How to define an identically equal expression. Identity transformations of expressions

Basic properties of addition and multiplication of numbers.

Commutative property of addition: when the terms are rearranged, the value of the sum does not change. For any numbers a and b, the equality is true

The associative property of addition: in order to add a third number to the sum of two numbers, you can add the sum of the second and third to the first number. For any numbers a, b and c the equality is true

Commutative property of multiplication: permutation of factors does not change the value of the product. For any numbers a, b and c, the equality is true

The associative property of multiplication: in order 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.

For any numbers a, b and c, the equality is true

Distributive property: To multiply a number by a sum, you can multiply that number by each term and add the results. For any numbers a, b and c the equality is true

It follows from the commutative and associative properties of addition that in any sum you can rearrange the terms as you like and combine them into groups in an arbitrary way.

Example 1 Let's calculate the sum 1.23+13.5+4.27.

To do this, it is convenient to combine the first term with the third. We get:

1,23+13,5+4,27=(1,23+4,27)+13,5=5,5+13,5=19.

It follows from the commutative and associative properties of multiplication: in any product, you can rearrange the factors in any way and arbitrarily combine them into groups.

Example 2 Let's find the value of the product 1.8 0.25 64 0.5.

Combining the first factor with the fourth, and the second with the third, we will have:

1.8 0.25 64 0.5 \u003d (1.8 0.5) (0.25 64) \u003d 0.9 16 \u003d 14.4.

The distribution property is also valid when the number is multiplied by the sum of three or more terms.

For example, for any numbers a, b, c and d, the equality is true

a(b+c+d)=ab+ac+ad.

We know that subtraction can be replaced by addition by adding to the minuend the opposite number to the subtrahend:

This allows a numeric expression type a-b consider the sum of numbers a and -b, consider a numerical expression of the form a + b-c-d as the sum of numbers a, b, -c, -d, etc. The considered properties of actions are also valid for such sums.

Example 3 Let's find the value of the expression 3.27-6.5-2.5+1.73.

This expression is the sum of the numbers 3.27, -6.5, -2.5 and 1.73. Applying the addition properties, we get: 3.27-6.5-2.5+1.73=(3.27+1.73)+(-6.5-2.5)=5+(-9) = -four.

Example 4 Let's calculate the product 36·().

The multiplier can be thought of as the sum of the numbers and -. Using the distributive property of multiplication, we get:

36()=36-36=9-10=-1.

Identities

Definition. Two expressions whose corresponding values are equal for any values of the variables are said to be identically equal.

Definition. An equality that is true for any values of the variables is called an identity.

Let's find the values of the expressions 3(x+y) and 3x+3y for x=5, y=4:

3(x+y)=3(5+4)=3 9=27,

3x+3y=3 5+3 4=15+12=27.

We got the same result. It follows from the distributive property that, in general, for any values of the variables, the corresponding values of the expressions 3(x+y) and 3x+3y are equal.

Consider now the expressions 2x+y and 2xy. For x=1, y=2 they take equal values:

However, you can specify x and y values such that the values of these expressions are not equal. For example, if x=3, y=4, then

The expressions 3(x+y) and 3x+3y are identically equal, but the expressions 2x+y and 2xy are not identically equal.

The equality 3(x+y)=x+3y, true for any values of x and y, is an identity.

True numerical equalities are also considered identities.

So, identities are equalities expressing the main properties of actions on numbers:

a+b=b+a, (a+b)+c=a+(b+c),

ab=ba, (ab)c=a(bc), a(b+c)=ab+ac.

Other examples of identities can be given:

a+0=a, a+(-a)=0, a-b=a+(-b),

a 1=a, a (-b)=-ab, (-a)(-b)=ab.

Identity transformations of expressions

The replacement of one expression by another, identically equal to it, is called an identical transformation or simply a transformation of an expression.

Identical transformations of expressions with variables are performed based on the properties of operations on numbers.

To find the value of the expression xy-xz given the values x, y, z, you need to perform three steps. For example, with x=2.3, y=0.8, z=0.2 we get:

xy-xz=2.3 0.8-2.3 0.2=1.84-0.46=1.38.

This result can be obtained in only two steps, using the expression x(y-z), which is identically equal to the expression xy-xz:

xy-xz=2.3(0.8-0.2)=2.3 0.6=1.38.

