I. Expressions in which, along with letters, numbers, signs can be used arithmetic operations and brackets are called algebraic expressions.

Examples of algebraic expressions:

2m-n; 3 · (2a+b); 0.24x; 0.3a-b · (4a + 2b); a 2 - 2ab;

Since a letter in an algebraic expression can be replaced by some various numbers, then the letter is called a variable, and the algebraic expression itself is called an expression with a variable.

II. If in an algebraic expression letters (variables) are replaced by their values ​​and the specified actions are performed, then the resulting number is called the value of the algebraic expression.

Examples. Find the value of an expression:

1) a + 2b -c for a = -2; b = 10; c = -3.5.

2) |x| + |y| -|z| at x = -8; y=-5; z = 6.

Decision.

1) a + 2b -c for a = -2; b = 10; c = -3.5. Instead of variables, we substitute their values. We get:

— 2+ 2 · 10- (-3,5) = -2 + 20 +3,5 = 18 + 3,5 = 21,5.

2) |x| + |y| -|z| at x = -8; y=-5; z = 6. We substitute the specified values. Remember that the module negative number is equal to its opposite number, and the modulus positive number equal to that number. We get:

|-8| + |-5| -|6| = 8 + 5 -6 = 7.

III. The values ​​of a letter (variable) for which the algebraic expression makes sense are called valid values ​​of the letter (variable).

Examples. At what values variable expression doesn't make sense?

Decision. We know that it is impossible to divide by zero, therefore, each of these expressions will not make sense with the value of the letter (variable) that turns the denominator of the fraction to zero!

In example 1), this is the value a = 0. Indeed, if instead of a we substitute 0, then the number 6 will need to be divided by 0, but this cannot be done. Answer: expression 1) does not make sense when a = 0.

In example 2) the denominator x - 4 = 0 at x = 4, therefore, this value x = 4 and cannot be taken. Answer: expression 2) does not make sense for x = 4.

In example 3) the denominator is x + 2 = 0 for x = -2. Answer: expression 3) does not make sense at x = -2.

In example 4) the denominator is 5 -|x| = 0 for |x| = 5. And since |5| = 5 and |-5| \u003d 5, then you cannot take x \u003d 5 and x \u003d -5. Answer: expression 4) does not make sense for x = -5 and for x = 5.
IV. Two expressions are called identically equal if for any allowed values variables, the corresponding values ​​of these expressions are equal.

Example: 5 (a - b) and 5a - 5b are identical, since the equality 5 (a - b) = 5a - 5b will be true for any values ​​of a and b. Equality 5 (a - b) = 5a - 5b is an identity.

Identity is an equality that is valid for all admissible values ​​of the variables included in it. Examples of identities already known to you are, for example, the properties of addition and multiplication, distributive property.

The replacement of one expression by another, identically equal to it, is called an identical transformation or simply a transformation of an expression. Identity transformations expressions with variables are executed based on the properties of operations on numbers.

Examples.

a) convert the expression to identically equal using the distributive property of multiplication:

1) 10 (1.2x + 2.3y); 2) 1.5 (a -2b + 4c); 3) a·(6m -2n + k).

Decision. Recall the distributive property (law) of multiplication:

(a+b) c=a c+b c(distributive law of multiplication with respect to addition: in order to multiply the sum of two numbers by a third number, you can multiply each term by this number and add the results).
(a-b) c=a c-b c(distributive law of multiplication with respect to subtraction: in order to multiply the difference of two numbers by a third number, you can multiply by this number reduced and subtracted separately and subtract the second from the first result).

1) 10 (1.2x + 2.3y) \u003d 10 1.2x + 10 2.3y \u003d 12x + 23y.

2) 1.5 (a -2b + 4c) = 1.5a -3b + 6c.

3) a (6m -2n + k) = 6am -2an +ak.

b) convert the expression to identically equal using the commutative and associative properties(laws of) addition:

4) x + 4.5 + 2x + 6.5; 5) (3a + 2.1) + 7.8; 6) 5.4s -3 -2.5 -2.3s.

Decision. We apply the laws (properties) of addition:

a+b=b+a(displacement: the sum does not change from the rearrangement of the terms).
(a+b)+c=a+(b+c)(associative: in order to add a third number to the sum of two terms, you can add the sum of the second and third to the first number).

4) x + 4.5 + 2x + 6.5 = (x + 2x) + (4.5 + 6.5) = 3x + 11.

