Start in science. Methods for solving equations

The equation representing quadratic trinomial, is usually called a quadratic equation. From an algebraic point of view, it is described by the formula a*x^2+b*x+c=0. In this formula, x is the unknown that needs to be found (it is called a free variable); a, b and c are numerical coefficients. There are a number of restrictions regarding the components indicated: for example, coefficient a should not be equal to 0.

Solving an equation: the concept of a discriminant

The value of the unknown x at which the quadratic equation turns into a true equality is called the root of such an equation. In order to solve a quadratic equation, you must first find the value of a special coefficient - the discriminant, which will show the number of roots of the equality in question. The discriminant is calculated using the formula D=b^2-4ac. In this case, the result of the calculation can be positive, negative or equal to zero.

It should be borne in mind that the concept requires that only the coefficient a be strictly different from 0. Consequently, the coefficient b can be equal to 0, and the equation itself in this case is of the form a*x^2+c=0. In such a situation, a coefficient value of 0 should be used in the formulas for calculating the discriminant and roots. So, the discriminant in this case will be calculated as D=-4ac.

Solving the equation with a positive discriminant

If the discriminant of the quadratic equation turns out to be positive, we can conclude that this equality has two roots. These roots can be calculated using the following formula: x=(-b±√(b^2-4ac))/2a=(-b±√D)/2a. Thus, to calculate the value of the roots of the quadratic equation at positive value discriminants are used known values coefficients available in . By using the sum and difference in the formula for calculating the roots, the result of the calculations will be two values that make the equality in question true.

Solving the equation with zero and negative discriminant

If the discriminant of the quadratic equation turns out to be equal to 0, we can conclude that specified equation has one root. Strictly speaking, in this situation the equation still has two roots, but due to the zero discriminant they will be equal to each other. In this case x=-b/2a. If, during the calculation process, the value of the discriminant turns out to be negative, it should be concluded that the quadratic equation in question does not have roots, that is, such values of x at which it turns into a true equality.

An equation is a mathematical expression that is an equality and contains an unknown. If an equality is true for any admissible values of the unknowns included in it, then it is called an identity; for example: a relation of the form (x – 1)2 = (x – 1)(x – 1) holds for all values of x.

If an equation containing an unknown x holds only for certain values of x and not for all values of x, as in the case of an identity, then it may be useful to determine those values of x for which the equation is valid. Such values of x are called roots or solutions of the equation. For example, the number 5 is the root of the equation 2x + 7= 17.

In the branch of mathematics called equation theory, the main subject of study is methods for solving equations. IN school course Algebra equations receive a lot of attention.

The history of the study of equations goes back many centuries. The most famous mathematicians who contributed to the development of the theory of equations were:

Archimedes (c. 287–212 BC) was an ancient Greek scientist, mathematician and mechanic. While studying a problem that reduced to a cubic equation, Archimedes discovered the role of the characteristic, which was later called the discriminant.

Francois Viet lived in the 16th century. He made great contributions to the study various problems mathematics. In particular, he introduced letter designations for the coefficients of the equation and established a connection between the roots of the quadratic equation.

Leonhard Euler (1707 – 1783) - mathematician, mechanic, physicist and astronomer. Author of St. 800 works on mathematical analysis, differential equations, geometry, number theory, approximate calculations, celestial mechanics, mathematics, optics, ballistics, shipbuilding, music theory, etc. He had a significant influence on the development of science. He derived formulas (Euler's formulas) expressing trigonometric functions variable x through an exponential function.

Lagrange Joseph Louis (1736 - 1813), French mathematician and mechanic. He has carried out outstanding research, including research on algebra (the symmetric function of the roots of an equation, on differential equations (theory special solutions, method of variation of constants).

J. Lagrange and A. Vandermonde are French mathematicians. In 1771, a method for solving systems of equations (the substitution method) was first used.

Gauss Karl Friedrich (1777 -1855) - German mathematician. He wrote a book outlining the theory of equations for dividing a circle (i.e., the equations xn - 1 = 0), which in many ways was a prototype of Galois theory. Besides common methods solving these equations, established a connection between them and the construction of regular polygons. For the first time since the ancient Greek scientists, he made a significant step forward in this matter, namely: he found all those values of n for which regular n-gon can be constructed with a compass and ruler. I studied the method of addition. I concluded that systems of equations can be added, divided, and multiplied.

