What are identical expressions? Identity transformations

§ 2. Identical expressions, identity. Identical transformation of an expression. Proofs of identities

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

We can come to the conclusion that the values ​​of the expressions 2(x - 1) 2x - 2 for each given value of the variable x are equal to each other. According to the distributive property of multiplication relative 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 relative to addition, since 2x + 3x = 5x).

Let us now consider the expressions 3x + 2y and 5xy. If x = 1 and b = 1, then the corresponding values ​​of these expressions are equal to each other:

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

However, you can specify values ​​of x and y 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 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 above, the identities, in particular, are the equalities: 2(x - 1) = 2x - 2 and 2x + 3x = 5x.

An identity is every equality that describes the known properties of operations on numbers. For example,

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

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

Identities include the following equalities:

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 combine similar terms in the expression -5x + 2x - 9, we get that 5x + 2x - 9 = 7x - 9. In this case, they say that the expression 5x + 2x - 9 was replaced by the identical expression 7x - 9.

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

Identical transformations have to be performed when simplifying an expression, that is, replacing a certain expression with an identically equal expression, which should make the notation 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, identical transformations of expressions are used.

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

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

Example 2. Prove the identity:

1) 2x - (x + 5) - 11 = x - 16;

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

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

R a s i z a n i .

1) Transform the left side of this equality:

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

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

2) Transform the right side of this equality:

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

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

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

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

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

By identical transformations, the left and right sides 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 kind of equality is called identity? Give an example of an identity. What is called an identity transformation of an expression? How to prove identity?

1. (Verbally) Or there are expressions that are 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 ∙ p and 2p ∙ a?

1. 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?

1. (Verbally) is the Lee identity equality:

1) 2a + 106 = 12ab;

2) 7р - 1 = -1 + 7р;

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

1. Open parenthesis:

1. Open parenthesis:

1. Combine similar terms:

1. Name several expressions identical to the expression 2a + 3a.
2. Simplify the expression using the permutation and connective properties of multiplication:

1) -2.5 x ∙ 4;

2) 4р ∙ (-1.5);

3) 0.2 x ∙ (0.3 g);

4)- x ∙<-7у).

1. Simplify the expression:

1) -2р ∙ 3.5;

2) 7a ∙ (-1.2);

3) 0.2 x ∙ (-3y);

4) - 1 m ∙ (-3n).

1. (Oral) Simplify the expression:

1) 2x - 9 + 5x;

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

4) 4a ∙ (-2b).

1. Combine similar 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(2р - 7) - 2(r - 3);

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

1. Open the brackets and combine similar 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) 0.6 x + 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.

1. Simplify the expression and find its meaning:

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

2) 1.7(y - 11) - 16.3, if b = 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.

1. Prove the identity:

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

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

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

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

1. 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.

1. 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 greater than it. Write down the perimeter of the triangle as an expression and simplify the expression.
2. The width of the rectangle is x cm, and the length is 3 cm greater than the width. Write down the perimeter of the rectangle as an expression and simplify the expression.

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

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

3) 4р - (3р - (2р - (r + 1)));

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

5) (6a - b) - (4 a – 33b);

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

1. Open the parentheses 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).

1. Prove the identity:

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

2) -(- 3p) - (-(8 - 5p)) = 2(4 - r);

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

1. Prove the identity:

1) 12a - ((8a - 16)) = -4(4 - 5a);

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

1. Prove that the meaning 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.

1. Prove that for any value of the variable the value of the expression

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

is the same number.

1. Prove that the sum of three consecutive even numbers is divisible by 6.
2. 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

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

1) square;

1. 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 was riding a bicycle, if it is 12 km/h more than the speed at which he was walking.

Interesting tasks for lazy students

1. 11 teams participate in the city football championship. Each team plays one match against the other. Prove that at any moment of the competition there is a team that will have played an even number of matches at that moment or has not yet played any.

After we have dealt with the concept of identities, we can move on to studying identically equal expressions. The purpose of this article is to explain what it is and to show with examples which expressions will be identically equal to others.

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Identically equal expressions: definition

The concept of identically equal expressions is usually studied together with the concept of identity itself as part of a school algebra course. Here is the basic definition taken from one textbook:

Definition 1

Identically equal each other there will be such expressions, the values ​​of which will be the same for any possible values ​​of the variables included in their composition.

Also, those numerical expressions to which the same values ​​will correspond are considered identically equal.

This is a fairly broad definition that will be true for all integer expressions whose meaning does not change when the values ​​of the variables change. However, later it becomes necessary to clarify this definition, since in addition to integers, there are other types of expressions that will not make sense with certain variables. This gives rise to the concept of admissibility and inadmissibility of certain variable values, as well as the need to determine the range of permissible values. Let us formulate a refined definition.

Definition 2

Identically equal expressions– these are those expressions whose values ​​are equal to each other for any permissible values ​​of the variables included in their composition. Numerical expressions will be identically equal to each other provided the values ​​are the same.

The phrase “for any valid values ​​of the variables” indicates all those values ​​of the variables for which both expressions will make sense. We will explain this point later when we give examples of identically equal expressions.