5) (3a + 2.1) + 7.8 = 3a + (2.1 + 7.8) = 3a + 9.9.

6) 6) 5.4s -3 -2.5 -2.3s = (5.4s -2.3s) + (-3 -2.5) = 3.1s -5.5.

in) transform the expression into identically equal using the commutative and associative properties (laws) of multiplication:

7) 4 · X · (-2,5); 8) -3,5 · 2y · (-one); 9) 3a · (-3) · 2s.

Decision. Let's apply the laws (properties) of multiplication:

a b=b a(displacement: permutation of factors does not change the product).
(a b) c=a (b c)(combinative: to multiply the product of two numbers by a third number, you can multiply the first number by the product of the second and third).

Let's solve the problem.

The student bought notebooks for 2 kopecks. for a notebook and a textbook for 8 kopecks. How much did he pay for the entire purchase?

To find out the cost of all notebooks, you need to multiply the price of one notebook by the number of notebooks. This means that the cost of notebooks will be equal to kopecks.

The cost of the entire purchase will be

Note that it is customary to omit the multiplication sign in front of a multiplier expressed by a letter, it is simply implied. Therefore, the previous entry can be represented as follows:

We have obtained a formula for solving the problem. It shows that to solve the problem it is necessary to multiply the price of a notebook by the number of purchased notebooks and add the cost of a textbook to the product.

Instead of the word "formula" for such entries, the name "algebraic expression" is also used.

An algebraic expression is a record consisting of numbers indicated by numbers or letters and connected by action signs.

For brevity, instead of "algebraic expression" they sometimes say simply "expression".

Here are some more examples of algebraic expressions:

From these examples, we see that an algebraic expression may consist of only one letter, or may not contain numbers at all, indicated by letters (two recent examples). In that last case expression is also called arithmetic expression.

Let's give the letter the value 5 in the algebraic expression we received (it means that the student bought 5 notebooks). Substituting the number 5 instead, we get:

which is equal to 18 (that is, 18 kopecks).

The number 18 is the value of this algebraic expression when

The value of an algebraic expression is the number that will be obtained if we substitute the data of their values ​​​​in this expression instead of letters and perform the indicated actions on the numbers.

For example, we can say: the value of the expression at is 12 (12 kopecks).

The value of the same expression for is 14 (14 kopecks), etc.

We see that the meaning of an algebraic expression depends on what values ​​we give to the letters included in it. True, sometimes it happens that the meaning of an expression does not depend on the meanings of the letters included in it. For example, the expression is equal to 6 for any values ​​of a.

Let's find in the form of an example numerical values expressions for different values letters a and b.

Substitute in given expression instead of a, the number 4, and instead of 6, the number 2 and calculate the resulting expression:

So, when the value of the expression For is equal to 16.

In the same way, we find that when the value of the expression is 29, when and it is equal to 2, etc.

The results of calculations can be written in the form of a table that will clearly show how the value of the expression changes depending on the change in the values ​​of the letters included in it.

Let's create a table with three rows. In the first line we will write the values ​​a, in the second - the values ​​6 and

in the third - the values ​​of the expression. We get such a table.

Algebra lessons introduce us to various types expressions. As new material arrives, the expressions become more complex. When you get acquainted with the powers, they are gradually added to the expression, complicating it. It also happens with fractions and other expressions.

To make the study of the material as convenient as possible, this is done by certain names in order to be able to highlight them. This article will give full review all basic school algebraic expressions.

Monomials and polynomials

Expressions monomials and polynomials are studied in school curriculum starting from 7th grade. Textbooks have given definitions of this kind.

Definition 1

monomials are numbers, variables, their degrees with natural indicator, any works made with their help.

Definition 2

polynomials is called the sum of monomials.

If we take, for example, the number 5, the variable x, the degree z 7, then the products of the form 5 x and 7 x 2 7 z 7 are considered single members. When the sum of monomials of the form is taken 5+x or z 7 + 7 + 7 x 2 7 z 7, then we get a polynomial.

To distinguish a monomial from a polynomial, pay attention to the degrees and their definitions. The concept of coefficient is important. When cast similar terms they are divided into the free term of the polynomial or the leading coefficient.

Most often, some actions are performed on monomials and polynomials, after which the expression is reduced to see a monomial. Addition, subtraction, multiplication, and division are performed, relying on an algorithm to perform operations on polynomials.

