Expression with one variable. Numeric, alphabetic and variable expressions: definitions, examples

A literal expression (or an expression with variables) is mathematical expression, which consists of numbers, letters and symbols of mathematical operations. For example, the following expression is literal:

a+b+4

Using alphabetic expressions you can write laws, formulas, equations and functions. The ability to manipulate letter expressions is the key to good knowledge of algebra and higher mathematics.

Any serious problem in mathematics comes down to solving equations. And in order to be able to solve equations, you need to be able to work with literal expressions.

To work with literal expressions, you need to be well-versed in basic arithmetic: addition, subtraction, multiplication, division, basic laws of mathematics, fractions, operations with fractions, proportions. And not just study, but understand thoroughly.

Lesson content

Variables

Letters that are contained in literal expressions are called variables. For example, in the expression a+b+ 4 variables are letters a And b. If we substitute any numbers instead of these variables, then the literal expression a+b+ 4 contact numeric expression, the value of which can be found.

Numbers that are substituted for variables are called values ​​of variables. For example, let's change the values ​​of the variables a And b. The equal sign is used to change values

a = 2, b = 3

We have changed the values ​​of the variables a And b. Variable a assigned a value 2 , variable b assigned a value 3 . As a result, the literal expression a+b+4 turns into a regular numeric expression 2+3+4 whose value can be found:

When variables are multiplied, they are written together. For example, record ab means the same as the entry a×b. If we substitute the variables a And b numbers 2 And 3 , then we get 6

You can also write together the multiplication of a number by an expression in parentheses. For example, instead of a×(b + c) can be written down a(b + c). Applying the distribution law of multiplication, we obtain a(b + c)=ab+ac.

Odds

In literal expressions you can often find a notation in which a number and a variable are written together, for example 3a. This is actually a shorthand for multiplying the number 3 by a variable. a and this entry looks like 3×a .

In other words, the expression 3a is the product of the number 3 and the variable a. Number 3 in this work they call coefficient. This coefficient shows how many times the variable will be increased a. This expression can be read as " a three times" or "three times A", or "increase the value of a variable a three times", but most often read as "three a«

For example, if the variable a equal to 5 , then the value of the expression 3a will be equal to 15.

3 × 5 = 15

Speaking in simple language, the coefficient is the number that comes before the letter (before the variable).

There can be several letters, for example 5abc. Here the coefficient is the number 5 . This coefficient shows that the product of variables abc increases fivefold. This expression can be read as " abc five times" or "increase the value of the expression abc five times" or "five abc«.

If instead of variables abc substitute the numbers 2, 3 and 4, then the value of the expression 5abc will be equal 120

5 × 2 × 3 × 4 = 120

You can mentally imagine how the numbers 2, 3 and 4 were first multiplied, and the resulting value increased fivefold:

The sign of the coefficient refers only to the coefficient and does not apply to the variables.

Consider the expression −6b. Minus before the coefficient 6 , applies only to the coefficient 6 , and does not belong to the variable b. Understanding this fact will allow you not to make mistakes in the future with signs.

Let's find the value of the expression −6b at b = 3.

−6b −6×b. For clarity, let us write the expression −6b in expanded form and substitute the value of the variable b

−6b = −6 × b = −6 × 3 = −18

Example 2. Find the value of an expression −6b at b = −5

Let's write down the expression −6b in expanded form

−6b = −6 × b = −6 × (−5) = 30

Example 3. Find the value of an expression −5a+b at a = 3 And b = 2

−5a+b This short form entries from −5 × a + b, so for clarity we write the expression −5×a+b in expanded form and substitute the values ​​of the variables a And b

−5a + b = −5 × a + b = −5 × 3 + 2 = −15 + 2 = −13

Sometimes letters are written without a coefficient, for example a or ab. In this case, the coefficient is unity:

but traditionally the unit is not written down, so they simply write a or ab

If there is a minus before the letter, then the coefficient is a number −1 . For example, the expression −a actually looks like −1a. This is the product of minus one and the variable a. It turned out like this:

−1 × a = −1a

There's a little catch here. In expression −a minus sign in front of the variable a actually refers to an "invisible unit" rather than a variable a. Therefore, you should be careful when solving problems.

For example, if given the expression −a and we are asked to find its value at a = 2, then at school we substituted a two instead of a variable a and received an answer −2 , without focusing too much on how it turned out. In fact, minus one was multiplied by positive number 2

−a = −1 × a

−1 × a = −1 × 2 = −2

If given the expression −a and you need to find its value at a = −2, then we substitute −2 instead of a variable a

−a = −1 × a

−1 × a = −1 × (−2) = 2

To avoid mistakes, at first invisible units can be written down explicitly.