We have simplified the calculations by replacing the expression xy-xz with the identical equal expression x(y-z).

Identity transformations of expressions are widely used in calculating the values of expressions and solving other problems. Some identical transformations have already been performed, for example, the reduction of similar terms, the opening of brackets. Recall the rules for performing these transformations:

to bring like terms, you need to add their coefficients and multiply the result by the common letter part;

if there is a plus sign in front of the brackets, then the brackets can be omitted, retaining the sign of each term enclosed in brackets;

if there is a minus sign before the brackets, then the brackets can be omitted by changing the sign of each term enclosed in brackets.

Example 1 Let's add like terms in the sum 5x+2x-3x.

We use the rule for reducing like terms:

5x+2x-3x=(5+2-3)x=4x.

This transformation is based on the distributive property of multiplication.

Example 2 Let's expand the brackets in the expression 2a+(b-3c).

Applying the rule for opening brackets preceded by a plus sign:

2a+(b-3c)=2a+b-3c.

The performed transformation is based on the associative property of addition.

Example 3 Let's expand the brackets in the expression a-(4b-c).

Let's use the rule for expanding brackets preceded by a minus sign:

a-(4b-c)=a-4b+c.

The performed transformation is based on the distributive property of multiplication and the associative property of addition. Let's show it. Let's represent the second term -(4b-c) in this expression as a product (-1)(4b-c):

a-(4b-c)=a+(-1)(4b-c).

Applying these properties of actions, we get:

a-(4b-c)=a+(-1)(4b-c)=a+(-4b+c)=a-4b+c.

§ 2. Identity expressions, identity. Identity transformation of an expression. Identity proofs

Let's find the values of the expressions 2(x - 1) 2x - 2 for the given values of the variable x. We write the results in a table:

It can be concluded that the values of the expressions 2(x - 1) 2x - 2 for each given value variable x are equal to each other. According to the distributive property of multiplication with respect to subtraction 2(x - 1) = 2x - 2. Therefore, for any other value of the variable x, the value of the expression 2(x - 1) 2x - 2 will also be equal to each other. Such expressions are called identically equal.

For example, the expressions 2x + 3x and 5x are synonyms, since for each value of the variable x, these expressions acquire the same values(this follows from the distributive property of multiplication with respect to addition, since 2x + 3x = 5x).

Consider now the expressions 3x + 2y and 5xy. If x \u003d 1 and b \u003d 1, then the corresponding values of these expressions are equal to each other:

3x + 2y \u003d 3 ∙ 1 + 2 ∙ 1 \u003d 5; 5xy = 5 ∙ 1 ∙ 1 = 5.

However, you can specify x and y values for which the values of these expressions will not be equal to each other. For example, if x = 2; y = 0, then

3x + 2y = 3 ∙ 2 + 2 ∙ 0 = 6, 5xy = 5 ∙ 20 = 0.

Consequently, there are such values of the variables for which the corresponding values of the expressions 3x + 2y and 5xy are not equal to each other. Therefore, the expressions 3x + 2y and 5xy are not identically equal.

Based on the foregoing, identities, in particular, are equalities: 2(x - 1) = 2x - 2 and 2x + 3x = 5x.

An identity is every equality, which is written known properties actions on numbers. For example,

a + b = b + a; (a + b) + c = a + (b + c); a(b + c) = ab + ac;

ab = ba; (ab)c = a(bc); a(b - c) = ab - ac.

There are also such equalities as identities:

a + 0 = a; a ∙ 0 = 0; a ∙ (-b) = -ab;

a + (-a) = 0; a ∙ 1 = a; a ∙ (-b) = ab.

1 + 2 + 3 = 6; 5 2 + 12 2 = 13 2 ; 12 ∙ (7 - 6) = 3 ∙ 4.

If we reduce similar terms in the expression -5x + 2x - 9, we get that 5x + 2x - 9 \u003d 7x - 9. In this case, they say that the expression 5x + 2x - 9 was replaced by the expression 7x - 9, which is identical to it.

Identical transformations of expressions with variables are performed by applying the properties of operations on numbers. In particular, identical transformations with the opening of brackets, the construction of similar terms, and the like.

Identical transformations have to be performed when simplifying the expression, that is, replacing some expression with an expression that is identically equal to it, which should be shorter.

Example 1. Simplify the expression:

1) -0.3 m ∙ 5n;

2) 2(3x - 4) + 3(-4x + 7);

3) 2 + 5a - (a - 2b) + (3b - a).

