Examples of systems of linear equations: solution method. Methodology for teaching solving equations based on the properties of equalities

More reliable than the graphical method discussed in the previous paragraph.

Substitution method

We used this method in 7th grade to solve systems linear equations. The algorithm that was developed in the 7th grade is quite suitable for solving systems of any two equations (not necessarily linear) with two variables x and y (of course, the variables can be designated by other letters, which does not matter). In fact, we used this algorithm in the previous paragraph, when the problem of a two-digit number led to a mathematical model, which is a system of equations. We solved this system of equations above using the substitution method (see example 1 from § 4).

An algorithm for using the substitution method when solving a system of two equations with two variables x, y.

1. Express y in terms of x from one equation of the system.
2. Substitute the resulting expression instead of y into another equation of the system.
3. Solve the resulting equation for x.
4. Substitute in turn each of the roots of the equation found in the third step instead of x into the expression y through x obtained in the first step.
5. Write the answer in the form of pairs of values (x; y), which were found in the third and fourth steps, respectively.

4) Substitute one by one each of the found values of y into the formula x = 5 - 3. If then
5) Pairs (2; 1) and solutions to a given system of equations.

Answer: (2; 1);

Algebraic addition method

This method, like the substitution method, is familiar to you from the 7th grade algebra course, where it was used to solve systems of linear equations. Let us recall the essence of the method using the following example.

Example 2. Solve system of equations

Let's multiply all terms of the first equation of the system by 3, and leave the second equation unchanged:
Subtract the second equation of the system from its first equation:

As a result of the algebraic addition of two equations of the original system, an equation was obtained that was simpler than the first and second equations of the given system. With this simpler equation we have the right to replace any equation of a given system, for example the second one. Then the given system of equations will be replaced by a simpler system:

This system can be solved using the substitution method. From the second equation we find. Substituting this expression instead of y into the first equation of the system, we get

It remains to substitute the found values of x into the formula

If x = 2 then

Thus, we found two solutions to the system:

Method for introducing new variables

You were introduced to the method of introducing a new variable when solving rational equations with one variable in the 8th grade algebra course. The essence of this method for solving systems of equations is the same, but from a technical point of view there are some features that we will discuss in the following examples.

Example 3. Solve system of equations

Let's introduce a new variable. Then the first equation of the system can be rewritten into a more in simple form: Let's solve this equation for the variable t:

Both of these values satisfy the condition and therefore are the roots of a rational equation with variable t. But that means either where we find that x = 2y, or
Thus, using the method of introducing a new variable, we managed to sort of “stratify” the first equation of the system, which was quite complex in appearance, into two simpler equations:

x = 2 y; y - 2x.

What's next? And then each of the two simple equations obtained must be considered in turn in a system with the equation x 2 - y 2 = 3, which we have not yet remembered. In other words, the problem comes down to solving two systems of equations:

We need to find solutions to the first system, the second system and include all the resulting pairs of values in the answer. Let's solve the first system of equations:

Let's use the substitution method, especially since everything is ready for it here: let's substitute the expression 2y instead of x into the second equation of the system. We get

Since x = 2y, we find, respectively, x 1 = 2, x 2 = 2. Thus, two solutions of the given system are obtained: (2; 1) and (-2; -1). Let's solve the second system of equations:

Let's use the substitution method again: substitute the expression 2x instead of y into the second equation of the system. We get

This equation has no roots, which means the system of equations has no solutions. Thus, only the solutions of the first system need to be included in the answer.

Answer: (2; 1); (-2;-1).

The method of introducing new variables when solving systems of two equations with two variables is used in two versions. First option: one new variable is introduced and used in only one equation of the system. This is exactly what happened in example 3. Second option: two new variables are introduced and used simultaneously in both equations of the system. This will be the case in example 4.

Example 4. Solve system of equations

Let's introduce two new variables:

Let's take into account that then

This will allow you to rewrite this system in a much simpler form, but relatively new variables a and b:

Since a = 1, then from the equation a + 6 = 2 we find: 1 + 6 = 2; 6=1. Thus, regarding the variables a and b, we got one solution:

Returning to the variables x and y, we obtain a system of equations

Let us apply the method to solve this system algebraic addition:

Since then from the equation 2x + y = 3 we find:
Thus, regarding the variables x and y, we got one solution:

Let us conclude this paragraph with a brief but rather serious theoretical conversation. You have already gained some experience in solving various equations: linear, quadratic, rational, irrational. You know that the main idea of solving an equation is to gradually move from one equation to another, simpler, but equivalent to the given one. In the previous paragraph we introduced the concept of equivalence for equations with two variables. This concept is also used for systems of equations.

Definition.

Two systems of equations with variables x and y are called equivalent if they have the same solutions or if both systems have no solutions.

All three methods (substitution, algebraic addition and introducing new variables) that we discussed in this section are absolutely correct from the point of view of equivalence. In other words, using these methods, we replace one system of equations with another, simpler, but equivalent to the original system.

Graphical method for solving systems of equations

We have already learned how to solve systems of equations in such common and reliable ways as the method of substitution, algebraic addition and the introduction of new variables. Now let's remember the method that you already studied in the previous lesson. That is, let's repeat what you know about the graphical solution method.

The method of solving systems of equations graphically involves constructing a graph for each of the specific equations that are included in a given system and are located in the same coordinate plane, as well as where it is necessary to find the intersections of the points of these graphs. To solve this system of equations are the coordinates of this point (x; y).

It should be remembered that it is typical for a graphical system of equations to have either one single right decision, either an infinite number of solutions, or no solutions at all.

Now let’s look at each of these solutions in more detail. And so, a system of equations can have a unique solution if the lines that are the graphs of the system’s equations intersect. If these lines are parallel, then such a system of equations has absolutely no solutions. If the direct graphs of the equations of the system coincide, then such a system allows one to find many solutions.

Well, now let’s look at the algorithm for solving a system of two equations with 2 unknowns using a graphical method:

Firstly, first we build a graph of the 1st equation;
The second step will be to construct a graph that relates to the second equation;
Thirdly, we need to find the intersection points of the graphs.
And as a result, we get the coordinates of each intersection point, which will be the solution to the system of equations.

Let's look at this method in more detail using an example. We are given a system of equations that needs to be solved:

Solving equations

1. First, we will build a graph of this equation: x2+y2=9.

But it should be noted that this graph of the equations will be a circle with a center at the origin, and its radius will be equal to three.

2. Our next step will be to graph an equation such as: y = x – 3.

In this case, we must construct a straight line and find the points (0;−3) and (3;0).

3. Let's see what we got. We see that the straight line intersects the circle at two of its points A and B.

Now we are looking for the coordinates of these points. We see that the coordinates (3;0) correspond to point A, and the coordinates (0;−3) correspond to point B.

And what do we get as a result?

The numbers (3;0) and (0;−3) obtained when the line intersects the circle are precisely the solutions to both equations of the system. And from this it follows that these numbers are also solutions to this system of equations.

That is, the answer to this solution is the numbers: (3;0) and (0;−3).

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Coursework

Formation of the concept of equation in primary school

1. History of the equation

2. Content and role of the line of equations in the modern school mathematics course

3. THEORETICAL BASIS FOR SOLVING EQUATIONS IN PRIMARY CLASSES

3.1 Equations in elementary grades

3.2 Methodology for working on the equation

CONCLUSION

LIST OF REFERENCES USED

APPENDIX A

solving equation math problem school

INTRODUCTION

Anytime modern system general education mathematics occupies one of the central places, which undoubtedly speaks of the uniqueness of this field of knowledge.

What is modern mathematics? Why is it needed? These and similar questions are often asked by children to teachers. And each time the answer will be different depending on the level of development of the child and his educational needs.

It is often said that mathematics is a language modern science. However, there appears to be a significant flaw in this statement. The language of mathematics is so widespread and so often effective precisely because mathematics cannot be reduced to it.

Outstanding mathematician A.N. Kolmogorov wrote: “Mathematics is not just one of the languages. Mathematics is language plus reasoning, it’s like language and logic together. Mathematics is a tool for thinking. It concentrates the results of the exact thinking of many people. With the help of mathematics you can connect one reasoning with another... The obvious complexities of nature with its strange laws and rules, each of which allows for a very different detailed explanation, are in fact closely related. However, if you do not want to use mathematics, then in this huge variety of facts you will not see that logic allows you to move “from one to another.”

