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What are identical transformations of expressions. Equations

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 actions: addition, subtraction, multiplication, exponentiation The difference between these concepts is that division by a variable is not performed in polynomials, and 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 $0$.

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 the numerator and denominator have common factor--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 reduction by the polynomial $x-2$ be possible, the denominator of the fraction should not be equal to $0$ (as well as the factor by which we reduce. In this example, the denominator and factor 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 performing actions: addition, subtraction, multiplication, taking a common factor out of the bracket, bringing algebraic fractions to a common denominator, reducing algebraic fractions, bringing like 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 we proved the identity using identical transformations, and it is true for all valid values ​​of the variable.

Equations

How to solve equations?

In this section, we will recall (or study - as anyone likes) the most elementary equations. So what is an equation? Speaking in human terms, this is some kind of mathematical expression, where there is an equals sign and an unknown. Which is usually denoted by the letter "X". solve the equation is to find such x-values ​​that, when substituting into original expression, will give us the correct identity. Let me remind you that identity is an expression that does not raise doubts even for a person who is absolutely not burdened with mathematical knowledge. Like 2=2, 0=0, ab=ab etc. So how do you solve equations? Let's figure it out.

There are all sorts of equations (I was surprised, right?). But all their infinite variety can be divided into only four types.

4. Other.)

All the rest, of course, most of all, yes ...) This includes cubic, and exponential, and logarithmic, and trigonometric, and all sorts of others. We will work closely with them in the relevant sections.

I must say right away that sometimes the equations of the first three types are so wound up that you don’t recognize them ... Nothing. We will learn how to unwind them.

And why do we need these four types? And then what linear equations solved in one way square others fractional rational - the third, a rest not solved at all! Well, it’s not that they don’t decide at all, I offended mathematics in vain.) It’s just that they have their own special techniques and methods.

But for any (I repeat - for any!) equations is a reliable and trouble-free basis for solving. Works everywhere and always. This base - Sounds scary, but the thing is very simple. And very (very!) important.

Actually, the solution of the equation consists of these same transformations. At 99%. Answer to the question: " How to solve equations?" lies, just in these transformations. Is the hint clear?)

Identity transformations of equations.

AT any equations to find the unknown, it is necessary to transform and simplify the original example. Moreover, so that when changing the appearance the essence of the equation has not changed. Such transformations are called identical or equivalent.

Note that these transformations are just for the equations. In mathematics, there are still identical transformations expressions. This is another topic.

Now we will repeat all-all-all basic identical transformations of equations.

Basic because they can be applied to any equations - linear, quadratic, fractional, trigonometric, exponential, logarithmic, etc. etc.

First identical transformation: both sides of any equation can be added (subtracted) any(but the same!) a number or an expression (including an expression with an unknown!). The essence of the equation does not change.

By the way, you constantly used this transformation, you only thought that you were transferring some terms from one part of the equation to another with a sign change. Type:

The matter is familiar, we move the deuce to the right, and we get:

Actually you taken away from both sides of the equation deuce. The result is the same:

x+2 - 2 = 3 - 2

The transfer of terms to the left-right with a change of sign is simply an abbreviated version of the first identical transformation. And why do we need such deep knowledge? - you ask. Nothing in the equations. Move it, for God's sake. Just don't forget to change the sign. But in inequalities, the habit of transference can lead to a dead end ....

Second identity transformation: both sides of the equation can be multiplied (divided) by the same non-zero number or expression. An understandable limitation already appears here: it is stupid to multiply by zero, but it is impossible to divide at all. This is the transformation you use when you decide something cool like

Understandably, X= 2. But how did you find it? Selection? Or just lit up? In order not to pick up and wait for insight, you need to understand that you are just divide both sides of the equation by 5. When dividing the left side (5x), the five was reduced, leaving a pure X. Which is what we needed. And when dividing the right side of (10) by five, it turned out, of course, a deuce.

That's all.

