# Algorithm of solving linear equations

### What is this article about, and why it is so important for the entire scientific community.

I. Even most outstanding mathematicians (scientists) solve linear equations according to the following principle: *“I do this transposition and not another not because I know the rule determining the sequence of necessary steps to be taken in order to find the solution, but because it is the way it should be done".* This leads to an awkward situation: if a student asks why we are doing this particular transposition, normally we respond: __because it is done this way in maths!!!__ In mathematics, which is deductive in its grounds, through connecting results with reasons.
Mathematics teacher at the school has only his own intuition of solving equations. His job is to train the students intuition, that within the universality of education, implies equally universal problems. __This article is a scientific announcement allowing formulation of a principle, which gives mathematicians and students a tool in form of the axiom of selecting a specific transposition in the equation.__With the introduction of additional – 5th transposition – enabling the indication of a scientific rule that tells what (and how many!) consecutive steps of solving are required to solve equations and transform formulas successfully each time. Statement: “Doing identity operations (transpositions) of number (a) compared to letter (x), i.e.: x = a, we may create any equation with one solution.*(X) Reverse order of these operations gives the initial equality. Transpositions that lead to a solution of the equation are made by means of elimination of redundant factors in the order reverse to the rule of the sequence of arithmetic operations, doing inverse operations:1. A factor present in a sum is eliminated by the transposition of subtraction 2. A factor present in a difference is eliminated by the transposition of summation 3. A factor present in a product is eliminated by the transposition of division
4. A factor present in the denominator of the quotient is eliminated by the transposition of multiplication. __5. A factor present in the power with the exponent of -1 (converse) is eliminated by the transposition of raising both sides of the equation to the power of -1.__

**II.** Dissemination of the content of this article (claim and part III) will lead to necessity to correct the curriculum of school mathematics and thus to the editorial changes of the content of mathematics books with need to identify the source text and the name of the magazine - in some sense, as said by one of the first readers of this publication : **“thanks to this article mankind will learn how to solve equations like 1225 : [ (13x-30)715-84] = 74.”**

**III.** Mathematisation of science and also scientific research and discoveries obliges scholars of all disciplines, even those very distant from mathematics in their fields of scientific interest, to use linear equations and formulas with one unknown freely, due to linearity of a number of phenomena that are described by the science. It is hard to expect that a social studies researcher could see and denote a linear dependence using an equation, if he/she cannot solve such equations independently. Such skill, that is training the algorithm presented in the article, is also the caesura in the school mathematics education. In the case of algorithm that organize the linear equations and second degree equations into the trivial general form – obviously indicating a solution - this issue is fundamental for education, thereby for the education of future scientists and finally, in the words of C.F Gauss," __for the dignity of the math itself."__ Without the ability to arrange equations in the trivial form, solving the equations, even if it is done as usual – that is intuitively – due to mathematization of everyday life, doing science, even in reconstructive field, is __impossible.__ Many scientific disciplines, not solely mathematical like Algebra, could not be created. On the other hand, mathematical education without this skill, at the level of problem understanding, would have to end on addition, multiplication, exponentiation, subtraction, division and nth root at the most. Why? __ Theorem about finding a solution of a linear equation (thereby the ability to transform formulas) is prior in relation to the theorem of Thales, Pythagoras and other tools of mathematics!__ Thus, without familiarizing with the theorem and skillfully using it, at least intuitively, as usual, not only teaching of mathematics, but also teaching of physics, chemistry and solving many of the elementary major technical issues could not take place

# V Transposition^{(I)} in solving equations - extraordinary omission of science.

Consider the equation:

Trivial? Over 95% of surveyed secondary school pupils from Poland are unable to solve this simple equation. 98% of forty million people of this European country don’t even know what to start with. And does science itself know it in the era of the omnipresent mathematisation? Is there any scientific rule that tells what (and how many!) consecutive steps of solving are required to solve equations and transform formulas successfully each time?