The high intensity of research and modeling in fields of mathematics, physics, biology and chemistry requires new computing resources. For the big computational complexity of such tasks computing time is large and costly. The most efficient way to increase efficiency is to adopt parallel principles. Purpose of this paper is to present the issue of parallel computing with emphasis on the analysis of parallel systems, the impact of communication delays on their efficiency and on overall execution time. Paper focuses is on finite algorithms for solving systems of linear equations, namely the matrix manipulation (Gauss elimination method GEM). Algorithms are designed for architectures with shared memory (openMP), distributed-memory (MPI) and for their combination (MPI+openMP). The properties of the algorithms were analytically determined and they were experimentally verified. The conclusions are drawn for theory and practice.