Example 4. Find the value of an expression abc at a=2 , b=3 And c=4

Expression abc 1×a×b×c. For clarity, let us write the expression abc a, b And c

1 × a × b × c = 1 × 2 × 3 × 4 = 24

Example 5. Find the value of an expression abc at a=−2 , b=−3 And c=−4

Let's write down the expression abc in expanded form and substitute the values ​​of the variables a, b And c

1 × a × b × c = 1 × (−2) × (−3) × (−4) = −24

Example 6. Find the value of an expression − abc at a=3 , b=5 and c=7

Expression − abc this is a short form for −1×a×b×c. For clarity, let us write the expression − abc in expanded form and substitute the values ​​of the variables a, b And c

−abc = −1 × a × b × c = −1 × 3 × 5 × 7 = −105

Example 7. Find the value of an expression − abc at a=−2 , b=−4 and c=−3

Let's write down the expression − abc in expanded form:

−abc = −1 × a × b × c

Let's substitute the values ​​of the variables a , b And c

−abc = −1 × a × b × c = −1 × (−2) × (−4) × (−3) = 24

How to determine the coefficient

Sometimes you need to solve a problem in which you need to determine the coefficient of an expression. In principle, this task is very simple. It is enough to be able to multiply numbers correctly.

To determine the coefficient in an expression, you need to separately multiply the numbers included in this expression and separately multiply the letters. The resulting numerical factor will be the coefficient.

Example 1. 7m×5a×(−3)×n

The expression consists of several factors. This can be clearly seen if you write the expression in expanded form. That is, the works 7m And 5a write it in the form 7×m And 5×a

7 × m × 5 × a × (−3) × n

Let's apply the associative law of multiplication, which allows you to multiply factors in any order. Namely, we will separately multiply the numbers and separately multiply the letters (variables):

−3 × 7 × 5 × m × a × n = −105man

The coefficient is −105 . After completion, it is advisable to arrange the letter part in alphabetical order:

−105amn

Example 2. Determine the coefficient in the expression: −a×(−3)×2

−a × (−3) × 2 = −3 × 2 × (−a) = −6 × (−a) = 6a

The coefficient is 6.

Example 3. Determine the coefficient in the expression:

Let's multiply numbers and letters separately:

The coefficient is −1. Please note that the unit is not written down, since it is customary not to write the coefficient 1.

These seemingly simplest tasks can play a very cruel joke on us. It often turns out that the sign of the coefficient is set incorrectly: either the minus is missing or, on the contrary, it was set in vain. To avoid these annoying mistakes, must be studied at a good level.

Addends in literal expressions

When adding several numbers, the sum of these numbers is obtained. Numbers that add are called addends. There can be several terms, for example:

1 + 2 + 3 + 4 + 5

When an expression consists of terms, it is much easier to evaluate because adding is easier than subtracting. But the expression can contain not only addition, but also subtraction, for example:

1 + 2 − 3 + 4 − 5

In this expression, the numbers 3 and 5 are subtrahends, not addends. But nothing prevents us from replacing subtraction with addition. Then we again get an expression consisting of terms:

1 + 2 + (−3) + 4 + (−5)

It doesn’t matter that the numbers −3 and −5 now have a minus sign. The main thing is that all the numbers in this expression are connected by an addition sign, that is, the expression is a sum.

Both expressions 1 + 2 − 3 + 4 − 5 And 1 + 2 + (−3) + 4 + (−5) equal to the same value - minus one

1 + 2 − 3 + 4 − 5 = −1

1 + 2 + (−3) + 4 + (−5) = −1

Thus, the meaning of the expression will not suffer if we replace subtraction with addition somewhere.

You can also replace subtraction with addition in literal expressions. For example, consider the following expression:

7a + 6b − 3c + 2d − 4s

7a + 6b + (−3c) + 2d + (−4s)

For any values ​​of variables a, b, c, d And s expressions 7a + 6b − 3c + 2d − 4s And 7a + 6b + (−3c) + 2d + (−4s) will be equal to the same value.

You must be prepared for the fact that a teacher at school or a teacher at an institute may call even numbers (or variables) that are not addends.

For example, if the difference is written on the board a − b, then the teacher will not say that a is a minuend, and b- subtractable. He will call both variables with one common word - terms. And all because the expression of the form a − b the mathematician sees how the sum a+(−b). In this case, the expression becomes a sum, and the variables a And (−b) become terms.