Thus, mathematics allows us to form certain forms of thinking necessary to study the world around us.

The mathematics course has an important influence on the formation various forms thinking: logical, spatial-geometric, algorithmic. Any creative process begins with the formulation of a hypothesis. Mathematics, with the appropriate organization of training, being a good school for constructing and testing hypotheses, teaches how to compare different hypotheses, find best option, set new tasks, look for ways to solve them. Among other things, she also develops the habit of methodical work, without which any creative process is unthinkable. By maximizing the possibilities of human thinking, mathematics is its highest achievement. It helps a person in self-awareness and the formation of his character.

This is a little of big list reasons why mathematical knowledge should become an integral part of general culture and a mandatory element of education and training r baby.

The study of the simplest equations and methods for solving them has become firmly established in the system of initial mathematical training.

The relevance of the topic of our work is that the study of equations by younger schoolchildren in elementary school prepares them for more successful study algebraic material in basic school. Equations are one of the means of modeling the fragments of reality being studied, and familiarity with them is an essential part of mathematical education.

Based on this, the goal course work study the process of forming the concept of an equation at the initial stage of teaching mathematics.

Object - the process of studying algebraic material using the example of equations in elementary school

The subject is the formation of the concept of an equation in the primary grades.

Hypothesis - the formation of a clear equation will be successful if the knowledge being studied is justified stumps m and in a way that is convincing to children.

To achieve the goal, I set the following tasks:

1. Study and analyze psychological, pedagogical and methodological literature on the research topic,

2. Reveal the process of forming the concept of an equation in teaching mathematics;

3. Consider the techniques used in forming the concept of an equation.

The course work consists of an introduction, three chapters, a conclusion and a list of references.

1. History of the equation

§ Algebra as the art of solving equations originated a long time ago in connection with the need for practice, as a result of the search for general techniques for solving similar problems. The earliest manuscripts that have reached us indicate that in Ancient Babylon and Ancient Egypt techniques for solving linear equations were known. The word “algebra” arose after the appearance of the treatise “Kitab al-jabr wal-mukabala” by the Khorezm mathematician and

§ astronomer Mohamed Ben Musa al Khwarizmi. The term "al-jerb", taken from the title of this book, later began to be used as algebra.

§ The equal sign was introduced in 1556 by the English mathematician Record, who explained it in such a way that nothing can be more equal than two parallel segments.

§ François Viète, seigneur de la Bigotière; 1540 - December 13, 1603) - an outstanding French mathematician, one of the founders of algebra

The creator of modern letter symbols is the French mathematician Francois Viète (1540 - 1603). Until the 16th century Algebra was presented mainly verbally. Letter symbols and mathematical symbols appeared gradually. The signs + - were first encountered by German algebraists of the 16th century. A little later, the * sign is introduced for multiplication. The division sign (:) was introduced only in the 17th century. A decisive step in the use of algebraic symbolism was made in the 16th century, when the French mathematician Francois Viète (1540-1603) and his contemporaries began to use letters to denote not only unknowns (which had been done before), but also any numbers. However, this symbolism was still different from the modern one. Thus, Viet used the letter N (Numerus-number) to denote the Unknown number, and the letters Q (Quadratus - square) and C (Cubus - cube) for the square and cube of the unknown. For example, writing the equation X cubed, minus 8X squared, plus 16X, equals 40 for Vieta would look like this: 1C-8Q+16N aequ. 40 (aequali - equal). Viet divides the presentation into two parts: general laws and their specific numerical implementations. That is, he first solves problems in general form, and only then gives numerical examples. In the general part, he denotes by letters not only the unknowns, which have already been encountered before, but also all other parameters for which he coined the term “coefficients” (literally: contributing). Viet used for this only capital letters-- vowels for unknowns, consonants for coefficients. Viet freely applies a variety of algebraic transformations - for example, changing variables or changing the sign of an expression when transferring it to another part of the equation.

The new system made it possible to simply, clearly and compactly describe the general laws of arithmetic and algorithms. The symbolism of Viet was immediately appreciated by scientists from different countries, who began to improve it. Diophantus (no earlier than the 3rd century AD) is the only ancient Greek mathematician known to us who studied algebra.

He solved various equations special attention devoted to indefinite equations, the theory of which is now called “Diophantine analysis.” Diophantus attempted to introduce alphabetic symbolism. Leaf from Arithmetic (14th century manuscript). The top line contains the equation:

The first book is preceded by an extensive introduction, which describes the notation used by Diophantus. Diophantus calls the unknown “number” (?syimt) and denotes it with the letter t, the square of the unknown with the symbol dn (short for denbmyt - “degree”). There are special signs for the following degrees unknown, up to the sixth, called cube-cube, and for the degrees opposite to them. Diophantus does not have an addition sign: he simply writes positive terms next to each other, and in each term the degree of the unknown is first written, and then the numerical coefficient

§ Evariste Galois (French: Иvariste Galois; October 25, 1811, October 25, 1811, Bourg-la-Reine, Hauts-de-Seine, France - May 31, 1832, France) - an outstanding French mathematician, the founder of modern higher algebra.

Heurist Galois (1811 - 1832) - this brilliant mathematician died in a duel set up by his enemies. The night before the duel, he wrote a letter in which he outlined his results, which gave rise to a whole science - “Galois theory”

§ Niels Henrik Abel (1802 - 1829) made important contributions to the theory of equations. In 1824 he published a proof of the undecidability in radicals of a general literal expression of the fifth degree.

“Abel left such a rich legacy for mathematicians that they will have something to do for the next 150 years” (Charles Hermite). Niels Henrik Abel (Norwegian Niels Henrik Abel; August 5, 1802, Fingø - April 6, 1829, Froland near Arendal) - famous Norwegian mathematician

1.From the history of the emergence of equations.

Algebra arose in connection with solving various problems using equations. Typically, problems require finding one or more unknowns, while knowing the results of some actions performed on the desired and given quantities. Such problems come down to solving one or a system of several equations, to finding the required ones using algebraic operations on given quantities. Algebra studies the general properties of operations on quantities.

Material related to equations makes up a significant part of the school mathematics course. This is explained by the fact that equations are widely used in various branches of mathematics and in solving important applied problems.

The origins of algebraic methods for solving practical problems are associated with the science of the ancient world. As is known from the history of mathematics, a significant part of the problems of a mathematical nature solved by Egyptian, Sumerian, Babylonian scribes-calculators (XX-VI centuries BC) were of a computational nature. However, even then, from time to time, problems arose in which the desired value of a quantity was specified by certain indirect conditions that, from our modern point of view, required the composition of an equation or system of equations. Initially, arithmetic methods were used to solve such problems. Subsequently, the beginnings of algebraic concepts began to form. For example, Babylonian calculators were able to solve problems that, from the point of view of modern classification, can be reduced to equations of the second degree. Thus, a method for solving word problems was created, which later served as the basis for isolating the algebraic component and its independent study.

This study was carried out in another era, first by Arab mathematicians (VI-X centuries AD), who identified the characteristic actions by which equations were reduced to standard view(bringing similar terms, transferring terms from one part of the equation to another with a change of sign), and then by European mathematicians of the Renaissance, who, as a result of a long search, created the language of modern algebra (the use of letters, the introduction of symbols for arithmetic operations, parentheses, etc.). At the turn of the XVI-XVII centuries. algebra as a specific part of mathematics, with its own subject, method, and areas of application, was already formed. Its further development, right up to our time, consisted of improving methods, expanding the scope of applications, clarifying concepts and their connections with concepts of other branches of mathematics. In this process, the importance of the role played by the concept of an equation in the system of algebraic concepts became increasingly clear.

Opening coordinate method(Descartes, 17th century) and the subsequent development of analytical geometry made it possible to apply algebra not only to problems related to the number system, but also to the study of various geometric shapes. This line of development of algebra strengthened the position of the equation as the leading algebraic concept, which was now associated with three main areas of its origin and functioning:

a) equation as a means of solving word problems;

b) an equation as a special kind of formula that serves as an object of study in algebra;

c) an equation as a formula that indirectly determines the numbers or coordinates of points in a plane (space) that serve as its solution.

Each of these ideas has proven useful in one way or another.

Thus, an equation as a general mathematical concept has many aspects, and none of the aspects can be excluded from consideration, especially if we're talking about about the problems of school mathematics education.