When there is one variable, it is possible to divide the polynomial into a polynomial, which are represented as a product. This action is called the factorization of a polynomial.

Rational (algebraic) fractions

The concept of rational fractions is studied in grade 8 high school. Some authors call them algebraic fractions.

Definition 3

Rational algebraic fraction They call a fraction in which polynomials or monomials, numbers, take the place of the numerator and denominator.

Consider the example of the record rational fractions of type 3 x + 2 , 2 a + 3 b 4 , x 2 + 1 x 2 - 2 and 2 2 x + - 5 1 5 y 3 x x 2 + 4 . Based on the definition, we can say that every fraction is considered a rational fraction.

Algebraic fractions can be added, subtracted, multiplied, divided, raised to a power. This is discussed in more detail in the section on operations with algebraic fractions. If it is necessary to convert a fraction, they often use the property of reduction and reduction to a common denominator.

Rational Expressions

AT school course the concept of irrational fractions is being studied, since it is necessary to work with rational expressions.

Definition 4

Rational Expressions are considered numerical and alphabetic expressions, where rational numbers and letters are used with addition, subtraction, multiplication, division, raising to an integer power.

Rational expressions may not have signs belonging to the function that lead to irrationality. Rational expressions do not contain roots, degrees with fractional irrational indicators, degrees with variables in the exponent, logarithmic expressions, trigonometric functions etc.

Based on the rule above, we will give examples of rational expressions. From the above definition, we have that both a numerical expression of the form 1 2 + 3 4, and 5, 2 + (- 0, 1) 2 2 - 3 5 - 4 3 4 + 2: 12 7 - 1 + 7 - 2 2 3 3 - 2 1 + 0 , 3 are considered rational. Expressions containing letter designations, also refer to rational a 2 + b 2 3 a - 0 , 5 b , with variables of the form a x 2 + b x + c and x 2 + x y - y 2 1 2 x - 1 .

All rational expressions subdivided into integers and fractions.

Integer rational expressions

Definition 5

Integer rational expressions are such expressions that do not contain division into expressions with variables of negative degree.

From the definition, we have that a whole rational expression is also an expression containing letters, for example, a + 1 , an expression containing several variables, for example, x 2 · y 3 − z + 3 2 and a + b 3 .

Expressions like x: (y − 1) and 2 x + 1 x 2 - 2 x + 7 - 4 cannot be rational integers, since they have division by an expression with variables.

Fractional rational expressions

Definition 6

Fractional rational expression is an expression that contains division by an expression with negative degree variables.

It follows from the definition that fractional rational expressions can be 1: x, 5 x 3 - y 3 + x + x 2 and 3 5 7 - a - 1 + a 2 - (a + 1) (a - 2) 2 .

If we consider expressions of this type (2 x - x 2): 4 and a 2 2 - b 3 3 + c 4 + 1 4, 2, then they are not considered fractional rational, since they do not have expressions with variables in the denominator.

Expressions with powers

Definition 7

Expressions that contain powers in any part of the notation are called power expressions or power expressions.

For the concept, we give an example of such an expression. They may not contain variables, for example, 2 3 , 32 - 1 5 + 1 . 5 3 . 5 · 5 - 2 5 - 1 . 5 . Power expressions of the form 3 · x 3 · x - 1 + 3 x , x · y 2 1 3 are also characteristic. In order to solve them, it is necessary to perform some transformations.

Irrational expressions, expressions with roots

The root, which has a place in the expression, gives it a different name. They are called irrational.

Definition 8

Irrational expressions name expressions that have signs of roots in the record.

It can be seen from the definition that these are expressions of the form 64 , x - 1 4 3 + 3 3 , 2 + 1 2 - 1 - 2 + 3 2 , a + 1 a 1 2 + 2 , x y , 3 x + 1 + 6 x 2 + 5 x and x + 6 + x - 2 3 + 1 4 x 2 3 + 3 - 1 1 3 . Each of them has at least one root icon. The roots and degrees are connected, so you can see expressions such as x 7 3 - 2 5, n 4 8 · m 3 5: 4 · m 2 n + 3.

Trigonometric expressions

Definition 9

trigonometric expression are expressions containing sin , cos , tg and ctg and their inverses - arcsin , arccos , arctg and arcctg .