It's funny, but these two (only two!) identical transformations underlie the solution all equations of mathematics. How! It makes sense to look at examples of what and how, right?)

Examples of identical transformations of equations. Main problems.

Let's start with first identical transformation. Move left-right.

An example for the little ones.)

Let's say we need to solve the following equation:

3-2x=5-3x

Let's remember the spell: "with X - to the left, without X - to the right!" This spell is an instruction for applying the first identity transformation.) What is the expression with the x on the right? 3x? The answer is wrong! On our right - 3x! Minus three x! Therefore, when shifting to the left, the sign will change to a plus. Get:

3-2x+3x=5

So, the X's were put together. Let's do the numbers. Three on the left. What sign? The answer "with none" is not accepted!) In front of the triple, indeed, nothing is drawn. And this means that in front of the triple is plus. So the mathematicians agreed. Nothing is written, so plus. Therefore, the triple will be transferred to the right side with a minus. We get:

-2x+3x=5-3

There are empty spaces left. On the left - give similar ones, on the right - count. The answer is immediately:

In this example, one identical transformation was enough. The second was not needed. Well, okay.)

An example for the elders.)

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

Along with the study of operations and their properties in algebra, they study such concepts as expression, equation, inequality . The initial acquaintance with them takes place in primary course mathematics. They are introduced, as a rule, without strict definitions, most often ostensively, which requires the teacher not only to be very careful in the use of terms denoting these concepts, but also to know a number of their properties. Therefore, the main task that we set when starting to study the material of this paragraph is to clarify and deepen knowledge about expressions (numerical and with variables), numerical equalities and numerical inequalities, equations and inequalities.

The study of these concepts is associated with the use of mathematical language, it refers to artificial languages which are created and developed together with this or that science. Like any other mathematical language, it has its own alphabet. In our course, it will be presented partially, due to the need to pay more attention to the relationship between algebra and arithmetic. This alphabet includes:

1) numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; with their help, numbers are written according to special rules;

2) signs of operations +, -, , :;

3) relationship signs<, >, =, M;

4) lowercase letters of the Latin alphabet, they are used to designate numbers;

5) brackets (round, curly, etc.), they are called technical signs.

Using this alphabet, words are formed in algebra, calling them expressions, and sentences are obtained from words - numerical equalities, numerical inequalities, equations, inequalities with variables.

As you know, records 3 + 7, 24: 8, 3 × 2 - 4, (25 + 3)× 2-17 are called numerical expressions. They are formed from numbers, action signs, brackets. If we perform all the actions indicated in the expression, we get a number called the value of a numeric expression . So, the value of the numeric expression is 3 × 2 - 4 is equal to 2.

There are numeric expressions whose values ​​cannot be found. Such expressions are said to be don't make sense .

for example, expression 8: (4 - 4) does not make sense, since its value cannot be found: 4 - 4 = 0, and division by zero is impossible. The expression 7-9 also makes no sense if we consider it on the set natural numbers, since the values ​​of the expression 7-9 cannot be found on this set.

Consider the notation 2a + 3. It is formed from numbers, action signs and the letter a. If instead of a we substitute numbers, then different numerical expressions will be obtained:

if a = 7, then 2 × 7 + 3;

if a = 0, then 2 × 0 + 3;

if a = - 4, then 2 × (- 4) + 3.

In the notation 2a + 3, such a letter a is called variable , and the entry itself 2a + 3 - variable expression.


A variable in mathematics is usually denoted by any lower case Latin alphabet. AT primary school to denote a variable, besides letters, other signs are used, for example, œ. Then the expression with a variable has the form: 2ל + 3.

Each expression with a variable corresponds to a set of numbers, substituting which results in a numerical expression that makes sense. This set is called expression scope .

For example, the domain of expression 5: (x - 7) consists of all real numbers, except for the number 7, since for x \u003d 7 the expression 5: (7 - 7) does not make sense.

