms for the effective solution of large-dimensional SLAE. Classical numerical methods for solving SLAE are successfully used to solve SLAE of small dimension, but they do not provide the required quality of solutions with an increase in the number of calculations. One of the new approaches to solving large-scale SLAE is the parallelization of the computational process, which makes it possible to maximize the use of existing computing resources. Approaches based on multi-agent systems and organizations are successfully used to effectively organize the distributed computing process. The agent-based approach is a tool that allows you to flexibly allocate available computing resources and improve the quality of solutions. Another promising approach is the use of optimization methods based on evolutionary calculations. In this paper, we propose a hybrid approach to solving large-dimensional SLAE based on the integration of evolutionary algorithms, Pareto optimization methods, and an agent-based approach to computing. It is proposed to use the architecture of the software subsystem based on a service-oriented approach. The structure of a multilevel software subsystem based on an agent-based approach has been developed. A software implementation was performed and a series of computational experiments were conducted that confirmed the effectiveness of the proposed approach.