Examples of trigonometric functions are obvious: sin π 4 cos π 6 cos 6 x - 1 and 2 sin x t g 2 x + 3 , 4 3 t g π - arcsin - 3 5 .

To work with such functions, you need to use the properties, basic formulas direct and inverse functions. The article transformation of trigonometric functions will reveal this issue in more detail.

Logarithmic Expressions

After getting acquainted with logarithms, we can talk about complex logarithmic expressions.

Definition 10

Expressions that have logarithms are called logarithmic.

An example of such functions would be log 3 9 + ln e , log 2 (4 a b) , log 7 2 (x 7 3) log 3 2 x - 3 5 + log x 2 + 1 (x 4 + 2) .

You can find such expressions where there are degrees and logarithms. This is understandable, since from the definition of the logarithm it follows that this is an exponent. Then we get expressions like x l g x - 10 , log 3 3 x 2 + 2 x - 3 , log x + 1 (x 2 + 2 x + 1) 5 x - 2 .

To deepen the study of the material, you should refer to the material on the transformation of logarithmic expressions.

Fractions

There are expressions special kind, which are called fractions. Since they have a numerator and a denominator, they can contain not just numeric values, but also expressions of any type. Consider the definition of a fraction.

Definition 11

Shot they call such an expression that has a numerator and a denominator, in which there are both numerical and alphabetic designations or expressions.

Examples of fractions that have numbers in the numerator and denominator look like this 1 4 , 2 , 2 - 6 2 7 , π 2 , - e π , (− 15) (− 2) . The numerator and denominator can contain both numerical and literal expressions of the form (a + 1) 3 , (a + b + c) (a 2 + b 2) , 1 3 + 1 - 1 3 - 1 1 1 + 1 1 + 1 5 , cos 2 α - sin 2 α 1 + 3 t g α , 2 + ln 5 ln x .

Although expressions such as 2 5 − 3 7 , x x 2 + 1: 5 are not fractions, however, they do have a fraction in their notation.

General expression

Senior classes consider tasks of increased difficulty, which contains all the combined tasks of group C in the USE. These expressions are particularly complex and have various combinations of roots, logarithms, powers, and trigonometric functions. These are jobs like x 2 - 1 sin x + π 3 or sin a r c t g x - a x 1 + x 2 .

Their appearance indicates that it can be attributed to any of the above species. Most often they are not classified as any, since they have a specific combined solution. They are considered as expressions general view, and no additional clarifications or expressions are used for the description.

When solving such an algebraic expression, it is always necessary to pay attention to its notation, the presence of fractions, powers, or additional expressions. This is necessary in order to accurately determine the way to solve it. If you are not sure about its name, then it is recommended to call it an expression general type and decide according to the above algorithm.

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Degree properties:

(1) a m ⋅ a n = a m + n

Example:

$$(a^2) \cdot (a^5) = (a^7)$$ (2) a m a n = a m − n

Example:

$$\frac(((a^4)))(((a^3))) = (a^(4 - 3)) = (a^1) = a$$ (3) (a ⋅ b) n = a n ⋅ b n

Example:

$$((a \cdot b)^3) = (a^3) \cdot (b^3)$$ (4) (a b) n = a n b n

Example:

$$(\left((\frac(a)(b)) \right)^8) = \frac(((a^8)))(((b^8)))$$ (5) (a m ) n = a m ⋅ n

Example:

$$(((a^2))^5) = (a^(2 \cdot 5)) = (a^(10))$$ (6) a − n = 1 a n

Examples:

$$(a^( - 2)) = \frac(1)(((a^2)));\;\;\;\;(a^( - 1)) = \frac(1)(( (a^1))) = \frac(1)(a).$$

Properties square root:

(1) a b = a ⋅ b , for a ≥ 0 , b ≥ 0

Example:

18 = 9 ⋅ 2 = 9 ⋅ 2 = 3 2

(2) a b = a b , for a ≥ 0 , b > 0

Example:

4 81 = 4 81 = 2 9

(3) (a) 2 = a , for a ≥ 0

Example:

(4) a 2 = | a | for any a

Examples:

(− 3) 2 = | − 3 | = 3 , 4 2 = | 4 | = 4 .

Rational and irrational numbers

Rational numbers are numbers that can be represented as common fraction m n

Examples of rational numbers:

1 2 ;   − 9 4 ;   0,3333 … = 1 3 ;   8 ;   − 1236.

