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After we studied the concept of equalities, namely, one of their species - numerical equality, one can go to another important form - equations. As part of this material, we explain what equation and its root, we formulate the main definitions and give various examples of equations and finding their roots.
The concept of equation
Usually the concept of the equation is studied at the very beginning of the school year of algebra. Then it is determined like this:
Definition 1.
Equation It is called equality with an unknown number to find.
It is customary to designate unknown to small Latin letters, for example, t, r, m, etc., but the X, Y, Z is most often used. In other words, the equation determines the form of its record, that is, equality will be equation only when it is shown to a certain type - it should be the letter, the value that needs to be found.
Let us give a few examples of the simplest equations. These may be equalities of the form x \u003d 5, y \u003d 6, etc., as well as those that include arithmetic action, for example, x + 7 \u003d 38, z - 4 \u003d 2, 8 · t \u003d 4, 6: X \u003d 3.
After the concept of brackets is studied, the concept of equations with brackets appears. These include 7 · (x - 1) \u003d 19, x + 6 · (x + 6 · (x - 8)) \u003d 3, etc. The letter that needs to be found may occur more than once, but a few, like For example, in equation X + 2 + 4 · X - 2 - x \u003d 10. Also unknown can be located not only to the left, but also to the right or in both parts at the same time, for example, x · (8 + 1) - 7 \u003d 8, 3 - 3 \u003d z + 3 or 8 · x - 9 \u003d 2 · (x + 17).
Next, after students get acquainted with the concept of integer, valid, rational, natural numbers, as well as logarithms, roots and degrees, new equations appear, including all these objects. Examples of such expressions, we devoted a separate article.
In the program for grade 7, for the first time, the concept of variables arises. These are the letters that can take different values \u200b\u200b(see the article on numerical, alphabone expressions and expressions with variables). Based on this concept, we can give a new definition of equation:
Definition 2.
The equation - It is equality that includes a variable whose value must be calculated.
That is, for example, the expression X + 3 \u003d 6 · X + 7 is an equation with a variable x, a 3 · y - 1 + y \u003d 0 - equation from the variable y.
In one equation, there may be not one variable, but two or more. They are called respectively equations with two, three variables, etc. We write down the definition:
Definition 3.
Equations with two (three, four or more) variables are called equations that include the appropriate number of unknown.
For example, the equality of the form 3, 7 · x + 0, 6 \u003d 1 is an equation with one variable x, and x - z \u003d 5 is an equation with two variables x and z. An example of an equation with three variables may be an expression x 2 + (y - 6) 2 + (Z + 0, 6) 2 \u003d 26.
Root of the equation
When we talk about the equation, immediately arises the need to decide on the concept of its root. Let's try to explain what it means.
Example 1.
We are given a certain equation that includes one variable. If we substitute instead of an unknown letter, the equation will become a numerical equality - faithful or incorrect. So, if we replace the letter 2 in equation A + 1 \u003d 5, then the equality becomes incorrect, and if 4, then the faithful equality is 4 + 1 \u003d 5.
We are more interested in exactly the meanings with which the variable will turn into faithful equality. They are called roots or solutions. We write down.
Definition 4.
The root of the equation Call such a value of a variable that draws this equation to true equality.
The root can also be called a solution, or vice versa - both of these concepts mean the same thing.
Example 2.
Take an example to explain this definition. Above, we led the equation a + 1 \u003d 5. According to the definition, the root in this case will be 4, because in the substitution instead of the letter it gives the right numerical equality, and the deuce will not be a solution, since it is responsible for the incorrect equality 2 + 1 \u003d 5.
How many roots may have one equation? Is any equation have a root? Reply to these questions.
Equations that do not have a single root also exist. An example may be 0 · X \u003d 5. We can substitute in it infinitely many different numbers, but none of them turn it into faithful equality, since multiplication by 0 always gives 0.
There are also equations having several roots. They can have both finite, and infinitely a large number of roots.
Example 3.
So, in equation x - 2 \u003d 4 there is only one root - six, in x 2 \u003d 9 two roots - three and minus three, in x · (x - 1) · (x - 2) \u003d 0 three roots - zero, One and two, in the equation x \u003d x the roots are infinitely a lot.
Now explain how to record the roots of the equation correctly. If they are not, then we are writing: "The root equation does not have." You can also indicate the sign of an empty set ∅. If the roots are, then we write them through the comma or indicate as elements of the set, enclosing in curly brackets. So, if some equation has three roots - 2, 1 and 5, then we write - 2, 1, 5 or (- 2, 1, 5).
It is allowed to record the roots in the form of simple equalities. So, unknown in the equation is indicated by the letter Y, and the roots are 2 and 7, then we write y \u003d 2 and y \u003d 7. Sometimes lower indexes are added to the letters, for example, x 1 \u003d 3, x 2 \u003d 5. So we point to the roots numbers. If the solutions at the equation are infinitely a lot, then we record an answer as a numerical gap or use generally accepted designations: the set of natural numbers is denoted by n, integers - z, valid - R. Let's say if we need to record that the solution of the equation is any integer, then we write that x ∈ Z, and if any actual from one to nine, then y ∈ 1, 9.