Similar terms

Similar terms- these are terms that have the same letter part. For example, consider the expression 7a + 6b + 2a. Components 7a And 2a have the same letter part - variable a. So the terms 7a And 2a are similar.

Typically, similar terms are added to simplify an expression or solve an equation. This operation is called bringing similar terms.

To bring similar terms, you need to add the coefficients of these terms, and multiply the resulting result by the common letter part.

For example, let us present similar terms in the expression 3a + 4a + 5a. IN in this case, all terms are similar. Let's add up their coefficients and multiply the result by the common letter part - by the variable a

3a + 4a + 5a = (3 + 4 + 5)×a = 12a

Similar terms are usually brought up in mind and the result is written down immediately:

3a + 4a + 5a = 12a

Also, one can reason as follows:

There were 3 variables a , 4 more variables a and 5 more variables a were added to them. As a result, we got 12 variables a

Let's look at several examples of bringing similar terms. Considering that this topic is very important, at first we will write down every little detail in detail. Even though everything is very simple here, most people make many mistakes. Mainly due to inattention, not ignorance.

Example 1. 3a + 2a + 6a + 8 a

Let's add up the coefficients in this expression and multiply the resulting result by the common letter part:

3a + 2a + 6a + 8a = (3 + 2 + 6 + 8) × a = 19a

design (3 + 2 + 6 + 8)×a You don’t have to write it down, so we’ll write down the answer right away

3a + 2a + 6a + 8a = 19a

Example 2. Give similar terms in the expression 2a+a

Second term a written without a coefficient, but in fact there is a coefficient in front of it 1 , which we do not see because it is not recorded. So the expression looks like this:

2a + 1a

Now let's present similar terms. That is, we add up the coefficients and multiply the result by the common letter part:

2a + 1a = (2 + 1) × a = 3a

Let's write down the solution briefly:

2a + a = 3a

2a+a, you can think differently:

Example 3. Give similar terms in the expression 2a−a

Let's replace subtraction with addition:

2a + (−a)

Second term (−a) written without a coefficient, but in reality it looks like (−1a). Coefficient −1 again invisible due to the fact that it is not recorded. So the expression looks like this:

2a + (−1a)

Now let's present similar terms. Let's add the coefficients and multiply the result by the total letter part:

2a + (−1a) = (2 + (−1)) × a = 1a = a

Usually written shorter:

2a − a = a

Giving similar terms in the expression 2a−a You can think differently:

There were 2 variables a, subtract one variable a, in the end there was only one variable a left

Example 4. Give similar terms in the expression 6a − 3a + 4a − 8a

6a − 3a + 4a − 8a = 6a + (−3a) + 4a + (−8a)

Now let's present similar terms. Let's add the coefficients and multiply the result by the total letter part

(6 + (−3) + 4 + (−8)) × a = −1a = −a

Let's write down the solution briefly:

6a − 3a + 4a − 8a = −a

There are expressions that contain several various groups similar terms. For example, 3a + 3b + 7a + 2b. For such expressions, the same rules apply as for the others, namely, adding the coefficients and multiplying the result by the common letter part. But to avoid mistakes, it’s convenient different groups The terms are highlighted with different lines.

For example, in the expression 3a + 3b + 7a + 2b those terms that contain a variable a, can be underlined with one line, and those terms that contain a variable b, can be emphasized with two lines:

Now we can present similar terms. That is, add the coefficients and multiply the resulting result by the total letter part. This must be done for both groups of terms: for terms containing a variable a and for terms containing a variable b.

3a + 3b + 7a + 2b = (3+7)×a + (3 + 2)×b = 10a + 5b

Again, we repeat, the expression is simple, and similar terms can be given in mind:

3a + 3b + 7a + 2b = 10a + 5b

Example 5. Give similar terms in the expression 5a − 6a −7b + b

Let's replace subtraction with addition where possible:

5a − 6a −7b + b = 5a + (−6a) + (−7b) + b

Let us underline similar terms with different lines. Terms containing variables a we underline with one line, and the terms are the contents of the variables b, underline with two lines:

Now we can present similar terms. That is, add the coefficients and multiply the resulting result by the common letter part:

5a + (−6a) + (−7b) + b = (5 + (−6))×a + ((−7) + 1)×b = −a + (−6b)

If the expression contains ordinary numbers without letter factors, then they are added separately.