O. I. Somov - enriched various parts of mathematics with important and numerous works, among them the theory of certain algebraic equations higher degrees.

Galois Evariste (1811-1832) - French mathematician. His main merit is the formulation of a set of ideas that he came to in connection with the continuation of research on the solvability of algebraic equations, begun by J. Lagrange, N. Abel, and others, and created the theory of algebraic equations of higher degrees with one unknown.

A. V. Pogorelov (1919 – 1981) - His work involves geometric methods with analytical methods theory of partial differential equations. His works also had a significant impact on the theory of nonlinear differential equations.

P. Ruffini - Italian mathematician. He devoted a number of works to proving the unsolvability of equations of degree 5, systematically using the closedness of the set of substitutions.

Despite the fact that scientists have been studying equations for a long time, science does not know how and when people needed to use equations. It is only known that people have been solving problems leading to the solution of the simplest equations since the time they became human. Another 3 - 4 thousand years BC. e. The Egyptians and Babylonians knew how to solve equations. The rule for solving these equations coincides with the modern one, but it is unknown how they got there.

IN Ancient Egypt and Babylon, the method of false position was used. An equation of the first degree with one unknown can always be reduced to the form ax + b = c, in which a, b, c are integers. According to the rules arithmetic operations ax = c - b,

If b > c, then c b is a negative number. Negative numbers were unknown to the Egyptians and many other later peoples (along with positive numbers they began to be used in mathematics only in the seventeenth century). To solve problems that we now solve with equations of the first degree, the false position method was invented. In the Ahmes papyrus, 15 problems are solved by this method. The Egyptians had a special sign for an unknown number, which until recently was read "how" and translated as "heap" ("heap" or "unknown number" of units). Now they read a little less inaccurately: “yeah.” The solution method used by Ahmes is called the method of one false position. Using this method, equations of the form ax = b are solved. This method involves dividing each side of the equation by a. It was used by both the Egyptians and Babylonians. U different nations The method of two false positions was used. The Arabs mechanized this method and obtained the form in which it was transferred to the textbooks of European peoples, including Magnitsky’s Arithmetic. Magnitsky calls the solution a “false rule” and writes in the part of his book outlining this method:

This part is very cunning, because you can put everything with it. Not only what is in citizenship, but also the higher sciences in space, which are listed in the sphere of heaven, as the wise have needs.

The content of Magnitsky's poems can be briefly summarized as follows: this part of arithmetic is very tricky. With its help, you can calculate not only what is needed in everyday practice, but it also solves the “higher” questions that confront the “wise.” Magnitsky uses the “false rule” in the form that the Arabs gave it, calling it “the arithmetic of two errors” or the “method of scales.” Indian mathematicians often gave problems in verse. Lotus problem:

Above the quiet lake, half a measure above the water, the color of the lotus was visible. He grew up alone, and the wind, like a wave, Bent him to the side, and no longer

Flower over water. The fisherman's eye found him two meters from the place where he grew up. How deep is the lake water here? I'll ask you a question.

Types of equations

Linear equations

Linear equations are equations of the form: ax + b = 0, where a and b are some constants. If a is not equal to zero, then the equation has one single root: x = - b: a (ax + b; ax = - b; x = - b: a.).

For example: decide linear equation: 4x + 12 = 0.

Solution: Since a = 4, and b = 12, then x = - 12: 4; x = - 3.

Check: 4 (- 3) + 12 = 0; 0 = 0.

Since 0 = 0, then -3 is the root of the original equation.

Answer. x = -3

If a is equal to zero and b is equal to zero, then the root of the equation ax + b = 0 is any number.

For example:

0 = 0. Since 0 is equal to 0, then the root of the equation 0x + 0 = 0 is any number.

If a is equal to zero and b is not equal to zero, then the equation ax + b = 0 has no roots.

For example:

0 = 6. Since 0 is not equal to 6, then 0x – 6 = 0 has no roots.

Systems of linear equations.

A system of linear equations is a system in which all equations are linear.