Irrational numbers - numbers that cannot be represented as an ordinary fraction m n, these are infinite non-periodic decimal fractions.

Examples of irrational numbers:

e = 2.71828182845…

π = 3.1415926…

2 = 1,414213562…

3 = 1,7320508075…

Simply put, irrational numbers are numbers that contain the square root sign in their notation. But not everything is so simple. Some rational numbers disguise themselves as irrational ones, for example, the number 4 contains a square root sign in its notation, but we are well aware that we can simplify the notation 4 = 2. This means that the number 4 is a rational number.

Similarly, the number 4 81 = 4 81 = 2 9 is a rational number.

Some problems require you to determine which numbers are rational and which are irrational. The task is to understand which numbers are irrational and which are disguised as them. To do this, you need to be able to perform the operations of taking the factor out from under the square root sign and introducing the factor under the root sign.

Insertion and removal of the factor for the sign of the square root

By taking the factor out of the square root sign, you can significantly simplify some mathematical expressions.

Example:

Simplify expression 2 8 2 .

1 way (taking out the multiplier from under the root sign): 2 8 2 = 2 4 ⋅ 2 2 = 2 4 ⋅ 2 2 = 2 ⋅ 2 = 4

Method 2 (introducing a multiplier under the root sign): 2 8 2 = 2 2 8 2 = 4 ⋅ 8 2 = 4 ⋅ 8 2 = 16 = 4

Abbreviated multiplication formulas (FSU)

sum square

(1) (a + b) 2 = a 2 + 2 a b + b 2

Example:

(3 x + 4 y) 2 = (3 x) 2 + 2 ⋅ 3 x ⋅ 4 y + (4 y) 2 = 9 x 2 + 24 x y + 16 y 2

The square of the difference

(2) (a − b) 2 = a 2 − 2 a b + b 2

Example:

(5 x − 2 y) 2 = (5 x) 2 − 2 ⋅ 5 x ⋅ 2 y + (2 y) 2 = 25 x 2 − 20 x y + 4 y 2

Sum of squares does not factor

a 2 + b 2 ≠

Difference of squares

(3) a 2 − b 2 = (a − b) (a + b)

Example:

25 x 2 - 4 y 2 = (5 x) 2 - (2 y) 2 = (5 x - 2 y) (5 x + 2 y)

sum cube

(4) (a + b) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3

Example:

(x + 3 y) 3 = (x) 3 + 3 ⋅ (x) 2 ⋅ (3 y) + 3 ⋅ (x) ⋅ (3 y) 2 + (3 y) 3 = x 3 + 3 ⋅ x 2 ⋅ 3 y + 3 ⋅ x ⋅ 9 y 2 + 27 y 3 = x 3 + 9 x 2 y + 27 x y 2 + 27 y 3

difference cube

(5) (a − b) 3 = a 3 − 3 a 2 b + 3 a b 2 − b 3

Example:

(x 2 − 2 y) 3 = (x 2) 3 − 3 ⋅ (x 2) 2 ⋅ (2 y) + 3 ⋅ (x 2) ⋅ (2 y) 2 − (2 y) 3 = x 2 ⋅ 3 − 3 ⋅ x 2 ⋅ 2 ⋅ 2 y + 3 ⋅ x 2 ⋅ 4 y 2 − 8 y 3 = x 6 − 6 x 4 y + 12 x 2 y 2 − 8 y 3

Sum of cubes

(6) a 3 + b 3 = (a + b) (a 2 − a b + b 2)

Example:

8 + x 3 = 2 3 + x 3 = (2 + x) (2 2 − 2 ⋅ x + x 2) = (x + 2) (4 − 2 x + x 2)

Difference of cubes

(7) a 3 − b 3 = (a − b) (a 2 + a b + b 2)

Example:

x 6 - 27 y 3 = (x 2) 3 - (3 y) 3 = (x 2 - 3 y) ((x 2) 2 + (x 2) (3 y) + (3 y) 2) = ( x 2 − 3 y) (x 4 + 3 x 2 y + 9 y 2)

Standard form of number

In order to understand how to bring arbitrary rational number to standard form, you need to know what the first significant digit of the number is.

First significant figure numbers call it the first non-zero digit on the left.

Examples:
2 5 ; 3, 05; 0 , 143 ; 0 , 00 1 2 . The first significant digit is highlighted in red.