Example 6. Give similar terms in the expression 4a + 3a − 5 + 2b + 7

Let's replace subtraction with addition where possible:

4a + 3a − 5 + 2b + 7 = 4a + 3a + (−5) + 2b + 7

Let us present similar terms. Numbers −5 And 7 do not have letter factors, but they are similar terms - they just need to be added. And the term 2b will remain unchanged, since it is the only one in this expression that has a letter factor b, and there is nothing to add it with:

4a + 3a + (−5) + 2b + 7 = (4 + 3)×a + 2b + (−5) + 7 = 7a + 2b + 2

Let's write down the solution briefly:

4a + 3a − 5 + 2b + 7 = 7a + 2b + 2

The terms can be ordered so that those terms that have the same letter part are located in the same part of the expression.

Example 7. Give similar terms in the expression 5t+2x+3x+5t+x

Since the expression is a sum of several terms, this allows us to evaluate it in any order. Therefore, the terms containing the variable t, can be written at the beginning of the expression, and the terms containing the variable x at the end of the expression:

5t + 5t + 2x + 3x + x

Now we can present similar terms:

5t + 5t + 2x + 3x + x = (5+5)×t + (2+3+1)×x = 10t + 6x

Let's write down the solution briefly:

5t + 2x + 3x + 5t + x = 10t + 6x

The sum of opposite numbers is zero. This rule also works for literal expressions. If the expression contains identical terms, but with opposite signs, then you can get rid of them at the stage of reducing similar terms. In other words, simply eliminate them from the expression, since their sum is zero.

Example 8. Give similar terms in the expression 3t − 4t − 3t + 2t

Let's replace subtraction with addition where possible:

3t − 4t − 3t + 2t = 3t + (−4t) + (−3t) + 2t

Components 3t And (−3t) are opposite. The sum of opposite terms is zero. If we remove this zero from the expression, the value of the expression will not change, so we will remove it. And we will remove it by simply crossing out the terms 3t And (−3t)

As a result, we will be left with the expression (−4t) + 2t. In this expression, you can add similar terms and get the final answer:

(−4t) + 2t = ((−4) + 2)×t = −2t

Let's write down the solution briefly:

Simplifying Expressions

"simplify the expression" and below is the expression that needs to be simplified. Simplify an expression means making it simpler and shorter.

In fact, we've already been simplifying expressions when we've reduced fractions. After reduction, the fraction became shorter and easier to understand.

Let's consider next example. Simplify the expression.

This task can literally be understood as follows: “Apply any valid actions to this expression, but make it simpler.” .

In this case, you can reduce the fraction, namely, divide the numerator and denominator of the fraction by 2:

What else can you do? You can calculate the resulting fraction. Then we get the decimal fraction 0.5

As a result, the fraction was simplified to 0.5.

The first question you need to ask yourself when solving such problems should be “What can be done?” . Because there are actions that you can do, and there are actions that you cannot do.

Another important point The thing to remember is that the value of the expression should not change after simplifying the expression. Let's return to the expression. This expression represents a division that can be performed. Having performed this division, we get the value of this expression, which is equal to 0.5

But we simplified the expression and got a new simplified expression. The value of the new simplified expression is still 0.5

But we also tried to simplify the expression by calculating it. As a result, we received a final answer of 0.5.

Thus, no matter how we simplify the expression, the value of the resulting expressions is still equal to 0.5. This means that the simplification was carried out correctly at every stage. This is exactly what we should strive for when simplifying expressions - the meaning of the expression should not suffer from our actions.

There is often a need to simplify literal expressions. The same simplification rules apply to them as for numerical expressions. You can perform any valid actions, as long as the value of the expression does not change.

Let's look at a few examples.

Example 1. Simplify an expression 5.21s × t × 2.5

To simplify this expression, you can multiply the numbers separately and multiply the letters separately. This task is very similar to the one we looked at when we learned to determine the coefficient:

5.21s × t × 2.5 = 5.21 × 2.5 × s × t = 13.025 × st = 13.025st

So the expression 5.21s × t × 2.5 simplified to 13,025st.