To solve a system means to find all its solutions.

Before solving a system of linear equations, you can determine the number of its solutions.

Let a system of equations be given: (a1x + b1y = c1, (a2x + b2y = c2.

If a1 divided by a2 is not equal to b1 divided by b2, then the system has one unique solution.

If a1 divided by a2 is equal to b1 divided by b2, but equal to c1 divided by c2, then the system has no solutions.

If a1 divided by a2 is equal to b1 divided by b2, and equal to c1 divided by c2, then the system has infinitely many solutions.

A system of equations that has at least one solution is called simultaneous.

A joint system is called definite if it has final number solutions, and indefinite if the set of its solutions is infinite.

A system that does not have a single solution is called inconsistent or contradictory.

Methods for solving linear equations

There are several ways to solve linear equations:

1) Selection method. This is the most the simplest way. It lies in the fact that everyone is selected valid values unknown by enumeration.

For example:

Solve the equation.

Let x = 1. Then

4 = 6. Since 4 is not equal to 6, then our assumption that x = 1 was incorrect.

Let x = 2.

6 = 6. Since 6 is equal to 6, then our assumption that x = 2 was correct.

Answer: x = 2.

2) Simplification method

This method consists in transferring all terms containing unknowns to the left side, and known ones to the right side. opposite sign, give similar ones, and divide both sides of the equation by the coefficient of the unknown.

For example:

Solve the equation.

5x – 4 = 11 + 2x;

5x – 2x = 11 + 4;

3x = 15; : (3) x = 5.

Answer. x = 5.

3) Graphic method.

It consists in constructing a graph of functions given equation. Since in a linear equation y = 0, the graph will be parallel to the y-axis. The point of intersection of the graph with the x-axis will be the solution to this equation.

For example:

Solve the equation.

Let y = 7. Then y = 2x + 3.

Let's plot the functions of both equations:

Methods for solving systems of linear equations

In seventh grade, they study three ways to solve systems of equations:

1) Substitution method.

This method consists in expressing one unknown in terms of another in one of the equations. The resulting expression is substituted into another equation, which then turns into an equation with one unknown, and then it is solved. The resulting value of this unknown is substituted into any equation of the original system and the value of the second unknown is found.

For example.

Solve the system of equations.

5x - 2y - 2 = 1.

3x + y = 4; y = 4 - 3x.

Let's substitute the resulting expression into another equation:

5x – 2(4 – 3x) -2 = 1;

5x – 8 + 6x = 1 + 2;

11x = 11; : (11) x = 1.

Let's substitute the resulting value into the equation 3x + y = 4.

3 1 + y = 4;

3 + y = 4; y = 4 – 3; y = 1.

Examination.

/3 1 + 1 = 4,

\5 · 1 – 2 · 1 – 2 = 1;

Answer: x = 1; y = 1.

2) Method of addition.

This method is that if this system consists of equations that, when added term by term, form an equation with one unknown, then by solving this equation, we obtain the value of one of the unknowns. The resulting value of this unknown is substituted into any equation of the original system and the value of the second unknown is found.

For example:

Solve the system of equations.

/3у – 2х = 5,

\5x – 3y = 4.

Let's solve the resulting equation.

3x = 9; : (3) x = 3.

Let's substitute the resulting value into the equation 3y – 2x = 5.

3у – 2 3 = 5;

3у = 11; : (3) y = 11/3; y = 3 2/3.

So x = 3; y = 3 2/3.

Examination.

/3 11/3 – 2 3 = 5,

\5 · 3 – 3 · 11/ 3 = 4;

Answer. x = 3; y = 3 2/3

3) Graphic method.

This method is based on the fact that equations are plotted in one coordinate system. If the graphs of an equation intersect, then the coordinates of the intersection point are the solution to this system. If the graphs of the equation are parallel lines, then this system has no solutions. If the graphs of the equations merge into one straight line, then the system has infinitely many solutions.

For example.

Solve the system of equations.

18x + 3y - 1 = 8.

2x - y = 5; 18x + 3y - 1 = 8;

Y = 5 - 2x; 3y = 9 - 18x; : (3) y = 2x - 5. y = 3 - 6x.