Due to the importance and vastness of the material associated with the concept of an equation, its study in modern methods of mathematics is organized into a content-methodological line - a line of equations and inequalities. Here we consider the formation of the concepts of equations and inequalities, general and particular methods for solving them, the relationship of the study of equations and inequalities with numerical, functional and other lines of the school mathematics course. The identified areas of origin and functioning of the concept of an equation in algebra correspond to three main directions of development of the line of equations and inequalities in school course mathematics.

a) The applied orientation of the line of equations is revealed mainly when studying the algebraic method of solving word problems. This method is widely used in school mathematics as it relates to the teaching of techniques used in applications of mathematics.

Currently, mathematical modeling occupies a leading position in applications of mathematics. Using this concept, we can say that the applied value of equations and their systems is determined by the fact that they are the main part of the mathematical tools used in mathematical modeling.

b) The theoretical and mathematical orientation of the line of equations is revealed in two aspects: firstly, in the study of the most important classes of equations and their systems and, secondly, in the study of generalized concepts and methods related to the line as a whole. Both of these aspects are necessary in a school mathematics course. The main classes of equations are associated with the simplest and at the same time the most important mathematical models. The use of generalized concepts and methods makes it possible to logically organize the study of a line as a whole, since they describe what is common in procedures and solution techniques related to individual classes of equations, inequalities, and systems. In turn, these general concepts and methods are based on basic logical concepts: the unknown, equality, equivalence, logical consequence, which must also be revealed in the line of equations

c) The line of equations is characterized by an orientation towards establishing connections with the rest of the content of the mathematics course

This line is closely related to the number line. The main idea implemented in the process of establishing the relationship of these lines is the idea of sequential expansion of the numerical system. All numerical areas considered in school algebra and the beginnings of analysis, with the exception of the area of all real numbers, arise in connection with the solution of some equations and their systems. The areas of irrational and logarithmic expressions are associated, respectively, with the equations xk = b (k is a natural number greater than 1) and ax = b.

The line of equations is also closely related to the functional line. One of the most important connections is the application of methods developed in the line of equations to the study of functions (for example, to tasks of finding the domain of definition of certain functions, their roots, intervals of constant sign, etc.). On the other hand, the functional line has a significant impact on both the content of the line of equations and inequalities and the style of its study. In particular, functional representations serve as the basis for attracting graphical clarity to the solution and study of equations, inequalities and their systems.

3. On the interpretation of the concept of equation.

The concept of an equation is one of the most important general mathematical concepts. That is why it is difficult to propose a definition of it that is both strict from a formal point of view and accessible to students starting to master the school algebra course.

The logical-mathematical definition of the equation can be given in the following form: let a set of algebraic operations be fixed on the set M, x is a variable on M; then an equation on the set M with respect to x is a predicate of the form a(x) = b (x), where a(x) and b(x) are terms with respect to given operations, the notation of which includes the symbol x. An equation in two variables, etc., is defined similarly.

The terms “term” and “predicate” accepted in logic correspond to the terms of school mathematics “expression” and “sentence with a variable.” Therefore, the closest to the given formal definition is following definition: “A proposition with a variable, having the form of equality between two expressions with this variable, is called an equation”

Analyzing the given mathematical definition of the equation, we can distinguish two components in it. The first is that an equation is a special kind of predicate. The second specifies exactly what kind: this is an equality connecting two terms, and the terms also have a certain special form. When studying material related to the line of equations and inequalities, both components play a significant role.

The first one is the semantic component, which is important primarily for understanding the concept of the root of an equation. In addition, the semantic component is almost always used to justify the correctness of a particular equation transformation.

The second component refers to the formal features of the notation depicting the equation. Let's call this component sign. It is important in cases where the recording of an equation is subject to various transformations: often such transformations are carried out purely mechanically, without reference to their meaning.

Possibility of use in schooling approach to the concept of an equation, including explicit mention of a sentence with a variable, depends on the presence of this term and the terms “true”, “false” in the required material of the mathematics course. If they do not exist, then it is impossible to give such a definition. In this case, the semantic component of the concept of equation goes into the definition of another concept, closely related to the concept equations - roots equations The result is a system of two terms: the term “equation” carries the characteristics of a sign component, and the term “root of an equation” takes into account the semantic component. This definition is given, for example, in the textbook by A. N. Kolmogorov.

Often, especially at the beginning of a systematic algebra course, the concept of an equation is introduced by isolating it from the algebraic method of solving problems. In this case, regardless of what the text of the definition is, the approach to the concept of an equation is essential, in which it represents an indirect form of specifying some unknown number, which has a specific interpretation in accordance with the plot of the problem. For example, the concept of an equation is introduced based on the text problem: “An envelope with a New Year’s card costs 170 soum. An envelope is cheaper than a postcard for 70 bags. Find the cost of the postcard." The transition to the definition of the equation is carried out on the basis of an analysis of some formal features of the notation.x + (x---70) = 170, which expresses the content of this problem in algebraic form. Using the same plot, the concept of the root of an equation is introduced. These definitions are: “An equality containing an unknown number, indicated by a letter, is called an equation. The root of an equation is the value of the unknown at which this equation turns into a true equality.” The indicated method of introducing the concept of an equation corresponds to another component of the concept of an equation - applied.

Another approach to defining the concept of an equation is obtained by comparing the domain of definition of the equation and the set of its roots. Usually the set of roots of an equation is a proper subset of its domain of definition. On the other hand, when solving equations it is necessary to use transformations that are based on identities, that is, on equalities that are true throughout the entire domain of definition. The contrast between identity and equation highlighted here can be used as the basis for the definition of an equation: “A literal equality, which does not necessarily turn into a true numerical equality with admissible sets of letters, is called an equation.”

Forming the concept of an equation requires the use of one more term: “solve the equation.” Various options its definitions differ from each other essentially only by the presence or absence of the term “set” in them.

Thus, when mastering the concept of an equation, it is necessary to use the terms “equation”, “root of the equation”, “what does it mean to solve an equation”. In this case, along with the components of the concept of an equation included in the text of the definition, it is necessary to include all its other components as the material of this line unfolds.

The definition of an equation uses one of two terms: “variable” or “unknown.” The difference between them is that the variable runs through a series of values without specifically highlighting any of them, and the unknown is a letter designation for a specific number (therefore, this term is convenient to use when composing equations for word problems). Issues related to the choice of one of these terms for use in school practice cannot currently be considered finally resolved. The choice of one or another of them entails certain differences in the development of the content of the line of equations and inequalities. Thus, the term “variable” is associated with the operation of substituting a number instead of a letter, so in the equation a(x) = b(x) you can substitute specific numbers instead of x and find roots among them. The term “unknown” means a fixed number; Substituting a number in place of a letter denoting an unknown is therefore illogical. Finding the roots of the equation a(x)=b(x) from this point of view should be carried out using actions in which this equality is considered as true and they try to bring it to the form x=x, where x is a numerical expression.

When describing the methodology, we will use the term “unknown”, which is closer than “variable” to the algebraic method of solving word problems and thus to the applied orientation of the line of equations and inequalities.

2. Equivalence and logical consequence.

Let's consider the logical tools used in the process of studying equations and inequalities. The most important among them is the concept of equivalence.

Let us recall that equations are called equivalent if the corresponding predicates are equivalent, that is, if the conditions are met: the domains of definition of the equations are identical and the sets of their roots are equal. There are two ways to establish the equivalence of the equations. First: using known sets of roots of equations, make sure they coincide. Second: using the peculiarities of writing equations, carry out a sequential transition from one record to another through transformations that do not violate equivalence.

Obviously, for most tasks the second path is more typical. This is understandable, because equivalence in the theory of equations is precisely used to indicate specific rules for solving equations. However, in teaching it is inappropriate to limit oneself to it, since it relates only to practical application equivalence and requires the former for its justification. At the same time, mastering the concept of equivalence as the equivalence of predicates requires a significant culture of thinking and cannot be learned on initial stages studying a school algebra course without special significant effort.

With regard to the formation of the concept of equivalence and its application to solving equations, algebra textbooks can be divided into two groups. The first includes those manuals in which the use of equivalent transformations is based on the explicit introduction and study of the concept of equivalence; the second includes those in which the application of equivalent transformations precedes the isolation of the concept itself. The methodology for working on the concept of equivalence has significant differences with these approaches.