Example 2. Simplify an expression −0.4 × (−6.3b) × 2

Second piece (−6.3b) can be translated into a form understandable to us, namely written in the form ( −6,3)×b , then multiply the numbers separately and multiply the letters separately:

− 0,4 × (−6.3b) × 2 = − 0,4 × (−6.3) × b × 2 = 5.04b

So the expression −0.4 × (−6.3b) × 2 simplified to 5.04b

Example 3. Simplify an expression

Let's write this expression in more detail to clearly see where the numbers are and where the letters are:

Now let’s multiply the numbers separately and multiply the letters separately:

So the expression simplified to −abc. This solution can be written briefly:

When simplifying expressions, fractions can be reduced during the solution process, and not at the very end, as we did with ordinary fractions. For example, if in the course of solving we come across an expression of the form , then it is not at all necessary to calculate the numerator and denominator and do something like this:

A fraction can be reduced by selecting a factor in the numerator and denominator and reducing these factors by their largest common divisor. In other words, use in which we do not describe in detail what the numerator and denominator were divided into.

For example, in the numerator the factor is 12 and in the denominator the factor 4 can be reduced by 4. We keep the four in our mind, and dividing 12 and 4 by this four, we write down the answers next to these numbers, having first crossed them out

Now you can multiply the resulting small factors. In this case, there are few of them and you can multiply them in your mind:

Over time, you may find that when solving a particular problem, expressions begin to “get fat,” so it is advisable to get used to quick calculations. What can be calculated in the mind must be calculated in the mind. What can be quickly reduced must be reduced quickly.

Example 4. Simplify an expression

So the expression simplified to

Example 5. Simplify an expression

Let's multiply the numbers separately and the letters separately:

So the expression simplified to mn.

Example 6. Simplify an expression

Let's write this expression in more detail to clearly see where the numbers are and where the letters are:

Now let’s multiply the numbers separately and the letters separately. For ease of calculation, the decimal fraction −6.4 and mixed number can be converted to ordinary fractions:

So the expression simplified to

The solution for this example can be written much shorter. It will look like this:

Example 7. Simplify an expression

Let's multiply numbers separately and letters separately. For ease of calculation, a mixed number and decimals 0.1 and 0.6 can be converted to ordinary fractions:

So the expression simplified to abcd. If you skip the details, then this decision can be written much shorter:

Notice how the fraction has been reduced. New factors that are obtained as a result of reduction of previous factors can also be reduced.

Now let's talk about what not to do. When simplifying expressions, it is strictly forbidden to multiply numbers and letters if the expression is a sum and not a product.

For example, if you want to simplify the expression 5a+4b, then you cannot write it like this:

This is the same as if we were asked to add two numbers and we multiplied them instead of adding them.

When substituting any variable values a And b expression 5a +4b turns into an ordinary numerical expression. Let's assume that the variables a And b have the following meanings:

a = 2, b = 3

Then the value of the expression will be equal to 22

5a + 4b = 5 × 2 + 4 × 3 = 10 + 12 = 22

First, multiplication is performed, and then the results are added. And if we tried to simplify this expression by multiplying numbers and letters, we would get the following:

5a + 4b = 5 × 4 × a × b = 20ab

20ab = 20 × 2 × 3 = 120

It turns out a completely different meaning of the expression. In the first case it worked 22 , in the second case 120 . This means that simplifying the expression 5a+4b was performed incorrectly.

After simplifying an expression, its value should not change with the same values ​​of the variables. If, when substituting any variable values ​​into the original expression, one value is obtained, then after simplifying the expression, the same value should be obtained as before the simplification.

With expression 5a+4b there's really nothing you can do. It doesn't simplify it.

If an expression contains similar terms, then they can be added if our goal is to simplify the expression.

Example 8. Simplify an expression 0.3a−0.4a+a

0.3a − 0.4a + a = 0.3a + (−0.4a) + a = (0.3 + (−0.4) + 1)×a = 0.9a

or shorter: 0.3a − 0.4a + a = 0.9a

So the expression 0.3a−0.4a+a simplified to 0.9a

Example 9. Simplify an expression −7.5a − 2.5b + 4a

To simplify this expression, we can add similar terms:

−7.5a − 2.5b + 4a = −7.5a + (−2.5b) + 4a = ((−7.5) + 4)×a + (−2.5b) = −3.5a + (−2.5b)

or shorter −7.5a − 2.5b + 4a = −3.5a + (−2.5b)

Term (−2.5b) remained unchanged because there was nothing to put it with.

Example 10. Simplify an expression

To simplify this expression, we can add similar terms:

The coefficient was for ease of calculation.

So the expression simplified to

Example 11. Simplify an expression

To simplify this expression, we can add similar terms:

So the expression simplified to .

In this example, it would be more appropriate to add the first and last coefficients first. In this case we would have a short solution. It would look like this:

Example 12. Simplify an expression

To simplify this expression, we can add similar terms:

So the expression simplified to .