Let's build graphs of the functions y = 2x - 5 and y = 3 - 6x on the same coordinate system.

The graphs of the functions y = 2x - 5 and y = 3 - 6x intersect at point A (1; -3).

Therefore, the solution to this system of equations will be x = 1 and y = -3.

Examination.

2 1 - (- 3) = 5,

18 1 + 3 (-3) - 1 = 8.

18 - 9 – 1 = 8;

Answer. x = 1; y = -3.

Conclusion

Based on all of the above, we can conclude that equations are necessary in modern world not only for solving practical problems, but also as a scientific tool. That is why so many scientists have studied this issue and continue to study it.

Ministry of General and vocational education RF

Municipal educational institution

Gymnasium No. 12

composition

on the topic: Equations and methods for solving them

Completed by: student of class 10 "A"

Krutko Evgeniy

Checked by: mathematics teacher Iskhakova Gulsum Akramovna

Tyumen 2001

Plan................................................. ........................................................ ................................ 1

Introduction........................................................ ........................................................ ........................ 2

Main part................................................ ........................................................ ............... 3

Conclusion................................................. ........................................................ ............... 25

Application................................................. ........................................................ ................ 26

List of used literature......................................................... ........................... 29

Plan.

Introduction.

Historical reference.

Equations. Algebraic equations.

a) Basic definitions.

b) Linear equation and method for solving it.

c) Quadratic equations and methods for solving them.

d) Binomial equations and how to solve them.

d) Cubic equations and ways to solve it.

e) Biquadratic equation and a way to solve it.

g) Equations of the fourth degree and methods for solving them.

g) Equations of high degrees and methods for solving them.

h) Rational algebraic equation and its method

And) Irrational equations and ways to solve it.

j) Equations containing an unknown under a sign.

absolute value and method for solving it.

Transcendental equations.

A) Exponential equations and a way to solve them.

b) Logarithmic equations and a way to solve them.

Introduction

Mathematics education received in secondary school, is essential component general education And general culture modern man. Almost everything that surrounds modern man is all somehow connected with mathematics. A latest achievements in physics, technology and information technology leave no doubt that in the future the state of affairs will remain the same. Therefore, solving many practical problems comes down to solving various types equations that you need to learn to solve.

This work is an attempt to summarize and systematize the studied material on the above topic. I have arranged the material in order of difficulty, starting with the simplest. It includes both the types of equations known to us from the school algebra course, and additional material. At the same time, I tried to show the types of equations that are not studied in the school course, but the knowledge of which may be needed when entering higher education educational institution. In my work, when solving equations, I did not limit myself only to the real solution, but also indicated the complex one, since I believe that otherwise the equation is simply unsolved. After all, if an equation has no real roots, this does not mean that it has no solutions. Unfortunately, due to lack of time, I was not able to present all the material I have, but even with the material presented here, many questions may arise. I hope that my knowledge is enough to answer most questions. So, I begin to present the material.

Mathematics... reveals order,

symmetry and certainty,

and this is most important species beautiful.

Aristotle.

Historical reference

In those distant times, when the sages first began to think about equalities containing unknown quantities, there were probably no coins or wallets. But there were heaps, as well as pots and baskets, which were perfect for the role of storage caches that could hold an unknown number of items. “We are looking for a heap which, together with two thirds, a half and one seventh of it, makes 37...”, taught in the 2nd millennium BC new era Egyptian scribe Ahmes. In the ancients mathematical problems Mesopotamia, India, China, Greece, unknown quantities expressed the number of peacocks in the garden, the number of bulls in the herd, the totality of things taken into account when dividing property. Scribes, officials and priests initiated into secret knowledge, well trained in the science of accounts, coped with such tasks quite successfully.

Sources that have reached us indicate that ancient scientists had some general techniques for solving problems with unknown quantities. However, not a single papyrus or clay tablet contains a description of these techniques. The authors only occasionally supplied their numerical calculations with skimpy comments such as: “Look!”, “Do this!”, “You found the right one.” In this sense, the exception is the “Arithmetic” of the Greek mathematician Diophantus of Alexandria (III century) - a collection of problems for composing equations with a systematic presentation of their solutions.