You can also provide the following definition:

Definition 3

Identically equal expressions are expressions located in the same identity on the left and right sides.

Examples of expressions that are identically equal to each other

Using the definitions given above, let's look at a few examples of such expressions.

Let's start with numerical expressions.

Example 1

Thus, 2 + 4 and 4 + 2 will be identically equal to each other, since their results will be equal (6 and 6).

Example 2

In the same way, the expressions 3 and 30 are identically equal: 10, (2 2) 3 and 2 6 (to calculate the value of the last expression you need to know the properties of the degree).

Example 3

But the expressions 4 - 2 and 9 - 1 will not be equal, since their values ​​are different.

Let's move on to examples of literal expressions. a + b and b + a will be identically equal, and this does not depend on the values ​​of the variables (the equality of expressions in this case is determined by the commutative property of addition).

Example 4

For example, if a is equal to 4 and b is equal to 5, then the results will still be the same.

Another example of identically equal expressions with letters is 0 · x · y · z and 0 . Whatever the values ​​of the variables in this case, when multiplied by 0, they will give 0. The unequal expressions are 6 · x and 8 · x, since they will not be equal for any x.

In the event that the areas of permissible values ​​of the variables coincide, for example, in the expressions a + 6 and 6 + a or a · b · 0 and 0, or x 4 and x, and the values ​​of the expressions themselves are equal for any variables, then such expressions are considered identically equal. So, a + 8 = 8 + a for any value of a, and a · b · 0 = 0 too, since multiplying any number by 0 results in 0. The expressions x 4 and x will be identically equal for any x from the interval [ 0 , + ∞) .

But the range of valid values ​​in one expression may be different from the range of another.

Example 5

For example, let's take two expressions: x − 1 and x - 1 · x x. For the first of them, the range of permissible values ​​of x will be the entire set of real numbers, and for the second - the set of all real numbers, with the exception of zero, because then we will get 0 in the denominator, and such a division is not defined. These two expressions have a common range of values ​​formed by the intersection of two separate ranges. We can conclude that both expressions x - 1 · x x and x − 1 will make sense for any real values ​​of the variables, with the exception of 0.

The basic property of the fraction also allows us to conclude that x - 1 · x x and x − 1 will be equal for any x that is not 0. This means that on the general range of permissible values ​​these expressions will be identically equal to each other, but for any real x we ​​cannot speak of identical equality.

If we replace one expression with another, which is identically equal to it, then this process is called an identity transformation. This concept is very important, and we will talk about it in detail in a separate material.

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Having gained an idea of ​​identities, it is logical to move on to getting acquainted with. In this article we will answer the question of what identically equal expressions are, and also use examples to understand 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 7th grade algebra class. In the textbook on algebra for 7th grade by the author Yu. N. Makarychev, the following formulation is given:

Definition.

– these are expressions whose values ​​are equal for any values ​​of the variables included in them. Numerical expressions that have identical values ​​are also called identically equal.

This definition is used up to grade 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 clarified. Let us explain what this is connected with.

In the 8th grade, the study of other types of expressions begins, which, unlike whole expressions, may not make sense for some values ​​of the variables. This forces us to introduce definitions of permissible and unacceptable values ​​of variables, as well as the range of permissible values ​​of the variable’s variable value, and, as a consequence, to clarify the definition of identically equal expressions.

Definition.

Two expressions whose values ​​are equal for all permissible values ​​of the variables included in them are called identically equal expressions. Two numerical expressions having the same values ​​are also called identically equal.

In this definition of identically equal expressions, it is worth clarifying the meaning of the phrase “for all permissible values ​​of the variables included in them.” It implies all such values ​​of variables for which both identically equal expressions make sense at the same time. We will explain this idea in the next paragraph by looking at examples.

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

Definition.

Identically equal expressions– these are expressions on the left and right sides of the identity.

The meaning of this and the previous definitions coincide.

Examples of identically equal expressions

The definitions introduced in the previous paragraph allow us to give examples of identically equal expressions.

Let's start with identically equal numerical expressions. The numerical expressions 1+2 and 2+1 are identically equal, since 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 latter expressions are equal by virtue of ). But the numerical expressions 3+2 and 3−2 are not identically equal, since they correspond to the values ​​5 and 1, respectively, and they are not equal.

Now let's 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 ​​(as 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 obtain 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, when x=1 their values ​​are not equal. Indeed, for x=1, the expression 2 x is equal to 2 x 1=2, and the expression 3 x is equal to 3 x 1=3.

When the ranges of permissible 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 the variables from these areas, then here everything is clear - these expressions are identically equal for all permissible 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 of ; ed. S. A. Telyakovsky. - 17th ed. - M.: Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.

While studying algebra, we came across the concepts of a polynomial (for example ($y-x$,$\ 2x^2-2x$, etc.) and algebraic fraction (for example $\frac(x+5)(x)$, $\frac(2x ^2)(2x^2-2x)$,$\ \frac(x-y)(y-x)$, etc.) The similarity of these concepts is that both polynomials and algebraic fractions contain variables and numerical values, and arithmetic is performed. actions: addition, subtraction, multiplication, exponentiation. The difference between these concepts is that in polynomials division by a variable is not performed, but in algebraic fractions division by a variable can be performed.