The term remained unchanged, since there was nothing to add it to.

This solution can be written much shorter. It will look like this:

The short solution skipped the steps of replacing subtraction with addition and detailing how fractions were reduced to a common denominator.

Another difference is that in detailed solution the answer looks like , but in short as . In fact, they are the same expression. The difference is that in the first case, subtraction is replaced by addition, since at the beginning when we wrote the solution in in detail, we replaced subtraction with addition wherever possible, and this replacement was preserved for the answer.

Identities. Identically equal expressions

Once we have simplified any expression, it becomes simpler and shorter. To check whether the simplified expression is correct, it is enough to substitute any variable values ​​first into the previous expression that needed to be simplified, and then into the new one that was simplified. If the value in both expressions is the same, then the simplified expression is true.

Let's consider simplest example. Let it be necessary to simplify the expression 2a×7b. To simplify this expression, you can multiply numbers and letters separately:

2a × 7b = 2 × 7 × a × b = 14ab

Let's check whether we simplified the expression correctly. To do this, let’s substitute any values ​​of the variables a And b first into the first expression that needed to be simplified, and then into the second, which was simplified.

Let the values ​​of the variables a , b will be as follows:

a = 4, b = 5

Let's substitute them into the first expression 2a×7b

Now let’s substitute the same variable values ​​into the expression that resulted from simplification 2a×7b, namely in the expression 14ab

14ab = 14 × 4 × 5 = 280

We see that when a=4 And b=5 value of the first expression 2a×7b and the meaning of the second expression 14ab equal

2a × 7b = 2 × 4 × 7 × 5 = 280

14ab = 14 × 4 × 5 = 280

The same will happen for any other values. For example, let a=1 And b=2

2a × 7b = 2 × 1 × 7 × 2 =28

14ab = 14 × 1 × 2 =28

Thus, for any values ​​of the expression variables 2a×7b And 14ab are equal to the same value. Such expressions are called identically equal.

We conclude that between the expressions 2a×7b And 14ab you can put an equal sign because they are equal to the same value.

2a × 7b = 14ab

An equality is any expression that is joined by an equal sign (=).

And equality of the form 2a×7b = 14ab called identity.

An identity is an equality that is true for any values ​​of the variables.

Other examples of identities:

a + b = b + a

a(b+c) = ab + ac

a(bc) = (ab)c

Yes, the laws of mathematics that we studied are identities.

Faithful numerical equalities are also identities. For example:

2 + 2 = 4

3 + 3 = 5 + 1

10 = 7 + 2 + 1

Deciding difficult task to make the calculation easier, complex expression replaced by a simpler expression identical to the previous one. This replacement is called identical transformation expressions or just transforming the expression.

For example, we simplified the expression 2a×7b, and got a simpler expression 14ab. This simplification can be called the identity transformation.

You can often find a task that says "prove that equality is an identity" and then the equality that needs to be proven is given. Usually this equality consists of two parts: the left and right parts of the equality. Our task is to perform identity transformations with one of the parts of the equality and obtain the other part. Or perform identical transformations on both sides of the equality and make sure that both sides of the equality contain the same expressions.

For example, let us prove that the equality 0.5a × 5b = 2.5ab is an identity.

Let's simplify the left side of this equality. To do this, multiply the numbers and letters separately:

0.5 × 5 × a × b = 2.5ab

2.5ab = 2.5ab

As a result of a small identity transformation, left side equality became equal to the right side of the equality. So we have proven that the equality 0.5a × 5b = 2.5ab is an identity.

From identical transformations we learned to add, subtract, multiply and divide numbers, reduce fractions, add similar terms, and also simplify some expressions.

But these are not all identical transformations that exist in mathematics. There are many more identical transformations. We will see this more than once in the future.

Tasks for independent solution:

Did you like the lesson?
Join our new group VKontakte and start receiving notifications about new lessons

Let's consider a small problem that is often found in various magazines and tricks.

The magician offers to wish you a certain number. Next he asks to multiply it by three, and add six to the result. Then he asks to divide the resulting amount by three and subtract the resulting number from the result. Then he tells you the correct answer.

How does this happen, is it really magic?

No, actually everything is simpler. Let us think of the number 5. Now let's perform all the actions that the magician suggested to us.

• 1. 5*3=15.
• 2. 15+6=21.
• 3. 21:3=7.
• 4. 7-5=2.

We received a two as an answer. We could write this same solution as a numerical expression (5*3+6): 3 - 5. And its value would be the number 2.