When the equation is two, three roots or more, then, as a rule, they are not talking about roots, but about solutions of the equation. We formulate the definition of solving an equation with several variables.
Definition 5.
The solution of the equation with two, three and more variables is two, three and more values \u200b\u200bof the variables that draw this equation to the correct numerical equality.
Let us explain the definition on the examples.
Example 4.
Suppose we have an expression X + Y \u003d 7, which is an equation with two variables. Substitute instead of the first unit, and instead of the second two. We will have incorrect equality, it means that this pair of values \u200b\u200bwill not be solved by this equation. If we take a couple of 3 and 4, the equality will be true, it means that we have found a solution.
Such equations may also not have roots or have an infinite number. If we need to write two, three, four or more values, we write them through the comma in parentheses. That is, in the example above the answer will look like (3, 4).
In practice, most often have to deal with equations containing one variable. The algorithm of their solutions will consider in detail in the article on solving equations.
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equation inequality mathematics
The concept of "equation" refers to the most important general imaging concepts.
There are various interpretations of the concept of "equation".
AND I. Vilenkin et al. Leads logical - mathematical definition of the equation. Let the set of algebraic operations, x - variable on m; Then the equation on the set M relative to X is the predicate of the species, where they are the terms relative to the specified operations, the record of which includes a symbol. Analogically determine the equation from two or more variables.
The terms adopted in the Logic terms "Term" and "Predicate" correspond to such terms of school mathematics as an "expression" and "proposal with a variable". Therefore, the following definition can be considered the following definition: "The proposal with a variable that has the form of equality between two expressions with this variable is called the equation." Such a definition is given in the textbook "Algebra and the beginning of Analysis" by A.N Kolmogorov, etc. Equality with a variable is called the equation. The value of the variable in which the equality with the variable refers to the correct numerical equality is called the root of the equation.
Often, especially at the beginning of the systematic course of algebra, the concept of the equation is introduced by means of the allocation of it from the algebraic method of solving problems. For example, in the textbook Sh.A.Alimova, etc. The concept of the equation is entered on the material of the text task. The transition to the concept of the equation is carried out on the basis of the analysis of some formal features of the record, expressing the content of this problem in an algebraic form: "Equality containing an unknown number indicated by the letter is called the equation." The indicated method of administering the concept of the equation corresponds to another component of the concept of the equation - applied.
Another approach to the concept of the equation is obtained in the preparation of the area of \u200b\u200bdetermining the equation and the set of its roots. For example, in the textbook D.K. Fadeev "the letter equality that does not necessarily turn into the correct numerical equality under the permissible sets of letters is called the equation."
You can meet the third version of the definition, the role of which is manifested when studying the graphical method of solving equations: "The equation is the equality of two functions."
Among everyone studied in the course of mathematics of the types of equations V.I. Mishin allocates a relatively limit number of basic types. These include: a linear equation with one unknown, a system of two linear equations with two unknowns, square equations, the simplest irrational and transcendental.
Yu.M. Kolyagin et al. Classify the type of functions representing the right and left parts of the equations:
The equation is called:
algebraic, if both - algebraic functions;
transcendental, if although one of the functions and transcendental;
rational algebraic (or simply rational), if algebraic functions and rational;
irrational algebraic (or simply irrational), if at least one of the algebraic functions and irrational;
whole rational if function and whole rational;
fractional rational, if at least one of the rational functions and fractional rational.
The equation, where - the polynomial of the standard species is called linear (first degree), square (into the second degree), cubic (third degree) and in general, if a polynomial, has the first, second, third and in general, the degree.
The school studies several types of equations. These include: linear equations with one non-known, square equations, irrational and transcendental equations, rational equations. These types of equations are studied with great care, it is indicated for them to automatize the execution of the solution algorithm, the form in which the answer should be recorded.
Types of equations and decision methods:
1) linear equation
An equation with one variable is called equality containing only one variable.
The equation is called the root (or by the solution) of the equation, in which the equation turns into a correct numerical equality.
Find all the roots of the equation or prove that they are not - it means to solve the equation.
Example 1: Solve equation.
2) Square equation
The square equation is the equation of the species where the coefficients A, B and C are any valid numbers, and hi? 0.
The roots of the square equation are the values \u200b\u200bof the variable in which the square equation refers to the correct numerical equality.
Solve the square equation - it means to find all its roots or establish that there are no roots.
Example 2: Solve equation
This equation can be solved either through the theorem of the Vieta, or through the discriminant.
Answer: x 1 \u003d -1, x 2 \u003d -2.
3) rational equations
rational equations - equations of the form
where and polynomials, attacks of the equation of the species, where are rational.
Example 3: Solve equation
4) irrational equations
Irrational equations are equations in which the variable is contained under the sign of the root or under the sign of the construction operation to a fractional degree.