In mathematics, expressions are considered that contain one, two or more variables.

For example, 2a + 3 is a one-variable expression, and (3x + 8y) × 2 is an expression with three variables. In order to obtain a numeric expression from an expression with three variables, instead of each variable, substitute the numbers that belong to the scope of the expression.

So, we have found out how numerical expressions and expressions with variables are formed from the alphabet of the mathematical language. If we draw an analogy with the Russian language, then expressions are the words of the mathematical language.

But, using the alphabet of the mathematical language, it is possible to form such, for example, records: (3 + 2)) - × 12 or 3x - y: +) 8, which cannot be called either a numeric expression or an expression with a variable. These examples indicate that the description - from which characters of the alphabet of the mathematical language expressions are formed, numerical and with variables, is not a definition of these concepts. Let's give a definition of a numeric expression (an expression with variables is defined similarly).

Definition.If f and q are numeric expressions, then (f) + (q), (f) - (q), (f) × (q), (f) (q) are numerical expressions. Each number is considered to be a numeric expression.

If this definition is followed exactly, then one would have to write too many brackets, for example, (7) + (5) or (6): (2). To shorten the notation, we agreed not to write brackets if several expressions are added or subtracted, and these operations are performed from left to right. In the same way, brackets are not written when several numbers are multiplied or divided, and these operations are performed in order from left to right.

for example, they write like this: 37 - 12 + 62 - 17 + 13 or 120:15-7:12.

In addition, we agreed to first perform the actions of the second stage (multiplication and division), and then the actions of the first stage (addition and subtraction). Therefore, the expression (12-4:3) + (5-8:2-7) is written as follows: 12 - 4: 3 + 5 - 8: 2 - 7.

Task. Find the value of the expression 3x (x - 2) + 4(x - 2) for x = 6.

Decision

1 way. Substitute the number 6 instead of a variable in this expression: 3 × 6-(6 - 2) + 4 × (6 - 2). To find the value of the resulting numerical expression, we perform all the indicated actions: 3 × 6 × (6 - 2) + 4 × (6-2) = 18 × 4 + 4 × 4 = 72 + 16 = 88. Therefore, when X= 6 the value of the expression 3x(x-2) + 4(x-2) is 88.

2 way. Before substituting the number 6 in this expression, let's simplify it: Zx (x - 2) + 4 (x - 2) = (X - 2)(3x + 4). And then, substituting in the resulting expression instead of X number 6, do the following: (6 - 2) × (3 × 6 + 4) = 4x (18 + 4) = 4x22 = 88.

Let us pay attention to the following: both in the first method of solving the problem, and in the second one, we replaced one expression with another.

for example, the expression 18 × 4 + 4 × 4 was replaced by the expression 72 + 16, and the expression 3x (x - 2) + 4(x - 2) - by the expression (X - 2)(3x + 4), and these substitutions lead to the same result. In mathematics, describing the solution of this problem, they say that we performed identical transformations expressions.

Definition.Two expressions are said to be identically equal if, for any values ​​of the variables from the domain of the expressions, their corresponding values ​​are equal.

Examples of identically equal expressions are the expressions 5(x + 2) and 5x+ 10, because for any real values X their values ​​are equal.

If two expressions that are identically equal on a certain set are joined by an equal sign, then we get a sentence called identity on this set.

for example, 5(x + 2) = 5x + 10 is an identity on the set of real numbers, because for all real numbers the values ​​of the expression 5(x + 2) and 5x + 10 are the same. Using the general quantifier notation, this identity can be written as follows: (" x н R) 5(x + 2) = 5x + 10. True numerical equalities are also considered identities.

Replacing an expression with another that is identically equal to it on some set is called the identical transformation of the given expression on this set.

So, replacing the expression 5(x + 2) with the expression 5x + 10, which is identically equal to it, we performed the identical transformation of the first expression. But how, given two expressions, to find out whether they are identically equal or not? Find the corresponding values ​​of expressions by substituting specific numbers for variables? Long and not always possible. But then what are the rules that must be followed when performing identical transformations of expressions? There are many of these rules, among them are the properties of algebraic operations.