However, the first manual for solving problems that became widely known was the work of the Baghdad scientist of the 9th century. Muhammad bin Musa al-Khwarizmi. The word "al-jabr" from the Arabic name of this treatise - "Kitab al-jaber wal-mukabala" ("Book of restoration and opposition") - over time turned into the well-known word "algebra", and al-Khwarizmi's work itself served the starting point in the development of the science of solving equations.

equations Algebraic equations

Basic definitions

In algebra, two types of equalities are considered - identities and equations.

Identity is an equality that holds for all (admissible) values of the letters included in it). To record an identity along with a sign

the sign is also used.

The equation is an equality that holds only for certain values of the letters included in it. The letters included in the equation, according to the conditions of the problem, can be unequal: some can take all their permissible values (they are called parameters or coefficients equations and are usually denoted by the first letters Latin alphabet:

, , ... - or the same letters provided with indices: , , ... or , , ...); others whose values need to be found are called unknown(they are usually designated by the last letters of the Latin alphabet: , , , ... - or the same letters provided with indices: , , ... or , , ...).

In general, the equation can be written as follows:

(, , ..., ).

Depending on the number of unknowns, the equation is called an equation with one, two, etc. unknowns.

Usually, equations appear in problems in which you need to find a certain quantity. The equation allows you to formulate the problem in the language of algebra. Having solved the equation, we obtain the value of the desired quantity, which is called the unknown. “Andrey has several rubles in his wallet. If you multiply this number by 2 and then subtract 5, you get 10. How much money does Andrey have?” Let's designate the unknown amount of money as x and write the equation: 2x-5=10.

To talk about ways to solve equations, first you need to define the basic concepts and become familiar with the generally accepted notations. For different types equations, there are various algorithms for solving them. The easiest way to solve equations is of the first degree with one unknown. Many people are familiar with the formula for solving from school quadratic equations. Techniques higher mathematics will help solve equations more high order. The set of numbers on which an equation is defined is closely related to its solutions. Also interesting is the relationship between equations and function graphs, since the representation of equations in graphical form Helps them great.

Description. An equation is a mathematical equality with one or more unknown quantities, for example 2x+3y=0.

Expressions on both sides of the equal sign are called left and right sides of the equation. The letters of the Latin alphabet indicate unknowns. Although there can be any number of unknowns, below we will only talk about equations with one unknown, which we will denote by x.

Degree of equation is the maximum power to which the unknown can be raised. For example,
$3x^4+6x-1=0$ is an equation of the fourth degree, $x-4x^2+6x=8$ is an equation of the second degree.

The numbers by which the unknown is multiplied are called coefficients. In the previous example, the unknown to the fourth power has a coefficient of 3. If, when replacing x with this number, given equality, then we say that this number satisfies the equation. It's called solving the equation, or its root. For example, 3 is the root, or solution, of the equation 2x+8=14, since 2*3+8=6+8=14.

Solving equations. Let's say we want to solve the equation 2x+5=11.

You can substitute some value x into it, for example x=2. Replace x with 2 and get: 2*2+5=4+5=9.

There's something wrong here because on the right side of the equation we should have gotten 11. Let's try x=3: 2*3+5=6+5=11.

The answer is correct. It turns out that if the unknown takes the value 3, then equality is satisfied. Therefore, we have shown that the number 3 is a solution to the equation.

The method we used to solve this equation is called selection method. Obviously it is inconvenient to use. Moreover, it cannot even be called a method. To verify this, just try to apply it to an equation of the form $x^4-5x^2+16=2365$.

Solution methods. There are so-called “rules of the game” that will be useful to familiarize yourself with. Our goal is to determine the value of the unknown that satisfies the equation. Therefore, it is necessary to identify the unknown in some way. To do this, it is necessary to transfer the terms of the equation from one part to another. The first rule for solving equations is...

1. When moving a member of an equation from one part to another, its sign changes to the opposite: plus changes to minus and vice versa. Consider as an example the equation 2x+5=11. Let's move 5 from the left side to the right: 2x=11-5. The equation will become 2x=6.

Let's move on to the second rule.
2. Both sides of the equation can be multiplied and divided by a number that is not equal to zero. Let's apply this rule to our equation: $x=\frac62=3$. On the left side of the equality, only the unknown x remained, therefore, we found its value and solved the equation.