In connection with the issue under consideration, three main stages can be distinguished in the study of the material of the line of equations and inequalities. The first stage covers the initial course of school mathematics and the beginning of the algebra course. Here you will get acquainted with in various ways solutions to individual, simplest classes of equations. The transformations used in this case receive inductive justification when considering specific examples. As experience is gained, inductive reasoning is increasingly replaced by one where equivalence is actually used, but the term itself is not used. The duration of this stage may vary; it depends on the methodological guidelines adopted in this textbook.

At the second stage, the concept of equivalence is isolated and its theoretical content is compared with the rules of transformation that are derived on its basis. The duration of this stage is insignificant, since it only involves the identification of this concept and its use in several theoretical examples.

At the third stage, based on the general concept of equivalence, the development and general theory, and theories of individual classes of equations. This style is typical for the algebra and elementary analysis courses studied in high school. It is also used in some algebra textbooks for junior high school.

In addition to equivalent ones, other, generally speaking, not equivalent transformations are also used to study the material of a line of equations. Most of them are not revealed in the school course, although they are used more or less significantly, in particular, in the study of equations. The only exception is the concept of logical consequence, which in some cases teaching aids is the subject of study. The methodology for working with the concept of logical implication (as well as the idea of it if the concept is not introduced) has many similarities with the methodology for studying equivalence and equivalent transformations.

Logical implication begins to be used much later than equivalence and is adopted as some kind of addition to it. When solving equations, all other things being equal, preference is given to an equivalent transformation; logical implication is used only when a corresponding equivalent transformation cannot be found. This, however, does not mean that the use of logical implication is a necessary measure. Often in the practice of teachers, logical following is used as a technique that simplifies the decision process if maintaining equivalence can be achieved at a relatively high cost.

Among the unequal transformations there are transformations that are not logical consequences. For example, transition to consideration of a particular case (example: transition from the equation a -b = 0 to consideration of the equation a = 0). Such transitions can be viewed as practical techniques that allow you to focus on individual steps in the process of solving an equation.

On the classification of transformations of equations and their systems.

There are three main types of such transformations:

1) Transform one of the parts of the equation.

2) Consistent transformation of both sides of the equation.

3) Transformation of the logical structure.

Transformations of the second type are relatively numerous. They form the core of the material studied in the line of equations.

Let us give examples of transformations of this type.

1) -Adding the same expression to both sides of the equation.

2) Multiplication (division) of both sides of the equation by the same expression.

3) Transition from the equation a=b to the equation ¦ (a)=¦ (b), where ¦ is some function, or the inverse transition.

The third type of transformations includes transformations of equations and their systems that change the logical structure of tasks. Let us clarify the term “logical structure” used. In each task, elementary predicates can be identified - individual equations. By the logical structure of a task we understand the way of connecting these elementary predicates through logical connectives of conjunction or disjunction.

Depending on the means used for transformations, two subtypes can be distinguished in this type: transformations carried out using arithmetic operations and using logical operations. The former can be called arithmetic transformations of the logical structure, the latter - logical transformations of the logical structure.

The study and use of transformations of equations and their systems, on the one hand, presuppose a fairly high logical culture of students, and on the other hand, in the process of studying and applying such transformations there are ample opportunities for the formation of a logical culture. Great value has clarification of issues related to the characterization of the transformations being made: are they equivalent or logical, is it necessary to consider several cases, is verification necessary? The difficulties that have to be overcome here are related to the fact that it is not always possible to characterize the same transformation unambiguously: in some cases it may turn out to be, for example, equivalent, in others the equivalence will be violated.

As a result of studying the material of the line of equations, students should not only master the application of algorithmic instructions to solve specific problems, but also learn to use logical means to justify decisions in cases where this is necessary.

4. Logical rationales when studying equations

When studying the material of a line of equations, considerable attention is paid to the issues of substantiating the process of solving specific tasks. At the initial stages of studying an algebra course and in the mathematics course of previous grades, these justifications are of an empirical, inductive nature. As experience is gained in solving equations and systems of various classes, the general properties of transformations become increasingly important. Finally, the achieved level of proficiency in various solution methods allows us to highlight the most frequently used transformations (equivalence and logical consequence). Algebra textbooks have significant differences in the methods of justification described. Nevertheless, all the indicated directions are highlighted, and in a common sequence for them. Let's briefly look at each of these areas.

Empirical rationale for the decision process. In this way, methods for solving the first classes of equations being studied are described. In particular, this is typical for equations of the 1st degree with one unknown. The technique for studying these equations consists of presenting an algorithm for solving such equations and analyzing several typical examples. Naturally, this algorithm is not formed immediately. Before this, several examples are analyzed, and the purpose of the consideration is to highlight in the sequence of actions the operations necessary to describe the algorithm. The teacher’s explanation may be as follows: “We need to solve the equation 5x+4=3x+10. We will try to collect all the terms containing the unknown in one part, and all the terms that do not contain the unknown in the other part of the equation. Let's add the number (--4) to both sides of the equation, this equation will take the form 5x=3x+10--4. Now we add (--3x) to both sides of the equation, we get the equation 5x--3x=10--4. We present similar terms on the left side of the equation, and on the right side we calculate the value of the expression; the equation becomes 2x=6. Divide both sides of the equation by 2, we get x=3.” This story is accompanied by a record of transformations appearing sequentially on the board:

By analyzing the solution, the teacher can come to the rules for solving 1st degree equations with one unknown. Let us draw attention to some formal gaps in this presentation. First of all, such a story does not focus on the fact that under the influence of transformations the equation is transformed into some new equation. Students seem to be dealing with the same equation all the time. If the emphasis was placed directly on the transition from one equation to another, then this would require a more careful analysis of the concepts associated with equivalence, which is not typical for the first stages of algebra training.

Further, the question of whether all the roots of the equation have been found is not raised here. Even if it arises during the discussion of the decision process, an answer to it, as a rule, is not given. The main role is played by the actions of transferring terms from one part of the equation to another, grouping similar terms.

Thus, the issues of justifying the solution of the equation are in the background, and the formation of strong transformation skills is in the first place. From this we can conclude: at this stage, checking the found root serves as a necessary part of justifying the correctness of the solution.

Outwardly, the difference between the two methods of justification (besides the fact that the first uses the term “set”) is manifested in the fact that in the first of them the properties of equalities with variables are used, and in the second - the properties of numerical equalities. The difficulty of learning any of these methods is approximately the same.

The transition to deductive justification can be made on various materials. For example, this can be done when studying a linear equation with two variables, a system of two linear equations with two unknowns, a linear equation with one unknown. It is necessary, however, to note that, whatever the method of justification, it is not an end in itself in the course of school mathematics. The purpose of learning rationales is to ensure that the decision process is informed. Once it has been achieved, further use is already reasonable reception leads to the formation of a skill that students use in the future, returning to the rationale for the technique only occasionally.

Introduction to justify the solution of equations and their systems of concepts of equivalence and logical implication. The considered methods of justification are based on the connection of the line of equations and inequalities with the numerical system. However, consistent application of these techniques is difficult due to the cumbersome nature of the reasoning. Therefore, at a certain stage of studying the content of an algebra course, a general logical system of justifications is identified. It has already been said that this system includes the concepts of equivalence and logical consequence

Let's turn to the parsed equation 5x+4=3x+10. Using equivalence, its solution is carried out as follows: “Since the transfer of terms of the equation from one part to another with a change in sign is an equivalent transformation, then, having carried it out, we arrive at an equation equivalent to the given one: 5x - 3x = 10 - 4. Simplifying the expressions on the left and right sides of the equation, we get 2x=6, whence x=3.”

In the absence of the concepts of equivalence and logical consequence, the description of the solution process also becomes gradually more and more compressed. The absence of these terms is manifested in the fact that the description of the solution itself does not contain elements of justification, which is quite difficult to produce under these conditions. For this reason, in manuals where equivalence and logical consequence appear late, relatively much attention is paid to the formation not of general techniques for solving equations, but of skills for solving equations of certain classes.