Both polynomials and algebraic fractions are called rational algebraic expressions in mathematics. But polynomials are whole rational expressions, and algebraic fractions are fractional rational expressions.

It is possible to obtain an entire algebraic expression from a fractional-rational expression using an identity transformation, which in this case will be the main property of a fraction - the reduction of fractions. Let's check this in practice:

Example 1

Convert:$\ \frac(x^2-4x+4)(x-2)$

Solution: This fractional rational equation can be transformed by using the basic property of fractional reduction, i.e. dividing the numerator and denominator by the same number or expression other than $0$.

This fraction cannot be reduced immediately; the numerator must be converted.

Let's transform the expression in the numerator of the fraction, for this we use the formula for the square of the difference: $a^2-2ab+b^2=((a-b))^2$

The fraction looks like

\[\frac(x^2-4x+4)(x-2)=\frac(x^2-4x+4)(x-2)=\frac(((x-2))^2)( x-2)=\frac(\left(x-2\right)(x-2))(x-2)\]

Now we see that the numerator and denominator have a common factor - this is the expression $x-2$, by which we will reduce the fraction

\[\frac(x^2-4x+4)(x-2)=\frac(x^2-4x+4)(x-2)=\frac(((x-2))^2)( x-2)=\frac(\left(x-2\right)(x-2))(x-2)=x-2\]

After reduction, we found that the original fractional rational expression $\frac(x^2-4x+4)(x-2)$ became a polynomial $x-2$, i.e. whole rational.

Now let us pay attention to the fact that the expressions $\frac(x^2-4x+4)(x-2)$ and $x-2\ $ can be considered identical not for all values ​​of the variable, because in order for a fractional rational expression to exist and to be able to reduce by the polynomial $x-2$, the denominator of the fraction must not be equal to $0$ (as well as the factor by which we are reducing. In this example, the denominator and the factor are the same, but This doesn't always happen).

The values ​​of the variable at which the algebraic fraction will exist are called the permissible values ​​of the variable.

Let's put a condition on the denominator of the fraction: $x-2≠0$, then $x≠2$.

This means that the expressions $\frac(x^2-4x+4)(x-2)$ and $x-2$ are identical for all values ​​of the variable except $2$.

Definition 1

Identically equal expressions are those that are equal for all valid values ​​of the variable.

An identical transformation is any replacement of the original expression with an identically equal one. Such transformations include performing actions: addition, subtraction, multiplication, putting a common factor out of brackets, bringing algebraic fractions to a common denominator, reducing algebraic fractions, bringing similar terms, etc. It is necessary to take into account that a number of transformations, such as reduction, reduction of similar terms, can change the permissible values ​​of the variable.

Techniques used to prove identities

Bring the left side of the identity to the right or vice versa using identity transformations

Reduce both sides to the same expression using identical transformations

Transfer the expressions in one part of the expression to another and prove that the resulting difference is equal to $0$

Which of the above techniques to use to prove a given identity depends on the original identity.

Example 2

Prove the identity $((a+b+c))^2- 2(ab+ac+bc)=a^2+b^2+c^2$

Solution: To prove this identity, we use the first of the above methods, namely, we will transform the left side of the identity until it is equal to the right.

Let's consider the left side of the identity: $\ ((a+b+c))^2- 2(ab+ac+bc)$ - it represents the difference of two polynomials. In this case, the first polynomial is the square of the sum of three terms. To square the sum of several terms, we use the formula:

\[((a+b+c))^2=a^2+b^2+c^2+2ab+2ac+2bc\]

To do this, we need to multiply a number by a polynomial. Remember that for this we need to multiply the common factor behind the brackets by each term of the polynomial in the brackets. Then we get:

$2(ab+ac+bc)=2ab+2ac+2bc$

Now let's return to the original polynomial, it will take the form:

$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2+2ab+2ac+2bc-(2ab+2ac+2bc)$

Please note that before the bracket there is a “-” sign, which means that when the brackets are opened, all the signs that were in the brackets change to the opposite.

$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2+2ab+2ac+2bc-(2ab+2ac+2bc)= a ^2+b^2+c^2+2ab+2ac+2bc-2ab-2ac-2bc$

Let us present similar terms, then we obtain that the monomials $2ab$, $2ac$,$\ 2bc$ and $-2ab$,$-2ac$, $-2bc$ cancel each other out, i.e. their sum is $0$.

$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2+2ab+2ac+2bc-(2ab+2ac+2bc)= a ^2+b^2+c^2+2ab+2ac+2bc-2ab-2ac-2bc=a^2+b^2+c^2$

This means that by means of identical transformations we have obtained an identical expression on the left side of the original identity

$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2$

Note that the resulting expression shows that the original identity is true.

Please note that in the original identity all values ​​of the variable are allowed, which means we proved the identity using identity transformations, and it is true for all possible values ​​of the variable.