1) -0.3 m ∙ 5n = -0.3 ∙ 5mn = -1.5 mn;

2) 2(3x4) + 3(-4 + 7) = 6 x - 8 - 1 2x+ 21 = 6x + 13;

3) 2 + 5a - (a - 2b) + (3b - a) = 2 + 5a - a + 2 b + 3 b - a= 3a + 5b + 2.

To prove that equality is an identity (in other words, to prove identity, one uses identity transformations of expressions.

You can prove the identity in one of the following ways:

perform identical transformations of its left side, thereby reducing it to the form of the right side;
perform identical transformations of its right side, thereby reducing it to the form of the left side;
perform identical transformations of both its parts, thereby raising both parts to the same expressions.

Example 2. Prove the identity:

1) 2x - (x + 5) - 11 \u003d x - 16;

2) 206 - 4a = 5(2a - 3b) - 7(2a - 5b);

3) 2(3x - 8) + 4(5x - 7) = 13(2x - 5) + 21.

Development

1) Let's transform the left side of this equality:

2x - (x + 5) - 11 = 2x - X- 5 - 11 = x - 16.

By identical transformations, the expression on the left side of the equality was reduced to the form of the right side and thus proved that this equality is an identity.

2) Let's transform the right side of this equality:

5(2a - 3b) - 7(2a - 5b) = 10a - 15 b - 14a + 35 b= 20b - 4a.

By identical transformations, the right side of the equality was reduced to the form of the left side and thus proved that this equality is an identity.

3) In this case, it is convenient to simplify both the left and right parts of the equality and compare the results:

2(3x - 8) + 4(5x - 7) = 6x - 16 + 20x- 28 \u003d 26x - 44;

13 (2x - 5) + 21 \u003d 26x - 65 + 21 \u003d 26x - 44.

By identical transformations, the left and right parts of the equality were reduced to the same form: 26x - 44. Therefore, this equality is an identity.

What expressions are called identical? Give an example of identical expressions. What equality is called identity? Give an example of identity. What is called the identity transformation of an expression? How to prove identity?

(Oral) Or there are expressions identically equal:

1) 2a + a and 3a;

2) 7x + 6 and 6 + 7x;

3) x + x + x and x 3;

4) 2(x - 2) and 2x - 4;

5) m - n and n - m;

6) 2a ∙ r and 2p ∙ a?

Are the expressions identically equal:

1) 7x - 2x and 5x;

2) 5a - 4 and 4 - 5a;

3) 4m + n and n + 4m;

4) a + a and a 2;

5) 3(a - 4) and 3a - 12;

6) 5m ∙ n and 5m + n?

(Verbally) Is the identity of equality:

1) 2a + 106 = 12ab;

2) 7r - 1 = -1 + 7r;

3) 3(x - y) = 3x - 5y?

Open parenthesis:

Open parenthesis:

Reduce like terms:

Name several expressions that are identical to expressions 2a + 3a.
Simplify the expression using the permuting and conjunctive properties of multiplication:

1) -2.5 x ∙ 4;

2) 4p ∙ (-1.5);

3) 0.2 x ∙ (0.3 g);

4)- x ∙<-7у).

Simplify the expression:

1) -2p ∙ 3.5;

2) 7a ∙ (-1.2);

3) 0.2 x ∙ (-3y);

4) - 1 m ∙ (-3n).

(Verbal) Simplify the expression:

1) 2x - 9 + 5x;

2) 7a - 3b + 2a + 3b;

4) 4a ∙ (-2b).

Reduce like terms:

1) 56 - 8a + 4b - a;

2) 17 - 2p + 3p + 19;

3) 1.8 a + 1.9 b + 2.8 a - 2.9 b;

4) 5 - 7s + 1.9 g + 6.9 s - 1.7 g.

1) 4(5x - 7) + 3x + 13;

2) 2(7 - 9a) - (4 - 18a);

3) 3(2p - 7) - 2(g - 3);

4) -(3m - 5) + 2(3m - 7).

Open the brackets and reduce like terms:

1) 3(8a - 4) + 6a;

2) 7p - 2(3p - 1);

3) 2(3x - 8) - 5(2x + 7);

4) 3(5m - 7) - (15m - 2).