Now, let’s say we thought of the number 3. The result would be a numerical expression (3*3+6): 3 - 3. And its value would be the number 2.

Two again. The thought arises that there is no trick here, and in any case the number 2 will be obtained. Let's try to check this. Let's denote the number we have in mind with the letter x, and write down all the actions that the magician asked to do in the required order.

• We get (x*3+6):3 -x.

• (x*3+6):3 –x = x+2-x=2.

It turns out that the number we have in mind does not play any role at all; it will be reduced in any case.

In the analysis of the problem, we received the expression (x * 3 + 6): 3 – x, which is written using a letter denoting any number, numbers 3 and 6, parentheses and action signs. Such an expression is called an algebraic expression or an expression with a variable.

Defining an Expression with a Variable

• An algebraic expression or an expression with a variable is called any meaningful notation consisting of letters denoting any number, numbers and action signs.

For example, the following entries would be algebraic expressions:

• 2*(x+y),
• 34*a-13*a*x,
• (123-65*a):3 +4.

If instead of each letter that is included in the algebraic expression we substitute some numeric value, and then perform all the actions, the result will be a certain number. This number is called meaning algebraic expression.

For example, the value of the algebraic expression 5*a+2*x-7 with a=2 and x=3 will be the number 9, since 5*2+2*3 -7 = 9.

In the problem that we considered at the beginning, the value of the algebraic expression (x*3+6):3 – x will be the number 2, for any value of the variable x.

Entries 2 A + 8, 3A + 5b, A 4 – bс are called expressions with variables. Putting numbers instead of letters, we get numerical expressions. General concept expressions with variables are defined in exactly the same way as the concept of a numerical expression, only, in addition to numbers, expressions with variables can also contain letters.

For expressions with a variable, simplifications are also used: do not put parentheses containing only a number or a letter, do not put a multiplication sign between letters, between numbers and letters, etc.

There are expressions with one, two, three, etc. variables. Designate A(X), IN(x, y), etc.

An expression with a variable cannot be called either a statement or a predicate. For example, about expression 2 A+ 5 it cannot be said whether it is true or false, therefore it is not a statement. If instead of a variable A substitute numbers, we get various numerical expressions, which are also not statements, therefore, this expression is also not a predicate.

Each expression with a variable corresponds to a set of numbers, the substitution of which produces a numerical expression that makes sense. This set is called the domain of definition of the expression.

Example. 8: (4 – X) – domain of definition R\(4), because at X= 4 expression 8: (4 – 4) does not make sense.

If the expression contains multiple variables, for example, X And at, then the domain of definition of this expression is understood as a set of pairs of numbers ( A; b) such that when replacing X on A And at on b the result is a numeric expression that has a value.

Example. , the domain of definition is the set of pairs ( A; b) │A≥ b.

Definition. Two expressions with a variable are said to be identically equal if for any value. Variables from the scope of expressions have their corresponding values ​​equal.

That. two expressions A(X), IN(X) are identically equal on the set X, If

1) sets acceptable values the variables in these expressions are the same;

2) for anyone X 0 of their sets of permissible values, the meanings of expressions at X 0 coincide, i.e. A(X 0) = IN(X 0) is a correct numerical equality.

Example. (2 X+ 5) 2 and 4 X 2 + 20X+ 25 – identically equal expressions.

Designate A(X) º IN(X). Note that if two expressions are identically equal on some set E, then they are identically equal on any subset E 1 М E. It should also be noted that the statement about the identity equality of two expressions with a variable is a statement.

If we connect two identically equal expressions on a certain set with an equal sign, we get a sentence that is called an identity on this set.

True numerical equalities are also considered identities. The identities are the laws of addition and multiplication real numbers, rules for subtracting a number from a sum and a sum from a number, rules for dividing a sum by a number, etc. Identities are also the rules for actions with zero and one.

Replacing an expression with another that is identically equal to it on a certain set is called an identical transformation of a given expression.

Example. 7 X + 2 + 3X = 10 X+ 2 - identical transformation, is not an identical transformation on R.

§ 5. Classification of expressions with a variable

1) An expression made up of variables and numbers using only the operations of addition, subtraction, multiplication, and exponentiation is called an integer expression or polynomial.

Example. (3X 2 + 5) ∙ (2X – 3at)

2) Rational is an expression constructed from variables and numbers using the operations of addition, subtraction, multiplication, division, and exponentiation. Rational expression can be represented as a ratio of two integer expressions, i.e. polynomials. Note that integer expressions are a special case of rational ones.