To convert a number to standard form:

1. Shift the comma so that it is immediately after the first significant digit.
2. Multiply the resulting number by 10 n, where n is a number, which is defined as follows:
3. n > 0 if the comma was shifted to the left (multiplying by 10 n indicates that the comma should actually be to the right);
4. n< 0 , если запятая сдвигалась вправо (умножение на 10 n , указывает, что на самом деле запятая должна стоять левее);
5. the absolute value of the number n is equal to the number of digits by which the comma was shifted.

Examples:

25 = 2 , 5 ← ​ , = 2,5 ⋅ 10 1

The comma has moved to the left by 1 digit. Since the decimal point is shifted to the left, the exponent is positive.

Already brought to the standard form, you do not need to do anything with it. It can be written as 3.05 ⋅ 10 0 , but since 10 0 = 1, we leave the number in its original form.

0,143 = 0, 1 → , 43 = 1,43 ⋅ 10 − 1

The comma has moved to the right by 1 digit. Since the decimal point is shifted to the right, the exponent is negative.

− 0,0012 = − 0, 0 → 0 → 1 → , 2 = − 1,2 ⋅ 10 − 3

The comma has moved three places to the right. Since the decimal point is shifted to the right, the exponent is negative.

Algebraic expressions begin to be studied in the 7th grade. They have a number of properties and are used in problem solving. Let's study this topic in more detail and consider an example of solving the problem.

Concept definition

What expressions are called algebraic? This is mathematical notation, composed of numbers, letters and signs of arithmetic operations. The presence of letters is the main difference between numerical and algebraic expressions. Examples:

• 4a+5;
• 6b-8;
• 5s:6*(8+5).

A letter in algebraic expressions represents a number. Therefore, it is called a variable - in the first example it is the letter a, in the second - b, and in the third - c. The algebraic expression itself is also called variable expression.

Expression value

Meaning of an algebraic expression is the number obtained as a result of performing all the arithmetic operations that are specified in this expression. But in order to get it, the letters must be replaced by numbers. Therefore, the examples always indicate which number corresponds to the letter. Consider how to find the value of the expression 8a-14*(5-a) if a=3.

Let's substitute the number 3 instead of the letter a. We get the following entry: 8*3-14*(5-3).

As in numerical expressions, the solution of an algebraic expression is carried out according to the rules for performing arithmetic operations. Let's solve everything in order.

• 5-3=2.
• 8*3=24.
• 14*2=28.
• 24-28=-4.

Thus, the value of the expression 8a-14*(5-a) for a=3 is -4.

The value of a variable is called valid if the expression makes sense for it, that is, it is possible to find its solution.

An example of a valid variable for the expression 5:2a is the number 1. Substituting it into the expression, we get 5:2*1=2.5.

The invalid variable for this expression is 0. If we substitute zero into the expression, we get 5:2*0, that is, 5:0. You can't divide by zero, so the expression doesn't make sense.

Identity expressions

If two expressions are equal for any values ​​of their constituent variables, they are called identical.
Example of identical expressions :
4(a+c) and 4a+4c.
Whatever values ​​the letters a and c take, the expressions will always be equal. Any expression can be replaced by another, identical to it. This process is called identity transformation.

An example of an identical transformation .
4*(5a+14c) - this expression can be replaced by an identical one by applying mathematical law multiplication. To multiply a number by the sum of two numbers, you need to multiply this number by each term and add the results.

• 4*5a=20a.
• 4*14s=64s.
• 20a + 64s.

Thus, the expression 4*(5a+14c) is identical to 20a+64c.

The number that precedes the literal variable in an algebraic expression is called a coefficient. Coefficient and variable are multipliers.

Problem solving

Algebraic expressions are used to solve problems and equations.
Let's consider the task. Petya came up with a number. In order for classmate Sasha to guess it, Petya told him: first I added 7 to the number, then subtracted 5 from it and multiplied by 2. As a result, I got the number 28. What number did I guess?

To solve the problem, you need to designate the hidden number with the letter a, and then perform all the indicated actions with it.

• (a+7)-5.
• ((a+7)-5)*2=28.

Now let's solve the resulting equation.

Petya guessed the number 12.

What have we learned?

An algebraic expression is a record made up of letters, numbers and signs of arithmetic operations. Each expression has a value that is found by doing all the arithmetic in the expression. The letter in an algebraic expression is called a variable, and the number in front of it is called a coefficient. Algebraic expressions are used to solve problems.