1) 0.6x + 0.4(x - 20) if x = 2.4;

2) 1.3 (2a - 1) - 16.4 if a = 10;

3) 1.2 (m - 5) - 1.8 (10 - m), if m = -3.7;

4) 2x - 3(x + y) + 4y if x = -1, y = 1.

Simplify the expression and find its value:

1) 0.7 x + 0.3(x - 4) if x = -0.7;

2) 1.7 (y - 11) - 16.3, if v \u003d 20;

3) 0.6 (2a - 14) - 0.4 (5a - 1), if a = -1;

4) 5(m - n) - 4m + 7n if m = 1.8; n = -0.9.

Prove the identity:

1) - (2x - y) \u003d y - 2x;

2) 2(x - 1) - 2x = -2;

3) 2(x - 3) + 3(x + 2) = 5x;

4) s - 2 = 5(s + 2) - 4(s + 3).

Prove the identity:

1) -(m - 3n) = 3n - m;

2) 7(2 - p) + 7p = 14;

3) 5a = 3(a - 4) + 2(a + 6);

4) 4(m - 3) + 3(m + 3) = 7m - 3.

The length of one of the sides of the triangle is a cm, and the length of each of the other two sides is 2 cm more than it. Write the perimeter of the triangle as an expression and simplify the expression.
The width of the rectangle is x cm and the length is 3 cm more than the width. Write the perimeter of the rectangle as an expression and simplify the expression.

1) x - (x - (2x - 3));

2) 5m - ((n - m) + 3n);

3) 4p - (3p - (2p - (r + 1)));

4) 5x - (2x - ((y - x) - 2y));

5) (6а - b) - (4 a - 33b);

6) - (2.7 m - 1.5 n) + (2n - 0.48 m).

Expand the brackets and simplify the expression:

1) a - (a - (3a - 1));

2) 12m - ((a - m) + 12a);

3) 5y - (6y - (7y - (8y - 1)));

6) (2.1 a - 2.8 b) - (1a - 1b).

Prove the identity:

1) 10x - (-(5x + 20)) = 5(3x + 4);

2) - (- 3p) - (-(8 - 5p)) \u003d 2 (4 - g);

3) 3(a - b - c) + 5(a - b) + 3c = 8(a - b).

Prove the identity:

1) 12a - ((8a - 16)) \u003d -4 (4 - 5a);

2) 4(x + y -<) + 5(х - t) - 4y - 9(х - t).

Prove that the value of the expression

1.8(m - 2) + 1.4(2 - m) + 0.2(1.7 - 2m) does not depend on the value of the variable.

Prove that for any value of the variable, the value of the expression

a - (a - (5a + 2)) - 5 (a - 8)

is the same number.

Prove that the sum of three consecutive even numbers is divisible by 6.
Prove that if n is a natural number, then the value of the expression -2(2.5 n - 7) + 2 (3n - 6) is an even number.

Exercises to repeat

An alloy weighing 1.6 kg contains 15% copper. How many kg of copper is contained in this alloy?
What percentage is the number 20 of its:

1) square;

The tourist walked for 2 hours and rode a bicycle for 3 hours. In total, the tourist covered 56 km. Find the speed at which the tourist rode a bicycle if it is 12 km/h more than the speed at which he walked.

Interesting tasks for lazy students

11 teams participate in the city football championship. Each team plays one match with the others. Prove that at any moment of the competition there is a team that has played an even number of matches or has not played any yet.

Consider two equalities:

1. a 12 * a 3 = a 7 * a 8

This equality will hold for any value of the variable a. The range of valid values for that equality will be the entire set of real numbers.

2. a 12: a 3 = a 2 * a 7 .

This inequality will hold for all values of the variable a, except for a equal to zero. The range of admissible values for this inequality will be the entire set of real numbers, except for zero.

About each of these equalities, it can be argued that it will be true for any admissible values of the variables a. Such equations in mathematics are called identities.

The concept of identity

An identity is an equality that is true for any admissible values of the variables. If any valid values are substituted into this equality instead of variables, then the correct numerical equality should be obtained.

It is worth noting that true numerical equalities are also identities. Identities, for example, will be properties of actions on numbers.

3. a + b = b + a;

4. a + (b + c) = (a + b) + c;

6. a*(b*c) = (a*b)*c;

7. a*(b + c) = a*b + a*c;

11. a*(-1) = -a.