The use of logical terminology when describing solutions allows, in parallel with finding the roots, to also obtain a logical justification.” The role of logical concepts is especially important in the final general repetition of the algebra course and the entire high school mathematics course. Since it is necessary to identify the structure of large parts of the studied material, there is no opportunity to again go through the entire path of finding methods for solving various classes of equations, inequalities and their systems. Logical concepts make it possible not only to quickly reconstruct the path to finding such techniques, but also at the same time to justify their correctness. This leads to the development of funds logical thinking students. Taking this into account, at the stages of general repetition it is advisable to formulate the properties of equivalence and logical consequence in a general form and illustrate them with tasks related to various classes of equations and their systems.

What is an equation?

RO system. We will consider a description of the methodology for working on constructing and solving equations by considering various definitions of the equation. school encyclopedia an equation is defined as “two expressions joined by an equal sign; these expressions involve one or more variables called an unknown. To solve an equation means to find all those values of the unknowns (roots or solutions to the equation) at which it turns into a true equality or to establish that there are no such values.” It also defines an equation as “an analytical representation of the problem of finding the values of the arguments for which the values of two functions are equal.” It is clear that by analytical notation we mean a notation of equality, the left or right parts of which contain an unknown (unknown) letter (or number). It is the literal expression that determines the function of the letters included in it, specified on admissible numeric values.

The introduction of a problem notation (about finding an unknown quantity) using an equation begins with a specific problem. Methods for composing and solving equations are based on the relationship of the whole and its parts, and not on the 6 rules for finding unknowns in addition, subtraction, multiplication, and division. In order to find a way to solve the equation, it is enough to determine first using the diagram, and then immediately using the formula, what the unknown quantity is: a part or a whole. If a known quantity is a whole, then to find it you need to add, and if it is a part, then you need to subtract the known parts from the whole. Thus, the child does not need to remember the rules for finding an unknown addend, minuend and subtrahend. The child’s success and his skill in solving equations will depend on whether the child can move from describing the relationship between quantities using a diagram to describing it using a formula and vice versa. It is this transition from an equation as one type of formula to a diagram and determining with the help of a diagram the nature (part or whole) of an unknown quantity that are the basic skills that make it possible to solve any equations containing the actions of addition and subtraction. In other words, children must understand that for the right choice To solve an equation, and therefore a problem, you need to be able to see the relationship between the whole and the parts, which is where the diagram will help. The diagram here acts as a means of solving the equation, and the equation, in turn, as a means of solving the problem. Therefore, most tasks are focused on composing equations according to a given scheme and solving word problems by drawing up a diagram and, with its help, composing an equation that allows you to find a solution to the problem. Traditional school. The study of equations in the elementary grades of a traditional school occurs in several stages. The traditional school program provides for introducing children to first degree equations with one unknown. Of great importance in terms of preparing for the introduction of equations are exercises for selecting the missing number in equalities, deformed examples, such as 4+=5, 4-=2, -7=3, etc. In the process of performing such exercises, children get used to the idea that not only the sum or difference, but also one of the terms (minued or subtracted) can be unknown. Up to grade 2, an unknown number is usually denoted as follows: , ?, *. Now, letters of the Latin alphabet are used to denote an unknown number. An equality of the form 4 + x = 5 is called an equation. An equality where there is a letter is called an equation. At the first stage, equations are solved based on the composition of the number. The teacher introduces the concept of the unknown, the concept of an equation, shows different forms of reading, teaches how to write equations from dictation, examines the concepts of “solving an equation,” “what is called a root,” “what is a solution to an equation,” and teaches how to check solved equations. At the second stage, solving equations occurs using dependencies between components. In this case, when finding an unknown number, you can use the technique of replacing this equation with an equivalent equation. The transition support can be a graph. I will give examples of equations and their replacement with equivalent equations based on graphs.

x: 5 = 7

x = 7 5

35: 5 = 7

After students learn to solve the simplest equations, more complex equations of the following types are included: 48 - x = 16 + 9, a - (60 - 14) = 27, 51 - (x + 15) = 20, the solution of which is also carried out on the basis of the relationship between the results and components of arithmetic operations, preparation is being made for solving problems by composing equations. To solve such equations, you need knowledge of the order of actions in the expression, as well as the ability to perform simple transformations of expressions. Equations of these types are introduced gradually. At first, the simplest equations are complicated by the fact that they right side is specified not by a number, but by an expression. Next, equations are included in which the known component is given by the expression. It is useful to learn to read these equations with the names of the components. Finally, they begin to solve such equations, where one of the components is an expression involving an unknown number, for example: 60 - (x + 7) = 25, (12 - x) + 10 = 18.

When solving equations of this type, you have to use the rules for finding unknown components twice. Let's consider. Learning to solve such equations requires long exercises in the analysis of expressions and a good knowledge of the rules for finding unknown components. At first, exercises in explaining solved equations are useful. In addition, you should solve such equations more often with a preliminary clarification of what is unknown and what rules need to be remembered in order to solve this equation. This type of work prevents mistakes and helps to master the ability to solve equations.

Particular attention should be paid to checking the solution of the equation. Students must clearly know and understand the sequence and meaning of the actions performed during the test: the found number is substituted for the letter in the expression, then the value of this expression is calculated and, finally, it is compared with the given value or with the calculated value of the expression in another part of the equation. If the numbers are equal, then the equation is solved correctly.

Children can perform the test orally or in writing, but its main parts must always be clearly identified: substitute..., calculate..., compare...

3. THEORITICAL FOUNDATIONS FOR SOLVING EQUATIONS IN PRIMARY CLASSES

3.1 Equations in elementary grades

We will consider a description of the methodology for working on constructing and solving equations by considering various definitions of an equation.

The school encyclopedia defines an equation as “two expressions joined by an equal sign; and these expressions include one or more variables called the unknown. To solve an equation means to find all those values of the unknowns (roots or solutions to the equation) at which it turns into a true equality or to establish that there are no such values” (Istomina 2008:155). There, a definition of an equation is given as “an analytical recording of the problem of finding the values of the arguments for which the values of two functions are equal (Istomina 2008:156).

It is clear that by analytical notation we mean a notation of equality, the left or right parts of which contain an unknown (unknown) letter (or number). It is the literal expression that determines the function of the letters included in it, specified on admissible numeric values.

The introduction of a problem entry (about finding an unknown quantity) using an equation begins with specific task. Methods for composing and solving equations are based on the relationship of the whole and its parts, and not on the 6 rules for finding unknowns in addition, subtraction, multiplication, and division.

In order to find a way to solve the equation, it is enough to determine first using the diagram, and then immediately using the formula, what the unknown quantity is: a part or a whole. If a known quantity is a whole, then to find it you need to add, and if it is a part, then you need to subtract the known parts from the whole. Thus, the child does not need to remember the rules for finding an unknown addend, minuend and subtrahend.

The child’s success and his skill in solving equations will depend on whether the child can move from describing the relationship between quantities using a diagram to describing it using a formula and vice versa. It is this transition from solving as one of the types of formulas to a diagram and determining with the help of a diagram the nature (part or whole) of an unknown quantity are the basic skills that make it possible to solve any equations containing the actions of addition and subtraction.

In other words, children must understand that in order to correctly choose a method for solving an equation, and therefore a problem, they need to be able to see the relationship between the whole and the parts, and this is where the diagram will help. The diagram here acts as a means of solving the equation, and the equation, in turn, as a means of solving the problem. Therefore, most tasks are focused on composing equations according to a given scheme and solving word problems by drawing up a diagram and, with its help, composing an equation that allows you to find solutions to the problem.

Studying equations in primary school occurs in several stages. The school program provides for introducing children to equations of the first degree with one unknown. Of great importance in terms of preparing for the introduction of equations are exercises for selecting the missing number in equalities, deformed examples, such as 4+ = 5, 4- = 2, -7 = 3, etc.

In the process of performing such exercises, children get used to the idea that not only the sum or difference, but also one of the terms (diminished or subtracted) can be unknown.

Up to grade 2, an unknown number is usually denoted as follows: , ?, *. Now, letters of the Latin alphabet are used to denote an unknown number. The equality 4+x=5c is called an equation. An equality where there are letters is called an equation (Appendix A)

In the first stage of Eq. decide based on the composition of the number. The teacher introduces the concept of the unknown, the concept of an equation, shows different forms of reading, teaches how to write equations from dictation, examines the concepts of “solving equations”, “what is called a root”, “what is a solution to an equation”, teaches how to check solved equations.

At the second stage, the equation is solved using the dependency between the components. In this case, when finding an unknown number, you can use the technique of replacing this equation with an equivalent equation. The support of the transition can be the count (Istomina 2008:161).

