Engineering, Energy, Transport AIC

Dubchak Victor – PhD, Associate Professor of the Department "Mathematics, physics and computer technologies" of Vinnytsia National Agrarian University (3, Solnechnaya St., Vinnytsia, 21008, Ukraine,
e-mail: viktor_dubchak@rambler.ru).

Novitskaya Lyudmila – PhD, Associate Professor of the Department "Mathematics, physics and computer technologies" of the Vinnytsia National Agrarian University (3, Solnechnaya St., Vinnytsia, 21008, Ukraine, e-mail: li@vsau.vin.ua)

In this article a modified approach for finding solutions of a system of equations similar to the classical Gaussian method are investigated. In the proposed modification as well as in the classical method, the essence of the transformations reduces to the possibility of first finding one, usually the last, unknown; here the initial system of the n-th order first reduces to solving some auxiliary system of equations (n-1) -th order, and through its solutions, for example, the last unknown of the initial algebraic system can be found. This method can be applied to any algebraic systems according to the number of their solutions-both joint and incompatible, both definite and indeterminate systems. The determination of the number of solutions is achieved automatically by the algorithm of the calculations given in the work. Similarly, the search for other unknowns that have not yet been found is realized. It can be carried out by the same method according to the recurrence principle, and the order of the next auxiliary algebraic system decreases by one every time. Also, an auxiliary system as to additional unknowns can be solved either by other classical methods, or by applying elementary algebraic transformations. The main advantage of this method is the reduction of the order of the auxiliary system by one.
As an illustration of the proposed method, examples of finding solutions to algebraic systems (definite and indefinite) are offered. The conclusions of the method are drawn in the article. The results of this work can be used to solve specific technical problems, in particular, in the energy sector, in transport tasks, etc.

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