Video tutorial “Simplification of expressions. How to simplify a mathematical expression

A literal expression (or expression with variables) is a mathematical expression that consists of numbers, letters, and signs of mathematical operations. For example, the following expression is literal:

a+b+4

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

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

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

Lesson content

Variables

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

Numbers that are substituted for variables are called variable values. For example, let's change the values ​​of the variables a and b. Use the equals sign 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 converts to a normal numeric expression 2+3+4 whose value can be found:

2 + 3 + 4 = 9

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

2 x 3 = 6

Together, you can also write the multiplication of a number by an expression in brackets. For example, instead of a×(b + c) can be written a(b + c). Applying the distributive 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. In fact, this is 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 is called 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 "increment the value of the variable a three times", but most often read as "three a«

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

3 x 5 = 15

In simple terms, 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 five times. 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 to 120

5 x 2 x 3 x 4 = 120

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

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

Consider the expression −6b. Minus in front of the coefficient 6 , applies only to the coefficient 6 , and does not apply 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, we 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 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 is the short form for −5 × a + b, therefore, 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 one:

but the unit is traditionally not written down, so they just 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 came out like this:

−1 × a = −1a

Here lies a small trick. In the expression -a minus before variable a actually refers to the "invisible unit" and not the variable a. Therefore, when solving problems, you should be careful.

For example, given the expression -a and we are asked to find its value at a = 2, then at school we substituted a deuce instead of a variable a and get an answer −2 , not really focusing on how it turned out. In fact, there was a multiplication of minus one by a positive number 2

-a = -1 × a

−1 × a = −1 × 2 = −2

If an expression is given -a and it is required to find its value at a = −2, then we substitute −2 instead of a variable a

-a = -1 × a

−1 × a = −1 × (−2) = 2

In order to avoid mistakes, at first invisible units can be written 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, we write the expression abc a , b and c

1 x a x b x c = 1 x 2 x 3 x 4 = 24

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

Let's write 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 is the short form for −1×a×b×c. For clarity, we 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 the expression − abc expanded:

−abc = −1 × a × b × c

Substitute the value of the variables a , b and c

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

How to determine the coefficient

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

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 the expression is written in expanded form. That is, works 7m and 5a write in the form 7×m and 5×a

7 × m × 5 × a × (−3) × n

We apply the associative law of multiplication, which allows us to multiply factors in any order. Namely, separately multiply the numbers and separately multiply the letters (variables):

−3 × 7 × 5 × m × a × n = −105man

The coefficient is −105 . After completion, the letter part is preferably arranged in alphabetical order:

−105 am

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 recorded, since the coefficient 1 is usually not recorded.

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

Terms in literal expressions

When you add several numbers, you get the sum of those numbers. Numbers that add up are called terms. There can be several terms, for example:

1 + 2 + 3 + 4 + 5

When an expression consists of terms, it is much easier to calculate it, since it is easier to add than to subtract. 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 subtracted, not added. 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 are now with a minus sign. The main thing is that all the numbers in this expression are connected by the addition sign, that is, the expression is a sum.

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

1 + 2 − 3 + 4 − 5 = −1

1 + 2 + (−3) + 4 + (−5) = −1

Thus, the value of the expression will not suffer from the fact that 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 can call terms even those numbers (or variables) that are not them.

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

Similar terms

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

Usually, like terms are added to simplify an expression or solve an equation. This operation is called reduction of like terms.

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

For example, we give similar terms in the expression 3a + 4a + 5a. In this case, all terms are similar. We add their coefficients and multiply the result by the common letter part - by the variable a

3a + 4a + 5a = (3 + 4 + 5)×a = 12a

Such terms are usually given in the mind and the result is recorded immediately:

3a + 4a + 5a = 12a

Also, you can argue like this:

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 consider several examples of reducing similar terms. Considering that this topic is very important, at first we will write down every detail in detail. Despite the fact that everything is very simple here, most people make a lot of mistakes. Mostly due to inattention, not ignorance.