If two expressions for any admissible variables are respectively equal, then such expressions are called identically equal. Below are some examples of identically equal expressions:

1. (a 2) 4 and a 8 ;

2. a*b*(-a^2*b) and -a 3 *b 2 ;

3. ((x 3 *x 8)/x) and x 10 .

We can always replace one expression with any other expression identically equal to the first one. Such a replacement will be an identical transformation.

Identity Examples

Example 1: Are the following equalities identities:

1. a + 5 = 5 + a;

2. a*(-b) = -a*b;

3. 3*a*3*b = 9*a*b;

Not all of the above expressions will be identities. Of these equalities, only 1,2 and 3 equalities are identities. Whatever numbers we substitute in them, instead of the variables a and b, we still get the correct numerical equalities.

But 4 equality is no longer an identity. Because not for all admissible values this equality will be fulfilled. For example, with the values a = 5 and b = 2, you get the following result:

This equality is not true, since the number 3 does not equal the number -3.

Identity conversions are the work we do with numeric and alphabetic expressions, as well as with expressions that contain variables. We carry out all these transformations in order to bring the original expression to a form that will be convenient for solving the problem. We will consider the main types of identical transformations in this topic.

Yandex.RTB R-A-339285-1

Identity transformation of an expression. What it is?

For the first time we meet with the concept of identical transformed we in algebra lessons in grade 7. Then we first get acquainted with the concept of identically equal expressions. Let's deal with the concepts and definitions to facilitate the assimilation of the topic.

Definition 1

Identity transformation of an expression are actions performed to replace the original expression with an expression that will be identically equal to the original one.

Often this definition is used in an abbreviated form, in which the word "identical" is omitted. It is assumed that in any case we carry out the transformation of the expression in such a way as to obtain an expression identical to the original one, and this does not need to be emphasized separately.

Let us illustrate this definition with examples.

Example 1

If we replace the expression x + 3 - 2 to the identically equal expression x+1, then we carry out the identical transformation of the expression x + 3 - 2.

Example 2

Replacing expression 2 a 6 with expression a 3 is the identity transformation, while the replacement of the expression x to the expression x2 is not an identical transformation, since the expressions x and x2 are not identically equal.

We draw your attention to the form of writing expressions when carrying out identical transformations. We usually write the original expression and the resulting expression as an equality. So, writing x + 1 + 2 = x + 3 means that the expression x + 1 + 2 has been reduced to the form x + 3 .

Sequential execution of actions leads us to a chain of equalities, which is several consecutive identical transformations. So, we understand the notation x + 1 + 2 = x + 3 = 3 + x as a sequential implementation of two transformations: first, the expression x + 1 + 2 was reduced to the form x + 3, and it was reduced to the form 3 + x.

Identity transformations and ODZ

A number of expressions that we begin to study in grade 8 do not make sense for any values of variables. Carrying out identical transformations in these cases requires us to pay attention to the region of admissible values of variables (ODV). Performing identical transformations may leave the ODZ unchanged or narrow it down.

Example 3

When performing a transition from the expression a + (−b) to the expression a-b range of allowed values of variables a and b stays the same.

Example 4

Transition from expression x to expression x 2 x leads to a narrowing of the range of acceptable values of the variable x from the set of all real numbers to the set of all real numbers, from which zero has been excluded.

Example 5

Identity transformation of an expression x 2 x expression x leads to the expansion of the range of acceptable values of the variable x from the set of all real numbers except zero to the set of all real numbers.

Narrowing or expanding the range of allowable values of variables when carrying out identical transformations is important in solving problems, since it can affect the accuracy of calculations and lead to errors.

Basic identity transformations

Let's now see what identical transformations are and how they are performed. Let us single out those types of identical transformations that we have to deal with most often into the main group.

In addition to the basic identity transformations, there are a number of transformations that relate to expressions of a particular type. For fractions, these are methods of reduction and reduction to a new denominator. For expressions with roots and powers, all actions that are performed based on the properties of roots and powers. For logarithmic expressions, actions that are performed based on the properties of logarithms. For trigonometric expressions, all actions using trigonometric formulas. All these particular transformations are discussed in detail in separate topics that can be found on our resource. For this reason, we will not dwell on them in this article.

Let us proceed to the consideration of the main identical transformations.