I will give examples of equations replacing them with equivalent equations based on graphs.

After students learn to solve the simplest equations, more complex equations of the following types are included:

48 - x = 16 + 9

a - (6o -14) = 27

51-(x +15) = 20,

the solution, which is also carried out on the basis of the relationship between the results and components of arithmetic operations, preparation is being made for solving problems by composing equations

To solve such equations, you need knowledge of the order of actions in the expression, as well as the ability to perform simple transformations of expressions. Equations of these types are introduced gradually. At first, the simplest equations are complicated by the fact that their right side is given not by a number, but by an expression.

Next, equations are included in which the known component is given by the expression. It is useful to learn to read these equations with the names of the components. Finally, they begin to solve such equations, where one of the components is an expression involving an unknown number, for example:

(12's) + 10 = 18.

When solving equations of this type, you have to use the rules for finding unknown components twice. Consider:

Learning to solve such equations requires long exercises in the analysis of expressions and a good knowledge of the rules for finding unknown components. At first, exercises in explaining solved equations are useful.

In addition, you should solve such equations more often with a preliminary clarification of what is unknown and what rules need to be remembered in order to solve this equation.

This type of work prevents mistakes and helps to master the ability to solve equations.

Particular attention should be paid to checking the solution of the equation. Students must clearly know and understand the sequence and meaning of the actions performed during verification: the found number is represented instead of a letter in an expression, then the value of this expression is calculated and, finally, it is compared with a given value or with the calculated value of an expression in another part of the equation.

If you get equal numbers, then the equation is solved, right.

Children can perform the test orally or in writing, but at the same time its main links must always be clearly identified: substitute..., calculate..., compare...

Equations are also used to solve problems. There is a rule for composing an equation:

1. It turns out what is known and what is unknown.

2.Oboziachepe unknown for x.

3. Drawing up an equation.

4. Solution of the equation

5. The resulting number is interpreted in accordance with the requirement of the problem (M.L. Bantova, P.V. Beltyukova .2006:222).

A necessary requirement for developing the ability to solve problems using equations is the ability to compose expressions according to their conditions.

Therefore, a record of problem solving is introduced in the form of an expression. Students practice explaining the meaning of expressions compiled according to the conditions of the problem; make up expressions themselves according to the given condition problems, and also make up problems based on their solution, written in the form of expressions.

One of the most difficult moments is writing a problem in the form of an equation, therefore, in the beginning, when drawing up an equation, visual aids are widely used: drawings, diagrams, drawings.

To develop students’ ability to solve problems algebraically, it is necessary that they can solve equations, compose expressions for the problem and understand the essence of the process of “equalizing inequalities,” that is, transforming an inequality into an equation.

Already in the first lessons, children, by comparing two sets, determine which of them contains more elements and what needs to be done so that both sets have the same number of elements.

At the same time, the possibilities of using the algebraic method for solving word problems in the elementary grades are limited, so the arithmetic method remains the main one in school.

Thus, we can conclude that the study of equations continues throughout all three years of primary school.

3 .2 Methodology for working on the equation

Do young children also need equations? It’s easy to understand an example when the answer is hidden in a mysterious “x”, which not everyone can read correctly, either “yake” or “ha”. Solving problems using equations is mysterious and interesting, and hiding secrets for inquisitive person harmful. Therefore, familiarization with equations should begin in the first grade. And you can do it as follows.

Let's solve by substitution method

Solving a system of equations using the substitution method

2x+5y=1 (1 equation)
x-10y=3 (2nd equation)

1. Express
It can be seen that in the second equation there is a variable x with a coefficient of 1, which means that it is easiest to express the variable x from the second equation.
x=3+10y

2.After we have expressed it, we substitute 3+10y into the first equation instead of the variable x.
2(3+10y)+5y=1

3. Solve the resulting equation with one variable.
2(3+10y)+5y=1 (open the brackets)
6+20y+5y=1
25y=1-6
25y=-5 |: (25)
y=-5:25
y=-0.2

The solution to the equation system is the intersection points of the graphs, therefore we need to find x and y, because the intersection point consists of x and y. Let's find x, in the first point where we expressed it, we substitute y there.
x=3+10y
x=3+10*(-0.2)=1

It is customary to write points in the first place we write the variable x, and in the second place the variable y.
Answer: (1; -0.2)

Example #2:

Let's solve using the term-by-term addition (subtraction) method.

Solving a system of equations using the addition method

3x-2y=1 (1 equation)
2x-3y=-10 (2nd equation)

1. We choose a variable, let’s say we choose x. In the first equation, the variable x has a coefficient of 3, in the second - 2. We need to make the coefficients the same, for this we have the right to multiply the equations or divide by any number. We multiply the first equation by 2, and the second by 3 and get a total coefficient of 6.

3x-2y=1 |*2
6x-4y=2

2x-3y=-10 |*3
6x-9y=-30

2. Subtract the second from the first equation to get rid of the variable x. Solve the linear equation.
__6x-4y=2

5y=32 | :5
y=6.4

3. Find x. We substitute the found y into any of the equations, let’s say into the first equation.
3x-2y=1
3x-2*6.4=1
3x-12.8=1
3x=1+12.8
3x=13.8 |:3
x=4.6

The intersection point will be x=4.6; y=6.4
Answer: (4.6; 6.4)

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Currently, educational standards are increasingly turning to competencies as the leading criterion for students’ preparedness for effective activities in a certain field. One of the competencies is the ability to navigate information, the ability to receive it, analyze it, etc., i.e. Students must master certain general operating techniques. Based on the topic “Systems of rational equations” (8th grade), we will consider one of these techniques concerning the analysis of a given system and the construction (choice) of a method for solving it depending on its type. In this case, such an important quality of knowledge is laid down, which is called generality.

Basic general methods solutions to systems of equations are practiced in high school when studying the topic “Systems of Linear Equations” in the 7th grade. This is a substitution method, a method of addition, and equalization of coefficients. Using the material of the topic “Systems of rational equations” (8th grade according to the textbook by S.M. Nikolsky and others), it is usually assumed to train the same methods on systems of a different type and sometimes introduce additional general methods for solving systems. In real practice, when a student faces with a system of equations, he needs to navigate himself and choose a method for solving it, but an analysis of textbooks has shown that in them the process of analyzing a system and choosing a solution method is not made the subject of special learning, but only trains the ability to apply the learned method to a given system. As a result, students do not always master complete system knowledge and skills, based on which you can choose (build) an adequate, most effective way to solve a given system. We made an attempt to form such a system of knowledge and skills. After all, solving systems of equations is important not only in terms of the content of the mathematics course; they are used in physics, chemistry, when solving technical and engineering problems, when working with models of economic, social, biological and other phenomena and processes.

Using the example of several lessons for the 8th grade, we will show how we plan to organize joint activities of students and teachers to highlight the content of the named skill. at the first stage. Since one of the components is the analysis of a given system for the presence of solutions, we will devote one of the first lessons to this issue. Then we will try to highlight a general technique for solving systems of equations and connect them with methods known to schoolchildren. For this purpose, we have developed recommendations and a special system of tasks. Principles of constructing a task system for the first stage of training the following:

– the order of tasks is fixed, it performs a guiding function, allowing schoolchildren, together with the teacher, to build an indicative basis for the activity of solving an arbitrary system of rational equations and ultimately create a scheme for solving it,

– each subsequent system is related to previous tasks and reasoning, but contains one or more new important ideas that logically develop the topic,

– variables in systems vary: not always familiar x And y(after all, when modeling real problems from a variety of fields with systems of equations, it is not always convenient to introduce the notation x And y),

– in addition to tasks where a system of equations is given, creative tasks related to the invention of certain systems.

Materials for lessons

1. Systems without solutions.

a) The case when the system has a contradictory equation (without solutions):

№ 1. Answer : .

One of the most obvious cases: you can immediately notice that the first equation has no solutions. If there were a non-negative number on the right side of the first equation, then such a system would have a solution. Having slightly complicated this system, together with schoolchildren you can “invent”, for example, the following systems that have no solutions:

etc. Answer : .

We draw the attention of schoolchildren to the fact that no matter what the second equation of the system is in this case, it will not have solutions (remember the definition of a solution to a system of equations).

№ 2. Answer : .

The first equation of the system has solutions. On the left side of the second equation is the sum of two non-negative numbers, and on the right is a negative number. Contradiction. Note that at least one contradictory equation is sufficient to give the answer.

