Having received a general idea of equalities, and having become acquainted with one of their types - numerical equalities, you can start talking about another type of equalities that is very important from a practical point of view - equations. In this article we will look at what is an equation, and what is called the root of the equation. Here we will give the corresponding definitions and also present various examples equations and their roots.
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What is an equation?
Targeted introduction to equations usually begins in mathematics lessons in 2nd grade. At this time the following is given equation definition:
Definition.
The equation is an equality containing an unknown number that needs to be found.
Unknown numbers in equations are usually denoted using small Latin letters, for example, p, t, u, etc., but the letters x, y and z are most often used.
Thus, the equation is determined from the point of view of the form of writing. In other words, equality is an equation when it obeys specified rules records – contains the letter whose value needs to be found.
Let us give examples of the very first and most simple equations. Let's start with equations of the form x=8, y=3, etc. Equations that contain arithmetic signs along with numbers and letters look a little more complicated, for example, x+2=3, z−2=5, 3 t=9, 8:x=2.
The variety of equations grows after becoming familiar with - equations with brackets begin to appear, for example, 2·(x−1)=18 and x+3·(x+2·(x−2))=3. An unknown letter in an equation can appear several times, for example, x+3+3·x−2−x=9, also letters can be on the left side of the equation, on its right side, or on both sides of the equation, for example, x· (3+1)−4=8, 7−3=z+1 or 3·x−4=2·(x+12) .
Further after studying natural numbers acquaintance with integer, rational, real numbers occurs, new ones are learned mathematical objects: powers, roots, logarithms, etc., while more and more new types of equations containing these things appear. Examples of them can be seen in the article basic types of equations studying at school.
In 7th grade, along with the letters, which mean some specific numbers, begin to consider the letters that can take different meanings, they are called variables (see article). At the same time, the word “variable” is introduced into the definition of the equation, and it becomes like this:
Definition.
Equation called an equality containing a variable whose value needs to be found.
For example, the equation x+3=6·x+7 is an equation with the variable x, and 3·z−1+z=0 is an equation with the variable z.
During algebra lessons in the same 7th grade, we encounter equations containing not one, but two different unknown variables. They are called equations in two variables. In the future, the presence of three or more variables in the equations is allowed.
Definition.
Equations with one, two, three, etc. variables– these are equations containing in their writing one, two, three, ... unknown variables, respectively.
For example, the equation 3.2 x+0.5=1 is an equation with one variable x, in turn, an equation of the form x−y=3 is an equation with two variables x and y. And one more example: x 2 +(y−1) 2 +(z+0.5) 2 =27. It is clear that such an equation is an equation with three unknown variables x, y and z.
What is the root of an equation?
The definition of an equation is directly related to the definition of the root of this equation. Let's carry out some reasoning that will help us understand what the root of the equation is.
Let's say we have an equation with one letter (variable). If instead of a letter included in the entry of this equation we substitute a certain number, then the equation becomes numerical equality. Moreover, the resulting equality can be either true or false. For example, if you substitute the number 2 instead of the letter a in the equation a+1=5, you will get the incorrect numerical equality 2+1=5. If we substitute the number 4 instead of a in this equation, we get true equality 4+1=5 .
In practice, in the vast majority of cases, the interest is in those values of the variable whose substitution into the equation gives the correct equality; these values are called roots or solutions given equation.
Definition.
Root of the equation- this is the value of the letter (variable), upon substitution of which the equation turns into a correct numerical equality.
Note that the root of an equation in one variable is also called the solution of the equation. In other words, the solution to an equation and the root of the equation are the same thing.
Let us explain this definition with an example. To do this, let's return to the equation written above a+1=5. According to the stated definition of the root of an equation, the number 4 is the root of this equation, since when substituting this number instead of the letter a we get the correct equality 4+1=5, and the number 2 is not its root, since it corresponds to an incorrect equality of the form 2+1= 5 .
At this point, a number of natural questions arise: “Does any equation have a root, and how many roots does it have?” given equation"? We will answer them.
There are both equations that have roots and equations that do not have roots. For example, the equation x+1=5 has root 4, but the equation 0 x=5 has no roots, since no matter what number we substitute in this equation instead of the variable x, we will get the incorrect equality 0=5.
As for the number of roots of an equation, they exist as equations that have some final number roots (one, two, three, etc.), and equations having infinitely many roots. For example, the equation x−2=4 has a single root 6, the roots of the equation x 2 =9 are two numbers −3 and 3, the equation x·(x−1)·(x−2)=0 has three roots 0, 1 and 2, and the solution to the equation x=x is any number, that is, it has infinite set roots.
A few words should be said about the accepted notation for the roots of the equation. If the equation has no roots, then they usually write “the equation has no roots,” or use the sign empty set∅. If the equation has roots, then they are written separated by commas, or written as elements of the set in curly brackets. For example, if the roots of the equation are the numbers −1, 2 and 4, then write −1, 2, 4 or (−1, 2, 4). It is also permissible to write down the roots of the equation in the form of simple equalities. For example, if the equation includes the letter x, and the roots of this equation are the numbers 3 and 5, then you can write x=3, x=5, and subscripts x 1 =3, x 2 =5 are often added to the variable, as if indicating the numbers roots of the equation. An infinite set of roots of an equation is usually written in the form; if possible, the notation for sets of natural numbers N, integers Z, and real numbers R is also used. For example, if the root of an equation with a variable x is any integer, then write , and if the roots of an equation with a variable y are any real number from 1 to 9 inclusive, then write .