Rearrangement of terms, factors

Let's start by rearranging the terms. We deal with this identical transformation most often. And the following statement can be considered the main rule here: in any sum, the rearrangement of the terms in places does not affect the result.

This rule is based on the commutative and associative properties of addition. These properties allow us to rearrange the terms in places and at the same time obtain expressions that are identically equal to the original ones. That is why the rearrangement of terms in places in the sum is an identical transformation.

Example 6

We have the sum of three terms 3 + 5 + 7 . If we swap the terms 3 and 5, then the expression will take the form 5 + 3 + 7. There are several options for rearranging the terms in this case. All of them lead to obtaining expressions that are identically equal to the original one.

Not only numbers, but also expressions can act as terms in the sum. They, just like numbers, can be rearranged without affecting the final result of calculations.

Example 7

In the sum of three terms 1 a + b, a 2 + 2 a + 5 + a 7 a 3 and - 12 a of the form 1 a + b + a 2 + 2 a + 5 + a 7 a 3 + ( - 12) a terms can be rearranged, for example, like this (- 12) a + 1 a + b + a 2 + 2 a + 5 + a 7 a 3 . In turn, you can rearrange the terms in the denominator of the fraction 1 a + b, while the fraction will take the form 1 b + a. And the expression under the root sign a 2 + 2 a + 5 is also a sum in which the terms can be interchanged.

In the same way as the terms, in the original expressions one can interchange the factors and obtain identically correct equations. This action is governed by the following rule:

Definition 2

In the product, rearranging the factors in places does not affect the result of the calculation.

This rule is based on the commutative and associative properties of multiplication, which confirm the correctness of the identical transformation.

Example 8

Work 3 5 7 permutation of factors can be represented in one of the following forms: 5 3 7 , 5 7 3 , 7 3 5 , 7 5 3 or 3 7 5.

Example 9

Permuting the factors in the product x + 1 x 2 - x + 1 x will give x 2 - x + 1 x x + 1

Bracket expansion

Parentheses can contain entries of numeric expressions and expressions with variables. These expressions can be transformed into identically equal expressions, in which there will be no parentheses at all or there will be fewer of them than in the original expressions. This way of converting expressions is called parenthesis expansion.

Example 10

Let's carry out actions with brackets in an expression of the form 3 + x − 1 x in order to get the identically true expression 3 + x − 1 x.

The expression 3 · x - 1 + - 1 + x 1 - x can be converted to the identically equal expression without brackets 3 · x - 3 - 1 + x 1 - x .

We discussed in detail the rules for converting expressions with brackets in the topic "Bracket expansion", which is posted on our resource.

Grouping terms, factors

In cases where we are dealing with three or more terms, we can resort to such a type of identical transformations as a grouping of terms. By this method of transformation is meant the union of several terms into a group by rearranging them and placing them in brackets.

When grouping, the terms are interchanged in such a way that the grouped terms are in the expression record next to each other. After that, they can be enclosed in brackets.

Example 11

Take the expression 5 + 7 + 1 . If we group the first term with the third, we get (5 + 1) + 7 .

The grouping of factors is carried out similarly to the grouping of terms.

Example 12

In the work 2 3 4 5 it is possible to group the first factor with the third, and the second factor with the fourth, in this case we arrive at the expression (2 4) (3 5). And if we grouped the first, second and fourth factors, we would get the expression (2 3 5) 4.

The terms and factors that are grouped can be represented both by prime numbers and by expressions. The grouping rules were discussed in detail in the topic "Grouping terms and factors".

Replacing differences by sums, partial products and vice versa

The replacement of differences by sums became possible thanks to our acquaintance with opposite numbers. Now subtraction from a number a numbers b can be seen as an addition to the number a numbers −b. Equality a − b = a + (− b) can be considered fair and, on its basis, carry out the replacement of differences by sums.

Example 13

Take the expression 4 + 3 − 2 , in which the difference of numbers 3 − 2 we can write as the sum 3 + (− 2) . Get 4 + 3 + (− 2) .

Example 14

All differences in expression 5 + 2 x - x 2 - 3 x 3 - 0, 2 can be replaced by sums like 5 + 2 x + (− x 2) + (− 3 x 3) + (− 0 , 2).

We can proceed to sums from any differences. Similarly, we can make a reverse substitution.

The replacement of division by multiplication by the reciprocal of the divisor is made possible by the concept of reciprocal numbers. This transformation can be written as a: b = a (b − 1).