№ 3. Answer : .

The contradictory equation is somewhat more disguised. Here, in order to recognize it, you need to see in the second equation the formula for the square of the difference. The following is similar to number 1.

Assignment for schoolchildren: create more systems that do not have solutions.

№ 4. Answer : .

If you immediately notice or remember that a fraction with a non-zero numerator cannot be equal to zero, then the answer is obvious. If you replace zero in the condition with a non-zero number, then solutions to the system may appear.

№ 5. Answer : .

Result of division negative number a negative cannot be negative, so the first equation (and therefore the system) has no solutions.

b) The case when the system contains undefined expressions (ODZ is empty):

№ 6. Answer : .

In the first equation, under the root (radical) sign there is a negative expression, which means that such an arithmetic square root does not exist for any values x. We recall the definition of a solution to a system of equations and conclude that the system is inconsistent, i.e. has no solutions. Thus, the solution to this system (and a number of others) must begin with ODZ. After all, if it becomes clear that the ODZ is empty (as in this system), then the set of solutions will be empty.

This system is not a system of rational equations, i.e. is not included in the topic under consideration, but it contains a fundamentally important idea, so it is useful to give it at this stage. In addition, this will allow you to repeat and consolidate the definition of a system of rational equations. Eighth graders are already familiar with the definition and properties of the arithmetic square root.

Discussion: How else can you “construct” an inconsistent equation? What restrictions can there be on the meanings of expressions? From this part of the business lesson conclusion that an equation (and therefore the corresponding system of equations) has no solutions when:

a) equality cannot be satisfied due to certain properties:

sign restrictions: , , ,

fraction at ,

combinations: , , at and etc.

b) some expression included in it is not defined (that is, does not exist, does not make sense) (see task No. 4):

Does not exist when

Does not exist at .

Here it is worth drawing a parallel with tasks based on the same ideas. These are tasks to find the ODZ of variables in an expression, the domain of definition of a function (DOF), the set of values of a function (expression).

c) The case when one equation in a system contradicts another:

№ 7. Answer : .

One of the most obvious cases: we see that the left sides of both equations coincide, but the right ones do not. Contradiction.

№ 8. Answer : .

If you divide the second equation by 4 and move all the terms of each equation to one side, you will see that the equations contradict each other.

Here the question logically arises: what to do if you did not immediately notice that the system is incompatible? Answer: solve it using known methods. The answer will come naturally if you do everything correctly and understand about degenerate equations (0=0, 4=0, etc.), when faced with which many schoolchildren get confused, as school practice shows. Therefore, in order to overcome possible difficulties here, it is important to draw students’ attention to the fact that when solving any equations or systems, the question is always asked the same: “For what values of the unknown is the equality true?” or, respectively, “For which pairs (triples, quadruples, ...) of variables are all the equalities of the system simultaneously true?” Keeping this in mind, it is not difficult to understand that if during the solution the result is something like 0 = 4, then this “equation” and the original system have no solutions; and if it turns out, for example, 0 = 0 and there are no other contradictions, then the system has infinitely many solutions.

Assignment for schoolchildren: come up with several more systems that do not have solutions, such that when one number or sign is replaced in it by another, solutions appear. Enter the invented systems in pairs into the table:

A system without solutions	A system that has solutions

Thus, the result primary analysis systems can be one of three important conclusions:

1) (–) the system has no solutions, no further solution is needed,

2) (+) the system has a solution (solutions) need to be solved,

3) (?) the system may have solutions (or may not have them), you need to solve and remember what was said above.

After this part of the lesson, together with the schoolchildren, we conclude that solving the system must begin with its analysis, because If you can immediately understand that it has no solutions, then you won’t have to waste time on solving it, and you can immediately give the correct answer. There is also an educational effect in this regarding the importance of a preliminary analysis of a situation, object, phenomenon.

This material is used to develop the important skill of “looking” into a system and its components – equations. Note that the same skill can be practiced when solving equations (for example, by replacing the unknown). It is also useful when solving systems that have a solution.

It is worth drawing the attention of schoolchildren to various terms, used in relation to equations and systems that do not have solutions (inconsistent, contradictory). This is important for understanding mathematical problems and texts taken from various sources.

To consolidate the material, including terminology, and check the results of this part of the lesson, students are offered a small exercise: fill out the following table (in each cell, put the signs +, – or?, depending on whether what is indicated in the column header characterizes this system). Table columns: system | has solutions | answer: ? | some expression is not defined | contradictory | incompatible | joint

2. The case when one of the equations contains only one unknown.

№ 9. Answer : And .

There are no obvious contradictions in this system (unlike the previous ones). You can notice that in the first equation of the system there is only one variable ( d), so we can solve the first equation immediately. Its roots are -1 and 2. Let's substitute these values in turn into the second equation and find another unknown - z. Here we recall that the solution to a system of two equations with two unknowns is couples numbers.

When solving this system, schoolchildren have a reasonable question: “In what order should we write the numbers in the answer, because there is no x And y? Answer: in alphabetical order (as in the case of x And y).

3. The case when there is an explicit general expression in several equations, i.e. the generalized substitution leading to the answer is already prepared.

Remember the standard substitution method, known to schoolchildren from the 7th grade. We note that it works in any systems of equations, not only in systems of linear equations.

We consider the idea that you can substitute not only a variable into another equation, but also a certain expression. For this there must be identical expressions in several equations of the system. In this case it is true. Here a reasonable question may arise: “What to do if there are no identical expressions in the equations?”

So let's move on to generalized substitution method and touch on the idea of expression as generalized variable (this is where it starts unknown replacement method, used in solving equations and systems.). In this system, you can replace it with a new variable z. Then the system will take the form of an elementary system of linear equations. An analysis of textbooks and methods for solving systems of equations showed that a very wide class of systems offered in a school mathematics course is solved using a generalized substitution method, which can be called the central, main method. Let us try to solve all the systems proposed below using this method.

There are two options for carrying out a generalized substitution: b 2 and b 2 + u 2. The second one is more convenient in this case, although in order to apply it, the original system must be “prepared”: decomposed left side second factoring equation. The first requires more algebraic transformations, therefore, the likelihood of errors in solving increases. Thus, sometimes the substitution will have to be prepared (before executing).

Here we begin to identify and record techniques, allowing you to highlight common expressions in two equations. IN in this example– method of factorization. What other methods might there be? There can be a lot of them. This area can be called “creative”, because here you need to “invent” a way to make identical expressions appear, moreover, convenient for further solving the system. The “creative” area is very vast.

There are also two options for performing substitution. In the above version, another technique is used - multiplying both sides of one of the equations of the system by an unknown. A subtle point: you cannot multiply by zero. But this is exactly what ODZ is like here!

4. The case when there are no suitable general expressions for substitution in the equations, but they can easily be isolated.

Using this system as an example, you can “invent” a new method for 8th graders - division method. This system is solved by the division method and solved by the generalized substitution method, which in this case actually duplicates the division method in an implicit form.

Here we are faced with the fact that the system has not a finite number of solutions, but an infinite number. How to write down the answer in this case? Schoolchildren often have difficulties in such cases.r 2) or with fractions (if you substitute j 2). It is more convenient and reliable to work with integers, so it is better to choose the first option, although both will lead to the answer. Can be done here replacement, but there is no need.

Is it possible to apply the addition method to this system? Going straight to the original system is pointless, because both unknowns will remain. But if you multiply the equations by suitable numbers, then adding the resulting equations can eliminate one of the unknowns, which will help solve the system. We get generalized addition method or coefficient equalization method(called differently in the literature).

7. The case when it is convenient to replace the unknown

It is easy to notice identical expressions in the equations; replacing them with new unknowns will simplify the system. We come to unknown replacement method.

8. System of three equations with three unknowns.

№ 21. Answer : And .

The generalized substitution method still works here, but the substitution will need to be done multiple times. What if you try to add all the equations? It will work out a= 1. That is in this case, the addition method is very successful.

From the next part of the lesson we do conclusion :

The generalized substitution method allows you to solve a wide range of systems of equations. To solve using this method, you need to select suitable general expressions in several equations, express one of the expressions from some equation in terms of the remaining variables and substitute the system into other equalities in order to reduce the system to an equation with one unknown. At the same time standard method substitution is a special case of the generalized one, and the methods of addition (subtraction), equalization of coefficients, term-by-term division, and replacement of the unknown are “helpers” of the generalized substitution method, allowing one to somewhat simplify the calculations.

