Variability of methods for linear equations systems solving in historical tasks

The paper investigates the role of famous historical tasks in building knowledge of linear equations systems solving. The problems, which lead to systems of linear equations, are highlighted. The paper also gives an analysis of the authoring methods for linear equations systems solving, suggested by mathematicians of different times, as well as provides modern methods for solving these systems. This work is a follow-up study of different methods for solving historical tasks in algebra and number theory, including historical and genetic approach in training of teachers of mathematics. We suggest the system of historical tasks, which comprises two aspects. First, the variability of methods in different authoring problems, and second, solving the same problem in different ways. The analysis of creative and unconventional approaches to solving linear equations systems was conducted in the «Aryabhatiyam» treatise by Indian mathematician Aryabhata, the «De numeris datis» treatise by German mathematician Nemorarius, «Diversarum speculationum mathematicarum et physicarum liber» by Italian mathematician Giambattista Benedetti, in «Treatise of Algebra» by Scottish mathematician Colin Maclaurin. The variability of methods for solving the same problem was considered by Arab mathematician Alhazen, Indian mathematician Bhaskara II («Crown of treatises»), Italian mathematician Leonardo Pisano («Liber Abaci», section XI), Arab mathematician Beha-Eddin («Essence of Arithmetic»). At the conclusion, the suggested approach enables the creation of the bank of linear equations solving methods, the using of a creative approach to solving mathematical problems. Besides, the approach allows utilizing the educational and developmental potential of a history of mathematics. Learning of linear equations solving methods is included in Linear Algebra Curriculum for pedagogical specializations at universities (elimination of variables, Cramer's rule, matrix solution) and in Mathematics Curriculum for schools (addition, substitution, variable replacement, graphing method). Therefore, it is important to continue the research on the variability of methods for linear equations systems solving in historical tasks in mathematical education