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.

IN 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. and so on.

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.

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 in polynomials division by a variable is not performed, and 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 integer rational expressions, and algebraic fractional expressions are fractionally rational expressions.

It is possible to obtain a whole algebraic expression from a fractional-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:$\ \frac(x^2-4x+4)(x-2)$

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

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 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.

Let two algebraic expressions be given:

Let's make a table of the values ​​of each of these expressions for different numerical values ​​of the letter x.

We see that for all those values ​​that were given to the letter x, the values ​​of both expressions turned out to be equal. The same will be true for any other value of x.

To verify this, we transform the first expression. Based on the distribution law, we write:

Having performed the indicated operations on the numbers, we get:

So, the first expression, after its simplification, turned out to be exactly the same as the second expression.

Now it is clear that for any value of x, the values ​​of both expressions are equal.

Expressions whose values ​​are equal for any values ​​of the letters included in them are called identically equal or identical.

Hence, they are identical expressions.

Let's make one important remark. Let's take expressions:

Having compiled a table similar to the previous one, we will make sure that both expressions, for any value of x, except for have equal numerical values. Only when the second expression is equal to 6, and the first loses its meaning, since the denominator is zero. (Recall that you cannot divide by zero.) Can we say that these expressions are identical?

We agreed earlier that each expression will be considered only for admissible values ​​of letters, that is, for those values ​​for which the expression does not lose its meaning. This means that here, when comparing two expressions, we take into account only those letter values ​​that are valid for both expressions. Therefore, we must exclude the value. And since for all other values ​​of x both expressions have the same numerical value, we have the right to consider them identical.

Based on what has been said, we give the following definition of identical expressions:

1. Expressions are called identical if they have the same numerical values ​​for all admissible values ​​of the letters included in them.

If we connect two identical expressions with an equal sign, then we get an identity. Means:

2. An identity is an equality that is true for all admissible values ​​of the letters included in it.

We have already encountered identities before. So, for example, all equalities are identities, with which we expressed the basic laws of addition and multiplication.

For example, equalities expressing the commutative law of addition

and the associative law of multiplication

are valid for any values ​​of letters. Hence, these equalities are identities.

All true arithmetic equalities are also considered identities, for example:

In algebra, one often has to replace an expression with another that is identical to it. Let, for example, it is required to find the value of the expression

We will greatly facilitate the calculations if we replace the given expression with an expression that is identical to it. Based on the distribution law, we can write:

But the numbers in brackets add up to 100. So, we have an identity:

Substituting 6.53 instead of a on the right side of it, we immediately (in the mind) find the numerical value (653) of this expression.

Replacing one expression with another, identical to it, is called the identical transformation of this expression.

Recall that any algebraic expression for any admissible values ​​of letters is some

number. It follows from this that all the laws and properties of arithmetic operations that were given in the previous chapter are applicable to algebraic expressions. So, the application of the laws and properties of arithmetic operations transforms a given algebraic expression into an expression that is identical to it.

7th grade

“Identities. Identity transformation of expressions”.

Abdulkerimova Khadizhat Makhmudovna,

mathematic teacher

Lesson Objectives

to acquaint and initially consolidate the concepts of "identically equal expressions", "identity", "identical transformations";

to consider ways to prove identities, to contribute to the development of skills to prove identities;

to check the students' assimilation of the material covered, to form the skills of applying the studied for the perception of the new.

Lesson type: learning new material

Equipment : board, textbook, workbook.

P lan lesson

Organizing time

Checking homework

Knowledge update

The study of new material (Introduction and primary consolidation of the concepts of "identity", "identical transformations").

Training exercises (Formation of the concepts of "identity", "identical transformations").

Reflection of the lesson (Summarize the theoretical information obtained in the lesson).

Homework message (Explain the content of homework)

During the classes

I. Organizational moment.

II . Checking homework. (Front)

III . Knowledge update.

Give an example of a numeric expression and an expression with variables

Compare the values ​​of the expressions x+3 and 3x at x=-4; 1.5; 5

What number cannot be divided by? (0)

Multiplication result? (Work)

Largest two digit number? (99)

What is the product from -200 to 200? (0)

The result of the subtraction. (Difference)

How many grams in a kilogram? (1000)

Commutative property of addition. (The sum does not change from the rearrangement of the places of the terms)

Commutative property of multiplication. (The product does not change from the permutation of the places of factors)

The associative property of addition. (In order to add a number to the sum of two numbers, you can add the sum of the second and third to the first number)

Associative property of multiplication. (to multiply the product of two numbers by the third number, you can multiply the first number by the product of the second and third)

distribution property. (To multiply a number by the sum of two numbers, you can multiply this number by each term and add the results)

IV. Explanation of the new topic:

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

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.

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

2x+y=2*1+2=4

2xy=2*1*2=4

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

2x+y=2*3+4=10

2xy=2*3*4=24

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

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) 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 that express the basic properties of actions on numbers (Students comment on each property by pronouncing it).

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 (Students comment on each property, pronouncing it).

a + 0 = a

a * 1 = a

a + (-a) = 0

A * (- b ) = - ab

a - b = a + (- b )

(- a ) * (- b ) = ab

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

Teacher:

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

Identity transformations of expressions are widely used in calculating the values ​​of expressions and solving other problems. You already had to perform some identical transformations, for example, reduction of similar terms, expansion of brackets. Recall the rules for these transformations:

Students:

To bring like terms, it is necessary 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.

Teacher:

Example 1. We present similar terms

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

What rule did we use?

Student:

We have used the rule of reduction of like terms. This transformation is based on the distributive property of multiplication.

Teacher:

Example 2. Expand the brackets in the expression 2a + (b-3 c) = 2 a + b – 3 c

We applied the rule of opening brackets preceded by a plus sign.

Student:

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

Teacher:

Example 3. Let's open the brackets in the expression a - (4b- c) =a – 4 b + c

We used the rule of opening brackets, which are preceded by a minus sign.

What property is this transformation based on?

Student:

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

V . Doing exercises.

№85 Orally

№86 Orally

№88 Orally

№93

№94

№90av

№96

№97

VI . Lesson reflection .

The teacher asks questions, and the students answer them as they wish.

What two expressions are called identically equal? Give examples.

What equality is called identity? Give an example.

What identical transformations do you know?

VII . Homework . p.5, No. 95, 98,100 (a, c)

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

Identity transformation of an expression. What it is?

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

Definition 1

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

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

Let us illustrate this definition with examples.

Example 1

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

Example 2

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

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

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

Identity transformations and ODZ

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

Example 3

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

Example 4

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

Example 5

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

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

Basic identity transformations

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

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

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

Rearrangement of terms, factors

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

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

Example 6

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

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

Example 7

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

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

Definition 2

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

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

Example 8

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

Example 9

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

Bracket expansion

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

Example 10

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

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

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

Grouping terms, factors

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

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

Example 11

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

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

Example 12

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

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

Replacing differences by sums, partial products and vice versa

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

Example 13

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

Example 14

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

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

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

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

Example 15

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

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

Example 16

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

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

Example 17

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

Performing actions with numbers

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

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

Example 18

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

Solution

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

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

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

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

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

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

Example 19

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

Solution

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

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

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

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

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

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

Bracketing the Common Factor

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

Example 20

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

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

Reduction of similar terms

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

Example 21

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

Replacing numbers and expressions with identically equal expressions

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

Example 22 Example 23

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

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

Example 24

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

Adding and subtracting the same number

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

Example 25

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

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