2408-932XResearch Result. Social Studies and Humanities2408-932X10.18413/2408-932X-2026-12-1-0-94118RESEARCHES<strong>Hermeneutic Foundations for Solving Equations and Their Systems</strong><strong>Hermeneutic Foundations for Solving Equations and Their Systems</strong>ЕryginaNelly S.ЕryginaNelly S.erygina_n@bsuedu.ruPenkovViktor E.PenkovViktor E.penkov@bsu.edu.ruBelgorod State National Research University202612100

This article addresses issues related to schoolchildren&#39;s ability to solve different types of equations. It emphasises the importance of understanding the origins of algorithms and rules for solving equations, rather than merely memorising them. With this in mind, the authors identify three hermeneutic principles.&nbsp; Firstly, performing the same operation on both sides of an equation maintains the equality. Secondly, when dividing or multiplying both parts of an inequality by a negative number, the inequality sign must be reversed. Thirdly, when solving non-rational equations and their systems, the original equations must be reduced to an elementary equation or a form with only one function, which can be reduced to a rational equation by replacing the variable. After solving this equation by inverse replacement, an elementary equation is obtained that can be solved in the simplest way. By knowing these rules, students will be able to solve any equation, from the most basic to the most complex, throughout their mathematics studies, from the first to the eleventh grade.

This article addresses issues related to schoolchildren&#39;s ability to solve different types of equations. It emphasises the importance of understanding the origins of algorithms and rules for solving equations, rather than merely memorising them. With this in mind, the authors identify three hermeneutic principles.&nbsp; Firstly, performing the same operation on both sides of an equation maintains the equality. Secondly, when dividing or multiplying both parts of an inequality by a negative number, the inequality sign must be reversed. Thirdly, when solving non-rational equations and their systems, the original equations must be reduced to an elementary equation or a form with only one function, which can be reduced to a rational equation by replacing the variable. After solving this equation by inverse replacement, an elementary equation is obtained that can be solved in the simplest way. By knowing these rules, students will be able to solve any equation, from the most basic to the most complex, throughout their mathematics studies, from the first to the eleventh grade.

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