Example 4: solve equation
Establish both parts per square:
5) indicative and logarithmic equations
With the solution of the indicative equations, two basic methods are used: a) the transition from the equation to the equation; b) the introduction of new variables. Sometimes you have to use artificial techniques.
Logarithmic equations - are solved by three methods, that is, the transition from the equation to the equation - the result; the method of introducing new variables of logarithming, that is, the transition from the equation to the equation.
As well as in many cases, when solving the logarithmic equation, it has to use the properties of the logarithm of the work, private, degree, root.
The algebra considers two types of equalities - identities and equations.
The identity is equality that is performed with all (permissible) values \u200b\u200bof the letters of it.
The equation is equality that is performed only with some values \u200b\u200bof the letters of it.
The letters included in the equation may be unequal: Some can take all their permissible values, which are called coefficients (sometimes - parameters) of equations, others, the values \u200b\u200bof which are required to find, are called unknowns of this equation (as a rule, they are denoted by the latest letters of the Latin alphabet x , y, z, u, v, w, or the same letters equipped with indexes.
Equations are:
Quadratic equations
Rational equations
Equations containing a variable under the sign of the module
Irrational equations
Indicative equations
Logarithmic equations
Systems of equations:
Systems of rational equations
Systems of nonlinear equations
Symmetric systems
Mixed systems
Foreign roots arising in the process of transformations can be identified by checking. Of course, if all transformations led us to a chain of equivalent equations, then the check is optional. However, this is not always possible to achieve, it is easier to ensure that each chain equation is a consequence of the previous one, i.e. In order not to lose the roots. In this case, the verification is an element of the solution. It should be noted that it is often easier to check what to justify that it is not necessary. In addition, the test is a means of controlling the correctness of the calculated calculations. Sometimes it is useful to do this: at each stage of solving the equation to determine the gaps in which the roots of the equation may be. All roots that do not belong to these gaps are outsiders and should be discarded. However, the rest of the roots still need to check the substitution to the initial equation.
Each algebraic equation always has at least one solution, valid or complex.
In analytical geometry, one equation with two unknown is interpreted using a curve on the plane, the coordinates of all points of which satisfy this equation. One equation with three unknowns is interpreted using a surface in three-dimensional space. With this interpretation, the solution system equation coincides with the task of the search for points of intersection of lines, surfaces, etc. The equation with a large number of unknowns is interpreted using manifolds in n-dimensional spaces.
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Equations of mathematical physics are differential equations with private derivatives, as well as some related equations of other types (integral, integro-differential, etc.), which leads mathematical analysis of physical phenomena. For the theory of the equation of mathematical physics, tasks are characterized in this form, as needed in the study of the physical phenomenon. The circle of the equation of mathematical physics with expansion of the scope of mathematical analysis is also steadily expanding. When systematizing the results obtained, there is a need to include in the theory of the equation of mathematical physics of the equation and the problem of more general species than those that appear when analyzing specific phenomena; However, for such equations and tasks, it is characteristic that their properties admit more or less visual physical interpretation.
Chemical equations are images of chemical reactions through chemical signs, chemical, numbers, and mathematical signs. The possibility of such a description of chemical reactions pointed out in 1789 A. Lavoisier, based on the preservation of the mass law; However, the universal use of the chemical equation received only in the 1st half of the 19th century.
Linear equation - This is an algebraic equation. In this equation, the total degree of components of its polynomials is equal to one.
Linear equations are presented in this form:
In general form: a. 1 x. 1 + a. 2 x. 2 + … + a n x n + b. = 0
In canonical form: a 1 x 1 + a 2 x 2 + ... + a n x n \u003d b.
Linear equation with one variable.
The linear equation with the 1st variable is given to the form:
aX.+ b.=0.
For example:
2x + 7 \u003d 0. Where a \u003d 2, b \u003d 7;
0.1x - 2.3 \u003d 0. Where a \u003d 0.1, b \u003d -2.3;
12x + 1/2 \u003d 0. Where a \u003d 12, B \u003d 1/2.
The number of roots depending on a. and b.:
When a.= b.=0 So, the equation has an unlimited number of solutions, since.
When a.=0 , b.≠ 0 It means that the equation has no roots, because.
When a. ≠ 0 It means that the equation has only one root.
Linear equation with two variables.
By the variable equation x. It is equality type A (x) \u003d b (x)where A (X) and B (x) - Expressions OT x.. When substituting the set T. Values x.the equation is obtained by true numerical equality called a lot of truth this equation either solution of a given equation, and all the values \u200b\u200bof the variable - root equations.
Linear equations of 2 variables are presented in this form:
In general form: aX + BY + C \u003d 0,
In canonical form: aX + BY \u003d -C,
In the form of a linear function: y \u003d kx + mwhere 
.
The solution or roots of this equation is such a pair of variable values (x; y)which turns it into identity. These solutions (roots) in a linear equation with 2 variables unlimited quantity. The geometric model (graph) of this equation is straight y \u003d kx + m.
If there is an X-in equation in a square, then such an equation is called