Task. Factor the expression ax - bx + ab - b 2 .

Decision. Let's group the members of this expression in two (the first with the second, the third with the fourth): ax - bx + ab - b 2 \u003d (ax-bx) + (ab-b 2). This transformation is possible based on the associativity property of addition of real numbers.

We take out the common factor in the resulting expression from each bracket: (ax - bx) + (ab - b 2) \u003d x (a - b) + b (a - b) - this transformation is possible based on the distributive property of multiplication with respect to the subtraction of real numbers.

In the resulting expression, the terms have a common factor, we take it out of brackets: x (a - b) + b (a - b) \u003d (a - b) (x - b). The basis of the performed transformation is the distributive property of multiplication with respect to addition.

So, ax - bx + ab - b 2 \u003d (a - b) (x - b).

In the initial course of mathematics, as a rule, only identical transformations are performed numeric expressions. Theoretical basis Such transformations are the properties of addition and multiplication, various rules: adding a sum to a number, a number to a sum, subtracting a number from a sum, etc.

for example, to find the product of 35 × 4, you need to perform transformations: 35 × 4 = (30 + 5) × 4 = 30 × 4 + 5 × 4 = 120 + 20 = 140. The performed transformations are based on: the distributive property of multiplication with respect to addition; the principle of writing numbers in the decimal number system (35 = 30 + 5); rules for multiplication and addition of natural numbers.

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.

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Slides captions:

Identities. Identity transformations of expressions. 7th grade.

Find the value of the expressions at x=5 and y=4 3(x+y)= 3(5+4)=3*9=27 3x+3y= 3*5+3*4=27 Find the value of the expressions at x=6 and y=5 3(x+y)= 3(6+5)=3*11=33 3x+3y= 3*6+3*5=33

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

Consider now the expressions 2x + y and 2xy. for x=1 and y=2 they take equal values: 2x+y=2*1+2=4 2x=2*1*2=4 at x=3, y=4 expression values ​​are different 2x+y=2*3+4=10 2x=2*3*4 =24

CONCLUSION: The expressions 3(x+y) and 3x+3y are identically equal, but the expressions 2x+y and 2xy are not identically equal. Definition: Two expressions whose values ​​are equal for any values ​​of the variables are said to be identically equal.

IDENTITY The equality 3(x+y) and 3x+3y is true for any values ​​of x and y. Such equalities are called identities. Definition: An equality that is true for any values ​​of the variables is called an identity. True numerical equalities are also considered identities. We have already met with identities.

Identities are equalities expressing the basic properties of actions on numbers. a + b = b + a ab = ba (a + b) + c = a + (b + c) (ab)c = a(bc) a(b + c) = ab + ac

Other examples of identities can be given: a + 0 = a a * 1 = a a + (-a) = 0 a * (- b) = - ab a- b = a + (- b) (-a) * ( -b) = ab Replacing one expression with another expression identically equal to it is called identity transformation or simply expression transformation.

To bring like terms, you need to add their coefficients and multiply the result by the common letter part. Example 1. We give like terms 5x + 2x-3x \u003d x (5 + 2-3) \u003d 4x

If there is a plus sign in front of the brackets, then the brackets can be omitted, preserving the sign of each term enclosed in brackets. Example 2. Expand the brackets in the expression 2a + (b -3 c) = 2 a + b - 3 c

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 3. Let's open the brackets in the expression a - (4 b - c) \u003d a - 4 b + c

Homework: p. 5, No. 91, 97, 99 Thank you for the lesson!


On the topic: methodological developments, presentations and notes

Methods of preparing students for the exam in the section "Expressions and transformation of expressions"

This project was developed with the aim of preparing students for state exams in grade 9 and further to a unified state exam in 11th grade...