We have just looked at the simplest problem - linear equation with one unknown. Equations of this type always have a solution, moreover, they can always be solved using the simplest operations: addition, subtraction, multiplication and division. Alas, not all equations are so simple. Moreover, their degree of complexity increases very quickly. For example, equations of the second degree can be easily solved by any student high school, but methods for solving systems of linear equations or equations of higher degrees are studied only in high school.

Etc., it is logical to get acquainted with equations of other types. Next in line are linear equations, the targeted study of which begins in algebra lessons in the 7th grade.

It is clear that first you need to explain what a linear equation is, give a definition of a linear equation, its coefficients, show it general form. Then you can figure out how many solutions a linear equation has depending on the values of the coefficients, and how the roots are found. This will allow you to move on to solving examples, and thereby consolidate the learned theory. In this article we will do this: we will dwell in detail on all the theoretical and practical points relating to linear equations and their solutions.

Let’s say right away that here we will consider only linear equations with one variable, and in a separate article we will study the principles of solution linear equations with two variables.

Page navigation.

What is a linear equation?

The definition of a linear equation is given by the way it is written. Moreover, in different mathematics and algebra textbooks, the formulations of the definitions of linear equations have some differences that do not affect the essence of the issue.

For example, in the algebra textbook for grade 7 by Yu. N. Makarychev et al., a linear equation is defined as follows:

Definition.

Equation of the form a x=b, where x is a variable, a and b are some numbers, is called linear equation with one variable.

Let us give examples of linear equations that meet the stated definition. For example, 5 x = 10 is a linear equation with one variable x, here the coefficient a is 5, and the number b is 10. Another example: −2.3·y=0 is also a linear equation, but with a variable y, in which a=−2.3 and b=0. And in linear equations x=−2 and −x=3.33 a are not present explicitly and are equal to 1 and −1, respectively, while in the first equation b=−2, and in the second - b=3.33.

And a year earlier, in the textbook of mathematics by N. Ya. Vilenkin, linear equations with one unknown, in addition to equations of the form a x = b, also considered equations that can be brought to this form by transferring terms from one part of the equation to another with the opposite sign, as well as using casting similar terms. According to this definition, equations of the form 5 x = 2 x + 6, etc. also linear.

In turn, in the algebra textbook for grade 7 by A. G. Mordkovich the following definition is given:

Definition.

Linear equation with one variable x is an equation of the form a·x+b=0, where a and b are some numbers called coefficients of the linear equation.

For example, linear equations of this type are 2 x−12=0, here the coefficient a is 2, and b is equal to −12, and 0.2 y+4.6=0 with coefficients a=0.2 and b =4.6. But at the same time, there are examples of linear equations that have the form not a·x+b=0, but a·x=b, for example, 3·x=12.

Let us, so that we do not have any discrepancies in the future, by a linear equation with one variable x and coefficients a and b we mean an equation of the form a x + b = 0. This type of linear equation seems to be the most justified, since linear equations are algebraic equations first degree. And all the other equations mentioned above, as well as equations that, using equivalent transformations are reduced to the form a·x+b=0 , we will call equations that reduce to linear equations. With this approach, the equation 2 x+6=0 is a linear equation, and 2 x=−6, 4+25 y=6+24 y, 4 (x+5)=12, etc. - These are equations that reduce to linear ones.

How to solve linear equations?

Now it’s time to figure out how linear equations a·x+b=0 are solved. In other words, it's time to find out whether a linear equation has roots, and if so, how many of them and how to find them.

The presence of roots of a linear equation depends on the values of the coefficients a and b. In this case, the linear equation a x+b=0 has

the only root for a≠0,
has no roots for a=0 and b≠0,
has infinitely many roots for a=0 and b=0, in which case any number is a root of a linear equation.

Let us explain how these results were obtained.

We know that to solve equations we can move from the original equation to equivalent equations, that is, to equations with the same roots or, like the original one, without roots. To do this, you can use the following equivalent transformations:

transferring a term from one side of the equation to another with the opposite sign,
as well as multiplying or dividing both sides of an equation by the same non-zero number.