This rule was the basis of the rule for dividing ordinary fractions.

Example 15

Private 1 2: 3 5 can be replaced by a product of the form 1 2 5 3.

Similarly, by analogy, division can be replaced by multiplication.

Example 16

In the case of the expression 1+5:x:(x+3) replace division with x can be multiplied by 1 x. Division by x + 3 we can replace by multiplying with 1 x + 3. The transformation allows us to obtain an expression that is identical to the original one: 1 + 5 1 x 1 x + 3 .

Replacing multiplication by division is carried out according to the scheme a b = a: (b − 1).

Example 17

In the expression 5 x x 2 + 1 - 3, multiplication can be replaced by division as 5: x 2 + 1 x - 3.

Performing actions with numbers

Performing operations with numbers is subject to the rule of order of operations. First, operations are performed with powers of numbers and roots of numbers. After that, we replace logarithms, trigonometric and other functions with their values. Then the actions in parentheses are performed. And then you can already carry out all the other actions from left to right. It is important to remember that multiplication and division are carried out before addition and subtraction.

Operations with numbers allow you to transform the original expression into an identical one equal to it.

Example 18

Let's transform the expression 3 · 2 3 - 1 · a + 4 · x 2 + 5 · x by performing all possible operations with numbers.

Solution

First, let's look at the degree 2 3 and root 4 and calculate their values: 2 3 = 8 and 4 = 2 2 = 2 .

Substitute the obtained values into the original expression and get: 3 (8 - 1) a + 2 (x 2 + 5 x) .

Now let's do the parentheses: 8 − 1 = 7 . And let's move on to the expression 3 7 a + 2 (x 2 + 5 x) .

We just have to do the multiplication 3 and 7 . We get: 21 a + 2 (x 2 + 5 x) .

Answer: 3 2 3 - 1 a + 4 x 2 + 5 x = 21 a + 2 (x 2 + 5 x)

Operations with numbers may be preceded by other kinds of identity transformations, such as number grouping or parenthesis expansion.

Example 19

Take the expression 3 + 2 (6: 3) x (y 3 4) − 2 + 11.

Solution

First of all, we will change the quotient in parentheses 6: 3 on its meaning 2 . We get: 3 + 2 2 x (y 3 4) − 2 + 11 .

Let's expand the brackets: 3 + 2 2 x (y 3 4) − 2 + 11 = 3 + 2 2 x y 3 4 − 2 + 11.

Let's group the numerical factors in the product, as well as the terms that are numbers: (3 − 2 + 11) + (2 2 4) x y 3.

Let's do the parentheses: (3 − 2 + 11) + (2 2 4) x y 3 = 12 + 16 x y 3

Answer:3 + 2 (6: 3) x (y 3 4) − 2 + 11 = 12 + 16 x y 3

If we work with numerical expressions, then the purpose of our work will be to find the value of the expression. If we transform expressions with variables, then the goal of our actions will be to simplify the expression.

Bracketing the Common Factor

In cases where the terms in the expression have the same factor, then we can take this common factor out of brackets. To do this, we first need to represent the original expression as the product of a common factor and an expression in brackets, which consists of the original terms without a common factor.

Example 20

Numerically 2 7 + 2 3 we can take out the common factor 2 outside the brackets and get an identically correct expression of the form 2 (7 + 3).

You can refresh the memory of the rules for putting the common factor out of brackets in the corresponding section of our resource. The material discusses in detail the rules for taking the common factor out of brackets and provides numerous examples.

Reduction of similar terms

Now let's move on to sums that contain like terms. Two options are possible here: sums containing the same terms, and sums whose terms differ by a numerical coefficient. Operations with sums containing like terms is called reduction of like terms. It is carried out as follows: we put the common letter part out of brackets and calculate the sum of numerical coefficients in brackets.

Example 21

Consider the expression 1 + 4 x − 2 x. We can take the literal part of x out of brackets and get the expression 1 + x (4 − 2). Let's calculate the value of the expression in brackets and get the sum of the form 1 + x · 2 .

Replacing numbers and expressions with identically equal expressions

The numbers and expressions that make up the original expression can be replaced by expressions that are identically equal to them. Such a transformation of the original expression leads to an expression that is identically equal to it.

Example 22 Example 23

Consider the expression 1 + a5, in which we can replace the degree a 5 with a product identically equal to it, for example, of the form a 4. This will give us the expression 1 + a 4.

