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 fractional expressions are fractionally rational expressions.
It is possible to obtain a whole algebraic expression from a fractionally rational 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: This fractional-rational equation can be transformed by using the basic property of the fraction-cancellation, i.e. dividing the numerator and denominator by the same number or expression other than $0$.
Straightaway given fraction cannot be reduced, the numerator must be converted.
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 have obtained 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.
Consider two equalities:
1. a 12 * a 3 = a 7 * a 8
This equality will hold for any value of the variable a. area allowed values for that equality there will be the whole 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 acceptable 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 would be identity 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.
After we have dealt with the concept of identities, we can proceed to the study of 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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Identical Equal Expressions: Definition
The concept of identically equal expressions is usually studied together with the concept of identity itself in the framework of a school algebra course. Here is a basic definition taken from one textbook:
Definition 1
identically equal to each other there will be such expressions, the values of which will be the same for any possible values variables included in them.
Also, such numerical expressions are considered identically equal, which will correspond to the same values.
This is a fairly broad definition, which will be true for all integer expressions, the meaning of which does not change when the values of the variables change. However, later it becomes necessary to clarify this definition, because in addition to integers, there are other kinds of expressions that will not make sense with certain variables. This gives rise to the concept of the admissibility and inadmissibility of certain values of variables, as well as the need to determine the range of admissible values. Let us formulate a refined definition.
Definition 2
Identical equal expressions are those expressions whose values are equal to each other for any valid values of the variables included in their composition. Numeric expressions will be identically equal to each other provided the same values.
The phrase "for any admissible values of the variables" indicates all those values of the variables for which both expressions will make sense. We will explain this position later, when we give examples of identically equal expressions.
You can also specify the following definition:
Definition 3
Identical 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, consider a few examples of such expressions.
Let's start with numeric expressions.
Example 1
Thus, 2 + 4 and 4 + 2 will be identically equal to each other, since their results will be equal to (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 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 4 and b is 5, 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 because they won't be equal for any x .
In the event that the ranges of allowable values of the variables will 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 will be 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 scope of a valid value in one expression may differ from the scope 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 acceptable x values will be the entire set of real numbers, and for the second, the set of all real numbers, except for zero, because then we will get 0 in the denominator, and such a division is not defined. These two expressions have a common range, formed by the intersection of two separate ranges. It can be concluded that both expressions x - 1 · x x and x − 1 will make sense for any real values of the variables, except for 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 . So on general area admissible values, these expressions will be identically equal to each other, and for any real x one cannot speak of identical equality.
If we replace one expression with another that is identically equal to it, then this process is called identity transformation. This concept is very important, and we will talk about it in detail in a separate article.
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§ 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 of the 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 \u003d 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 that records known properties of 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.