In the next lessons - checking this task, discussing the proposed schemes and creating one common schema for the class , reflecting the completeness of the approximate basis for the analysis and solution of this system of equations. Further work will be aimed at organizing the assimilation of the scheme for solving systems of equations identified and recorded together with schoolchildren.

Before introducing the concept of “equation”, it is necessary to repeat the concepts: equality, true equality, the meaning of an expression. And also check the level of development of the skill to read letter expressions.

Studying equations in the early grades should prepare students to solve equations in middle and high school. Solving equations contributes to the formation of knowledge about the properties of arithmetic operations and the formation of computational skills, as well as the development of students’ thinking.

Learning objectives in this topic:

to form in students an understanding of the equation at the level of recognition;
develop the ability to understand the meaning of the task “solve an equation”;
teach to read, write, solve equations of the complexity determined by the program;
teach how to solve problems using equations (algebraic method of solving).

Basic approaches to teaching solving equations:

1) Early familiarization of children with the equation and methods of solving it (M.I. Moreau, M.A. Bantova, I.E. Arginskaya, L.G. Peterson, etc.) - from grades 1-2.

Stages of learning equations:

1) Preparatory

Preparatory exercises:

1. Which entries are correct?

3 + 5 = 8 7 + 2 = 10 10 – 4 = 5

How can I change the result so that the records become correct??

2. Read the expression: 15 - century. Find the value of the expression if b = 3, 4, 10, 11, 16.

3. Among the numbers written on the right, underline the number whose substitution in the box will result in a correct equality.

3+ □ =9 4, 5, 6 , 7

□ - 2 = 4 1, 2, 3, 4, 5, 6

2) Introduction to the concept of "equation"

Students are informed that in mathematics, instead of □, latin letters(x, y, a, b, c) and such records are called an equation: 3 + x = 6, 10: x = 5, etc.

It is important at this stage to strengthen students’ ability to recognize an equation among mathematical expressions: “Find the equation among the proposed entries: x+5=6, x-2, 9=x+2, 3+2=5.”

3) Formation of the ability to solve equations

Ways to solve equations:

In the mathematics course of the educational educational complex "School of Russia":

selection (its use in the first stages is necessary for students to understand the essence of solving the equation);
based on knowledge of the relationship between the components and the result of an arithmetic operation.

According to the program of I.I. Arginskaya (training system of L.V. Zankov):

selection;
using a number series, for example: x+3=8
according to the addition table;
based on decimal composition, for example: 20+x=25. The number 20 contains 2 tens, 25 is 2 tens and 5 units, which means x = 5 units;
based on the relationship between components and the result of actions;
based on the basic properties of equalities: 15●(x+2) = 6● (2x+7)

a) use the rule for multiplying a number by a sum: 15x+30=12x+42 (distribution law);

b) subtract 30 from both sides of the equation: 15x=12x+12;

c) subtract 12x from both sides of the equation: 3x=12;

d) find the unknown factor: x=12: 3; x=4.

In the mathematics course by L.G. Peterson (“School 2000...”) students are introduced to the following methods of solving equations:

· selection;

· based on the relationship between components and the result of actions (between part and whole);

· based on the concepts of “part-whole”, using a diagram in the form of a segment:

· using a number model;

· using a number beam;

Based on the relationship between the area of a rectangle and its sides.

In the course of mathematics by V.N. Rudnitskaya (“ Primary school XXI century"), graphs are widely used in the process of solving equations. For example: x+3=6, x:3=18

When checking an equation, show students that the value on the left side of the equation must be compared to the value on the right side. It is necessary to ensure that the check is carried out consciously.

4) Developing the ability to solve problems using equations.

The process of solving a word problem using equations consists of the following steps:

1. Perception of the task text and primary analysis of its content.

2. Finding a solution:

· identification of unknown numbers;

· selection of the unknown, which is appropriately designated by a letter;

· reformulation of the text of the problem with accepted notation;

· recording the received text.

3. Drawing up an equation, solving it, checking it, translating the found value of the variable into the language of the problem text.

4. Checking the solution to the problem using any known method.

5. Formulating an answer to the problem question.

Task: Two plants smelted 8,430 tons of steel per day. The first plant produced twice as much steel as the second. How much steel was smelted at the first plant and how much at the second?

2x t + x t= 8430t

x tons of steel were smelted by the second plant, 2x tons of steel were smelted by the first plant, (x+2x) tons of steel – two plants together. According to the condition, it is known that this is equal to 8430t.

Check: 2810+2●2810 = 8430

2810 tons of steel were smelted by the second plant, then 2810●2=5620 tons of steel were smelted by the first plant.

Answer: 2810 tons of steel were smelted by the second plant, 5620 tons of steel were smelted by the first plant.

Types of exercises aimed at teaching younger schoolchildren how to solve equations in mathematics textbooks of the educational educational complex “School of Russia”

	Type of exercise	Example assignment
	Tasks with “windows” and missing numbers	2) What numbers are missing? 3) Fill in the blanks so that the equations become true. 12+□=20 8+7-□=14 11-□=5 □-6=7
	Finding equations among other mathematical notations	1) Find equations among the following entries, write them down and solve. 30+x>40 45-5=40 60+x=90 80's 38-8<50 х-8=10 2) Find the extra entry: x+3=15 9+b=12 s-3 15-d=7
	Solving the equation by selection	1) From the numbers 7, 5, 1, 3, select for each equation a value of x that will give the correct equality. 9+x=14 7-x=2 x-1=0 x+5=6 x+7=10 5's=4 10's=5 x+3=4 2) Read the equation and select the value of the unknown that will give the correct equality. k+3 = 13 18=y+10 14=x+7 3) Selecting the values of x, solve the equations: x 6=12 4 x=12 12:x=3
	Finding the unknown component of an arithmetic operation	2) Solve the equations with explanation: 43+x=90 x-28=70 37x=50 Finish your conclusions: To find an unknown term, you need... To find an unknown minuend, you need... To find an unknown subtrahend, you need...
	Solving equations without indicating how to find the unknown	1) Solve the equations: 73x=70 35+x=40 k-6=24 2) Solve the equations and check: 28+x=39 94x=60 x-25=75 3) What is x equal to in the following equations? x+x+x=30 x-18=16-16 43 x=43:x x+20=12+8 4) Solve the equations with explanation: 18 x=54 x:16=3 57:x=3 5) Write down the equations and solve them: A) The unknown number was divided by 8 to get 120. B) What number must you divide 81 by to get 3?
	Solving equations without indicating how to find the unknown, but with an additional condition	1) Write down those equations whose solution is the number 10. x+8=18 47-y=40 y-8=2 y-3=7 50's=40 x+3=13 2) Find the missing numbers and solve the equations: x+□=36 x-15=□ □-x=20 3) Write down equations that can be solved by subtraction and solve them: x-24=46 x+35=60 39+x=59 72-x=40 x-35=60
	Explanation of already solved equations, search for errors	1) Explain the solution of equations and verification: 76:x=38 x 7=84 x=76:38 x=84:7 x=2 x=12 2) Find equations solved incorrectly and solve them: 768x=700 x+10=190 x-380=100 x=768-700 x=190+10 x=380-100 x=68 x=200 x=280
	Comparison of equations without calculation and with calculation of the value of the unknown, comparison of solutions to equations	1) Compare the equations of each pair and say, without calculating, in which of them the x value will be greater: x+34=68 96's=15 x+38=68 96's=18 2) Compare the equations of each pair and their solutions: x 3=120 x+90=160 75 x=75 x:3=120 x-90=160 75+x=75
	Solving problems algebraically	1) Solve the problems by creating an equation: A) The product of the intended number and the number 8 is equal to the difference between the numbers 11288 and 2920. B) The quotient of the numbers 2082 and 6 is equal to the sum of the intended number and the number 48. 2) Solve the problem: “The book has 48 pages. Dasha read the book for three days, 9 pages daily. How many pages does she have left to read?

2) Later familiarization of younger schoolchildren with the equation and methods for solving it (grade 4). Long preparatory period (N.B. Istomina). The focus of the tasks is on the development of basic techniques of mental activity (analysis, synthesis, comparison, classification, generalization).