The transformation performed is artificial. It only makes sense in preparation for other transformations.

Example 24

Consider the transformation of the sum 4 x 3 + 2 x 2. Here the term 4x3 we can represent as a product 2 x 2 x 2 x. As a result, the original expression takes the form 2 x 2 2 x + 2 x 2. Now we can isolate the common factor 2x2 and take it out of the brackets: 2 x 2 (2 x + 1).

Adding and subtracting the same number

Adding and subtracting the same number or expression at the same time is an artificial expression transformation technique.

Example 25

Consider the expression x 2 + 2 x. We can add or subtract one from it, which will allow us to subsequently carry out another identical transformation - to select the square of the binomial: x 2 + 2 x = x 2 + 2 x + 1 - 1 = (x + 1) 2 - 1.

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Having got an idea about identities , it is logical to move on to acquaintance with . In this article, we will answer the question of what identically equal expressions are, and also, using examples, we will figure out which expressions are identically equal and which are not.

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What are identically equal expressions?

The definition of identically equal expressions is given in parallel with the definition of identity. This happens in algebra class in 7th grade. In the textbook on algebra for 7 classes, the author Yu. N. Makarychev gives the following wording:

Definition.

are expressions whose values are equal for any values of the variables included in them. Numeric expressions that correspond to the same values are also called identically equal.

This definition is used up to class 8, it is valid for integer expressions, since they make sense for any values of the variables included in them. And in grade 8, the definition of identically equal expressions is specified. Let's explain what it is connected with.

In grade 8, the study of other types of expressions begins, which, unlike integer expressions, may not make sense for some values of variables. This makes it necessary to introduce definitions of admissible and invalid values of variables, as well as the range of admissible values of the ODV of a variable, and as a result, to clarify the definition of identically equal expressions.

Definition.

Two expressions whose values are equal for all admissible values of their variables are called identically equal expressions. Two numeric expressions that have the same value are also said to be identically equal.

In this definition of identically equal expressions, it is worth clarifying the meaning of the phrase "for all admissible values of the variables included in them." It implies all such values of variables for which both identically equal expressions simultaneously make sense. This idea will be clarified in the next section by considering examples.

The definition of identically equal expressions in A. G. Mordkovich's textbook is given a little differently:

Definition.

Identical equal expressions are expressions on the left and right sides of the identity.

In meaning, this and the previous definitions coincide.

Examples of identically equal expressions

The definitions introduced in the previous subsection allow us to bring examples of identically equal expressions.

Let's start with identically equal numerical expressions. The numeric expressions 1+2 and 2+1 are identically equal because they correspond to equal values 3 and 3 . The expressions 5 and 30:6 are also identically equal, as are the expressions (2 2) 3 and 2 6 (the values of the last expressions are equal due to ). But the numerical expressions 3+2 and 3−2 are not identically equal, since they correspond to the values 5 and 1, respectively, but they are not equal.

Now we give examples of identically equal expressions with variables. These are the expressions a+b and b+a . Indeed, for any values of the variables a and b, the written expressions take the same values (which follows from the numbers). For example, with a=1 and b=2 we have a+b=1+2=3 and b+a=2+1=3 . For any other values of the variables a and b, we will also get equal values of these expressions. The expressions 0·x·y·z and 0 are also identically equal for any values of the variables x , y and z . But the expressions 2 x and 3 x are not identically equal, since, for example, at x=1 their values are not equal. Indeed, for x=1, the expression 2 x is 2 1=2 , and the expression 3 x is 3 1=3 .

When the areas of allowable values of variables in expressions coincide, as, for example, in the expressions a+1 and 1+a , or a b 0 and 0 , or and , and the values of these expressions are equal for all values of variables from these areas, then here everything is clear - these expressions are identically equal for all admissible values of the variables included in them. So a+1≡1+a for any a , the expressions a b 0 and 0 are identically equal for any values of the variables a and b , and the expressions and are identically equal for all x from ; ed. S. A. Telyakovsky. - 17th ed. - M. : Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.

Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.

Mordkovich A. G. Algebra. 7th grade. At 2 pm Part 1. A textbook for students of educational institutions / A. G. Mordkovich. - 17th ed., add. - M.: Mnemozina, 2013. - 175 p.: ill. ISBN 978-5-346-02432-3.