For equations with two, three and big amount variables, as a rule, the term “root of the equation” is not used; in these cases they say “solution of the equation”. What is called solving equations with several variables? Let us give the corresponding definition.
Definition.
Solving an equation with two, three, etc. variables called a pair, three, etc. values of the variables, turning this equation into a correct numerical equality.
Let us show explanatory examples. Consider an equation with two variables x+y=7. Let's substitute the number 1 instead of x, and the number 2 instead of y, and we have the equality 1+2=7. Obviously, it is incorrect, therefore, the pair of values x=1, y=2 is not a solution to the written equation. If we take a pair of values x=4, y=3, then after substitution into the equation we will arrive at the correct equality 4+3=7, therefore, this pair of variable values, by definition, is a solution to the equation x+y=7.
Equations with several variables, like equations with one variable, may have no roots, may have a finite number of roots, or may have an infinite number of roots.
Pairs, triplets, quadruples, etc. The values of variables are often written briefly, listing their values separated by commas in parentheses. In this case, the numbers written in brackets correspond to the variables in alphabetical order. Let's clarify this point by returning to the previous equation x+y=7. The solution to this equation x=4, y=3 can be briefly written as (4, 3).
The greatest attention in the school course of mathematics, algebra and the beginnings of analysis is given to finding the roots of equations with one variable. We will discuss the rules of this process in great detail in the article. solving equations.
Bibliography.
- Mathematics. 2 classes Textbook for general education institutions with adj. per electron carrier. At 2 p.m. Part 1 / [M. I. Moro, M. A. Bantova, G. V. Beltyukova, etc.] - 3rd ed. - M.: Education, 2012. - 96 p.: ill. - (School of Russia). - ISBN 978-5-09-028297-0.
- Algebra: textbook for 7th grade general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 17th ed. - M.: Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
- Algebra: 9th grade: educational. for general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2009. - 271 p. : ill. - ISBN 978-5-09-021134-5.
After we have studied the concept of equalities, namely one of their types - numerical equalities, we can move on to another important view– equations. Within of this material we will explain what an equation is and its root, formulate basic definitions and give various examples equations and finding their roots.
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Concept of equation
Usually the concept of an equation is studied at the very beginning school course algebra. Then it is defined like this:
Definition 1
Equation called an equality with an unknown number that needs to be found.
It is customary to designate unknowns as small with Latin letters, for example, t, r, m etc., but most often x, y, z are used. In other words, the equation is determined by the form of its recording, that is, equality will be an equation only when it is reduced to a certain type– it must contain a letter, the meaning that needs to be found.
Let us give some examples of the simplest equations. These can be equalities of the form x = 5, y = 6, etc., as well as those that include arithmetic operations, for example, x + 7 = 38, z − 4 = 2, 8 · t = 4, 6: x = 3.
After the concept of brackets is learned, the concept of equations with brackets appears. These include 7 · (x − 1) = 19, x + 6 · (x + 6 · (x − 8)) = 3, etc. The letter that needs to be found can appear more than once, but several times, like, for example, in the equation x + 2 + 4 · x − 2 − x = 10 . Also, unknowns can be located not only on the left, but also on the right or in both parts at the same time, for example, x (8 + 1) − 7 = 8, 3 − 3 = z + 3 or 8 x − 9 = 2 (x + 17) .
Further, after students become familiar with the concepts of integers, reals, rationals, natural numbers, as well as logarithms, roots and powers, new equations appear that include all these objects. We have devoted a separate article to examples of such expressions.
In the 7th grade curriculum, the concept of variables appears for the first time. These are letters that can take different meanings(for more details, see the article on numerics, literal expressions and expressions with variables). Based on this concept, we can redefine the equation:
Definition 2
The equation is an equality involving a variable whose value needs to be calculated.
That is, for example, the expression x + 3 = 6 x + 7 is an equation with the variable x, and 3 y − 1 + y = 0 is an equation with the variable y.
One equation can have more than one variable, but two or more. They are called, respectively, equations with two, three variables, etc. Let us write down the definition:
Definition 3
Equations with two (three, four or more) variables are equations that include a corresponding number of unknowns.
For example, an equality of the form 3, 7 · x + 0, 6 = 1 is an equation with one variable x, and x − z = 5 is an equation with two variables x and z. An example of an equation with three variables would be x 2 + (y − 6) 2 + (z + 0, 6) 2 = 26.
Root of the equation
When we talk about an equation, the need immediately arises to define 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 a number for the unknown letter, the equation becomes a numerical equality - true or false. So, if in the equation a + 1 = 5 we replace the letter with the number 2, then the equality will become false, and if 4, then the correct equality will be 4 + 1 = 5.
We are more interested in precisely those values with which the variable will turn into a true equality. They are called roots or solutions. Let's write down the definition.
