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What does identically equal mean. Identical equal expressions: definition, examples

In the course of studying algebra, we came across the concepts of polynomial (for example ($y-x$ ,$\ 2x^2-2x$ and so on) 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 in polynomials and in algebraic fractions there are variables and numerical values, arithmetic operations: addition, subtraction, multiplication, exponentiation. The difference between these concepts is that division by a variable is not performed in polynomials, while division by a variable can be performed in algebraic fractions.

Both polynomials and algebraic fractions are called rational algebraic expressions in mathematics. But polynomials are integer rational expressions, and algebraic fractions are fractionally rational expressions.

You can get an integer from a fractional-rational expression algebraic expression using the identical transformation, which in this case will be the main property of a fraction - reduction of fractions. Let's check it out in practice:

Example 1

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

Decision: Convert Given fractional rational equation possible by using the main property fractions - abbreviations, i.e. dividing the numerator and denominator by the same number or expression other than $0$.

This fraction cannot be reduced immediately, it is necessary to convert the numerator.

We 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 has the form

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

Now we see that there is a common factor in the numerator and denominator - this is the expression $x-2$, on 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 get that the original fractional rational expression$\frac(x^2-4x+4)(x-2)$ has become a polynomial $x-2$, i.e. whole rational.

Now let's 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 for the reduction by the polynomial $x-2$ to be possible, the denominator of the fraction must not be equal to $0$ (as well as the factor by which we reduce. In this example the denominator and the multiplier are the same, but this is not always the case).

Variable values ​​for which the algebraic fraction will exist are called valid variable values.

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

So 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 possible values ​​of the variable.

An identical transformation is any replacement of the original expression with an identically equal one. Such transformations include the following actions: addition, subtraction, multiplication, parentheses algebraic fractions to a common denominator, reduction of algebraic fractions, reduction of similar terms, etc. It must be taken into account that a number of transformations, such as reduction, reduction of similar terms, can change the allowable values ​​of the variable.

Techniques used to prove identities

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

    Reduce both parts 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 methods to use to prove a given identity depends on the original identity.

Example 2

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

Decision: 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 side.

Consider the left side of the identity: $\ ((a+b+c))^2- 2(ab+ac+bc)$- it is 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. Recall that for this we need to multiply the common factor outside the brackets by each term of the polynomial in brackets. Then we get:

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

Now back to the original polynomial, it will take the form:

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

Note that there is a “-” sign in front of the bracket, which means that when the brackets are opened, all the signs that were in the brackets are reversed.

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

If we bring similar terms, then we get that the monomials $2ab$, $2ac$,$\ 2bc$ and $-2ab$,$-2ac$, $-2bc$ cancel each other out, i.e. their sum is equal to $0$.

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

So, by identical transformations, we obtained the 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.

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

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.

For example, in the expression 3+x, the number 3 can be replaced by the sum 1+2 , which results in the expression (1+2)+x , which is identically equal to the original expression. Another example: in the expression 1+a 5 the degree of a 5 can be replaced by a product identically equal to it, for example, of the form a·a 4 . This will give us the expression 1+a·a 4 .

This transformation is undoubtedly artificial, and is usually a preparation for some further transformation. For example, in the sum 4·x 3 +2·x 2 , taking into account the properties of the degree, the term 4·x 3 can be represented as a product 2·x 2 ·2·x . After such a transformation, the original expression will take the form 2·x 2 ·2·x+2·x 2 . Obviously, the terms in the resulting sum have a common factor 2 x 2, so we can perform the following transformation - parentheses. After it, we will come to the expression: 2 x 2 (2 x+1) .

Adding and subtracting the same number

Another artificial transformation of an expression is the addition and subtraction of the same number or expression at the same time. Such a transformation is identical, since it is, in fact, equivalent to adding zero, and adding zero does not change the value.

Consider an example. Let's take the expression x 2 +2 x . If you add one to it and subtract one, then this will allow you to perform another identical transformation in the future - select the square of the binomial: x 2 +2 x=x 2 +2 x+1−1=(x+1) 2 −1.

Bibliography.

  • Algebra: textbook for 7 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; 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 p.m. Part 1. Student's textbook educational institutions/ A. G. Mordkovich. - 17th ed., add. - M.: Mnemozina, 2013. - 175 p.: ill. ISBN 978-5-346-02432-3.

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) = -4.

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 identity transformation or simply by converting 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?

  1. (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?

  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 identity of equality:

1) 2a + 106 = 12ab;

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

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

  1. Open parenthesis:
  1. Open parenthesis:
  1. Reduce like terms:
  1. Name several expressions that are identical to expressions 2a + 3a.
  2. 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).

  1. Simplify the expression:

1) -2p ∙ 3.5;

2) 7a ∙ (-1.2);

3) 0.2 x ∙ (-3y);

4) - 1 m ∙ (-3n).

  1. (Verbal) Simplify the expression:

1) 2x - 9 + 5x;

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

4) 4a ∙ (-2b).

  1. 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).

  1. 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.

  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.

  1. 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 \u003d 5 (s + 2) - 4 (s + 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 more than it. Write 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 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).

  1. 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).

  1. 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).

  1. Prove the identity:

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

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

  1. 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.

  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 rode a bicycle if it is 12 km/h more than the speed at which he walked.

Interesting tasks for lazy students

  1. 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.

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