So, in a linear equation with one variable of the form a x+b=0 we can move the term b from the left side to right side with the opposite sign. In this case, the equation will take the form a·x=−b.

And then it begs the question of dividing both sides of the equation by the number a. But there is one thing: the number a can be equal to zero, in which case such division is impossible. To deal with this problem, we will first assume that the number a is non-zero, and we will consider the case of a being equal to zero separately a little later.

So, when a is not equal to zero, then we can divide both sides of the equation a·x=−b by a, after which it will be transformed to the form x=(−b):a, this result can be written using the fractional slash as.

Thus, for a≠0, the linear equation a·x+b=0 is equivalent to the equation, from which its root is visible.

It is easy to show that this root is unique, that is, the linear equation has no other roots. This allows you to do the opposite method.

Let's denote the root as x 1. Let us assume that there is another root of the linear equation, which we denote as x 2, and x 2 ≠x 1, which, due to definitions equal numbers through the difference is equivalent to the condition x 1 −x 2 ≠0. Since x 1 and x 2 are roots of the linear equation a·x+b=0, then the numerical equalities a·x 1 +b=0 and a·x 2 +b=0 hold. We can subtract the corresponding parts of these equalities, which the properties of numerical equalities allow us to do, we have a·x 1 +b−(a·x 2 +b)=0−0, from which a·(x 1 −x 2)+( b−b)=0 and then a·(x 1 −x 2)=0 . But this equality is impossible, since both a≠0 and x 1 − x 2 ≠0. So we came to a contradiction, which proves the uniqueness of the root of the linear equation a·x+b=0 for a≠0.

So we solved the linear equation a·x+b=0 for a≠0. The first result given at the beginning of this paragraph is justified. There are two more left that meet the condition a=0.

When a=0, the linear equation a·x+b=0 takes the form 0·x+b=0. From this equation and the property of multiplying numbers by zero it follows that no matter what number we take as x, when it is substituted into the equation 0 x + b=0, the numerical equality b=0 will be obtained. This equality is true when b=0, and in other cases when b≠0 this equality is false.

Consequently, with a=0 and b=0, any number is the root of the linear equation a·x+b=0, since under these conditions, substituting any number for x gives the correct numerical equality 0=0. And when a=0 and b≠0, the linear equation a·x+b=0 has no roots, since under these conditions, substituting any number instead of x leads to the incorrect numerical equality b=0.

The given justifications allow us to formulate a sequence of actions that allows us to solve any linear equation. So, algorithm for solving linear equation is:

First, by writing the linear equation, we find the values of the coefficients a and b.
If a=0 and b=0, then this equation has infinitely many roots, namely, any number is a root of this linear equation.
If a is nonzero, then

the coefficient b is transferred to the right side with the opposite sign, and the linear equation is transformed to the form a·x=−b,
after which both sides of the resulting equation are divided by a nonzero number a, which gives the desired root of the original linear equation.

The written algorithm is a comprehensive answer to the question of how to solve linear equations.

In conclusion of this point, it is worth saying that a similar algorithm is used to solve equations of the form a·x=b. Its difference is that when a≠0, both sides of the equation are immediately divided by this number; here b is already in the required part of the equation and there is no need to transfer it.

To solve equations of the form a x = b, the following algorithm is used:

If a=0 and b=0, then the equation has infinitely many roots, which are any numbers.
If a=0 and b≠0, then the original equation has no roots.
If a is non-zero, then both sides of the equation are divided by a non-zero number a, from which the only root of the equation is found, equal to b/a.

Examples of solving linear equations

Let's move on to practice. Let's look at how the algorithm for solving linear equations is used. Let us give solutions to typical examples corresponding to different meanings coefficients of linear equations.

Example.

Solve the linear equation 0·x−0=0.

Solution.

In this linear equation, a=0 and b=−0 , which is the same as b=0 . Therefore, this equation has infinitely many roots; any number is a root of this equation.

Answer:

x – any number.

Example.

Does the linear equation 0 x + 2.7 = 0 have solutions?

Solution.

IN in this case coefficient a is equal to zero, and coefficient b of this linear equation is equal to 2.7, that is, different from zero. Therefore, a linear equation has no roots.