Definition 4
Root of the equation They call the value of a variable that turns a given equation into a true equality.
The root can also be called a solution, or vice versa - both of these concepts mean the same thing.
Example 2
Let's take an example to clarify this definition. Above we gave the equation a + 1 = 5. According to the definition, the root is in this case will be 4, because when substituted instead of a letter it gives the correct numerical equality, and two will not be a solution, since it corresponds to the incorrect equality 2 + 1 = 5.
How many roots can one equation have? Does every equation have a root? Let's answer these questions.
Equations that do not have a single root also exist. An example would be 0 x = 5. We can substitute infinitely many different numbers, but none of them will turn it into a true equality, since multiplying by 0 always gives 0.
There are also equations that have several roots. They can be either finite or infinite a large number of roots.
Example 3
So, in the equation x − 2 = 4 there is only one root - six, in x 2 = 9 two roots - three and minus three, in x · (x − 1) · (x − 2) = 0 three roots - zero, one and two, there are infinitely many roots in the equation x=x.
Now let us explain how to correctly write the roots of the equation. If there are none, then we write: “the equation has no roots.” In this case, you can also indicate the sign of the empty set ∅. If there are roots, then we write them separated by commas or indicate them as elements of a set, enclosing them in curly braces. So, if any equation has three roots - 2, 1 and 5, then we write - 2, 1, 5 or (- 2, 1, 5).
It is allowed to write roots in the form of simple equalities. So, if the unknown in the equation is denoted by the letter y, and the roots are 2 and 7, then we write y = 2 and y = 7. Sometimes subscripts are added to letters, for example, x 1 = 3, x 2 = 5. In this way we point to the numbers of the roots. If the equation has infinitely many solutions, then we write the answer as numerical interval or we use generally accepted notations: the set of natural numbers is denoted by N, integers by Z, and real numbers by R. Let's say, if we need to write that the solution to the equation will be any integer, then we write that x ∈ Z, and if any real number from one to nine, then y ∈ 1, 9.
When an equation has two, three roots or more, then, as a rule, we talk not about roots, but about solutions to the equation. Let us formulate the definition of a solution to an equation with several variables.
Definition 5
The solution to an equation with two, three or more variables is two, three or more values of the variables that turn the given equation into a correct numerical equality.
Let us explain the definition with examples.
Example 4
Let's say we have the expression x + y = 7, which is an equation with two variables. Let's substitute one instead of the first, and two instead of the second. We will get an incorrect equality, which means that this pair of values will not be a solution to this equation. If we take the pair 3 and 4, then the equality becomes true, which means we have found a solution.
Such equations may also have no roots or an infinite number of them. If we need to write down two, three, four or more values, then we write them separated by commas in parentheses. That is, in the example above, the answer will look like (3, 4).
In practice, you most often have to deal with equations containing one variable. We will consider the algorithm for solving them in detail in the article devoted to solving equations.
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In algebra, there is the concept of two types of equalities - identities and equations. Identities are equalities that are valid for any values of the letters included in them. Equations are also equalities, but they are feasible only for certain values of the letters included in them.
According to the conditions of the problem, letters are usually unequal. This means that some of them can take on any acceptable values, called coefficients (or parameters), while others - they are called unknowns - take on values that need to be found in the solution process. As a rule, unknown quantities are denoted in equations by the last letters in (x.y.z, etc.), or by the same letters, but with an index (x 1, x 2, etc.), and known coefficients - by the first letters of that the same alphabet.
Based on the number of unknowns, equations with one, two and several unknowns are distinguished. Thus, all values of the unknowns for which the equation being solved turns into an identity are called solutions of the equations. An equation can be considered solved if all its solutions have been found or it has been proven that it does not have any. The task “solve an equation” is common in practice and means that you need to find the root of the equation.
Definition: the roots of an equation are those values of the unknowns from the admissible region at which the equation being solved turns into an identity.
The algorithm for solving absolutely all equations is the same, and its meaning is to use mathematical transformations this expression lead to more simple view.
Equations that have identical roots, in algebra are called equivalent.
The simplest example: 7x-49=0, root of the equation x=7;
x-7=0, similarly, the root x=7, therefore, the equations are equivalent. (In special cases equivalent equations may have no roots at all).
If the root of an equation is simultaneously the root of another, simpler equation obtained from the original one through transformations, then the latter is called a consequence of the previous equation.
If one of two equations is a consequence of the other, then they are considered equivalent. They are also called equivalent. The example above illustrates this.
Solving even the simplest equations in practice often causes difficulties. As a result of the solution, you can get one root of the equation, two or more, even infinite number- it depends on the type of equations. There are also those that have no roots, they are called unsolvable.
Examples:
1) 15x -20=10; x=2. This is the only root of the equation.
2) 7x - y=0. The equation has an infinite number of roots because each variable can have an infinite number of values.
3) x 2 = - 16. A number raised to the second power always gives positive result, so it is impossible to find the root of the equation. This is one of the unsolvable equations discussed above.
The correctness of the solution is checked by substituting the found roots instead of the letters and solving the resulting example. If the identity is satisfied, the solution is correct.