Example. .

3) Irrational is an expression constructed from variables and numbers using the operations of addition, subtraction, multiplication, division, exponentiation, as well as the operation of extracting the root n-th degree.

Expressions made up of numbers, action signs and parentheses are called numerical expressions. The number that is the result of performing all actions in numerical expression is called the value of a numeric expression. Numerical expressions that have no meaning are said to be don't make sense.

To compare numbers, use the signs ,,,,. In this case, double inequalities of the form can be used
etc. Inequalities that use signs And , called strict, in which signs are used And , –not strict.

Expressions made up of numbers, letters, action symbols and parentheses are called literal expressions or expressions with a variable or with variables. The set of values ​​of a variable for which an expression with a variable has a numerical value (makes sense) is called range of acceptable values variable of this expression.

Variable expressions are used to write numbers certain type. For example, record
means any three-digit number that has hundreds, tens and units, i.e.
. Using alphabetic expressions it is convenient to write down mathematical rules, laws, and definitions. For example, module definition(absolute value) numbers can be written like this:
.

Elements of statistics

A series of numbers obtained as a result of a statistical study is called statistical sampling or just sampling, and each number in this series is option samples. The number of numbers in a series is called volume samples. A sample record when the next option is not less than the previous one is called ordered series of data(or variation series).

Arithmetic mean of the sample is called the quotient of the sum of all variants of the sample and the quantity of the variant (i.e. the quotient of the sums of all variants and volume samples). The number of occurrences of the same variant in a sample is called frequency this options. The sample option with the highest frequency is called sampling mode. The difference between the largest and smallest sample options is called scope samples. If there is an odd number of options in an ordered series of data, then the average number of options is called median. If there is an even number of variants in an ordered series, then the arithmetic mean of the two average variants is called median.

Preparatory version

Entries 2 A+ 8, 3A+ 5b,A 4 –bWith are called expressions with variables. Putting numbers instead of letters, we get numerical expressions. The general concept of an expression with variables is defined in exactly the same way as the concept of a numerical expression, only, in addition to numbers, expressions with variables can also contain letters.

For expressions with a variable, simplifications are also used: do not put parentheses containing only a number or a letter, do not put a multiplication sign between letters, between numbers and letters, etc.

There are expressions with one, two, three, etc. variables. Designate A(X),IN(x, y), etc.

An expression with a variable cannot be called either a statement or a predicate. For example, about expression 2 A+ 5 it cannot be said whether it is true or false, therefore it is not a statement. If instead of a variable A substitute numbers, we get various numerical expressions, which are also not statements, therefore, this expression is also not a predicate.

Each expression with a variable corresponds to a set of numbers, the substitution of which produces a numerical expression that makes sense. This set is called the domain of definition of the expression.

Example. 8: (4 –X) – domain of definition R\(4), because at X= 4 expression 8: (4 – 4) does not make sense.

If the expression contains multiple variables, for example, X And at, then the domain of definition of this expression is understood as a set of pairs of numbers ( A;b) such that when replacing X on A And at on b the result is a numeric expression that has a value.

Example. , the domain of definition is the set of pairs ( A;b) │A≥b.

Definition. Two expressions with a variable are said to be identically equal if for any value. Variables from the scope of expressions have their corresponding values ​​equal.

That. two expressions A(X),IN(X) are identically equal on the set X, If

1) the sets of permissible values ​​of the variable in these expressions coincide;

2) for anyone X 0 of their sets of permissible values, the meanings of expressions at X 0 coincide, i.e. A(X 0) =IN(X 0) is a correct numerical equality.

Example. (2 X+ 5) 2 and 4 X 2 + 20X+ 25 – identically equal expressions.

Designate A(X) IN(X). Note that if two expressions are identically equal on some set E, then they are identically equal on any subset E 1 E. It should also be noted that the statement about the identity equality of two expressions with a variable is a statement.

If we connect two identically equal expressions on a certain set with an equal sign, we get a sentence that is called an identity on this set.

True numerical equalities are also considered identities. Identities are the laws of addition and multiplication of real numbers, rules for subtracting a number from a sum and a sum from a number, rules for dividing a sum by a number, etc. Identities are also the rules for actions with zero and one.

Replacing an expression with another that is identically equal to it on a certain set is called an identical transformation of a given expression.

Example. 7 X+ 2 + 3X= 10X+ 2 - identity transformation, is not an identity transformation on R.