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

We add the coefficients in this expression and multiply the result by the common letter part:

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

design (3 + 2 + 6 + 8)×a you can not write down, so we will immediately write down the answer

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

Example 2 Bring like terms in the expression 2a+a

Second term a written without a coefficient, but in fact it is preceded by a coefficient 1 , which we do not see due to the fact that it is not recorded. So the expression looks like this:

2a + 1a

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

2a + 1a = (2 + 1) × a = 3a

Let's write the solution in short:

2a + a = 3a

2a+a, you can argue in another way:

Example 3 Bring like terms in the expression 2a - a

Let's replace subtraction with addition:

2a + (−a)

Second term (−a) written without a coefficient, but in fact 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 we present similar terms. We add the coefficients and multiply the result by the common letter part:

2a + (−1a) = (2 + (−1)) × a = 1a = a

Usually written shorter:

2a − a = a

Bringing like terms in the expression 2a−a You can also argue in another way:

There were 2 variables a , subtracted one variable a , as a result there was only one variable a

Example 4 Bring like terms in the expression 6a - 3a + 4a - 8a

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

Now we present similar terms. We add the coefficients and multiply the result by the common letter part

(6 + (−3) + 4 + (−8)) × a = −1a = −a

Let's write the solution in short:

6a - 3a + 4a - 8a = -a

There are expressions that contain several different groups of similar terms. For example, 3a + 3b + 7a + 2b. For such expressions, the same rules apply as for the rest, namely, adding the coefficients and multiplying the result by the common letter part. But in order to avoid mistakes, it is convenient to underline different groups of terms 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 underlined with two lines:

Now we can bring like terms. That is, add the coefficients and multiply the result by the common letter part. This must be done for both groups of terms: for terms containing a variable a and for terms containing the 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 the mind:

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

Example 5 Bring like terms in the expression 5a - 6a - 7b + b

We replace subtraction with addition where possible:

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

Underline like terms with different lines. Terms containing variables a underline with one line, and the terms content are variables b, underlined with two lines:

Now we can bring like terms. That is, add the coefficients and multiply the 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 alphabetic factors, then they are added separately.

Example 6 Bring like 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 literal factors, but they are similar terms - you just need to add them up. 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 the solution in short:

4a + 3a − 5 + 2b + 7 = 7a + 2b + 2

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 Bring like terms in the expression 5t+2x+3x+5t+x

Since the expression is the 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 add like terms:

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

Let's write the solution in short:

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, just drop them from the expression because their sum is zero.

Example 8 Bring like terms in the expression 3t − 4t − 3t + 2t

Let's replace subtraction with addition where possible:

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

Terms 3t and (−3t) are opposite. The sum of opposite terms is equal to zero. If we remove this zero from the expression, then the value of the expression will not change, so we will remove it. And we will remove it by the usual deletion of the terms 3t and (−3t)

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

(−4t) + 2t = ((−4) + 2)×t = −2t

Let's write the solution in short:

Expression simplification

"simplify the expression" and the following is the expression to be simplified. Simplify Expression means to make it simpler and shorter.

In fact, we have already dealt with the simplification of expressions when reducing fractions. After the reduction, the fraction became shorter and easier to read.

Consider the following example. Simplify the expression.

This task can be literally understood as follows: "Do whatever you can do with 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 be done? You can calculate the resulting fraction. Then we get the decimal 0.5

As a result, the fraction was simplified to 0.5.

The first question to ask yourself when solving such problems should be “what can be done?” . Because there are things you can do and there are things you can't do.

Another important point to keep in mind is that the value of an expression must not change after the expression is simplified. 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, the final answer was 0.5.

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

It is often necessary to simplify literal expressions. For them, the same simplification rules apply as for numerical expressions. You can perform any valid action, as long as the value of the expression does not change.

Let's look at a few examples.

Example 1 Simplify 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 considered 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 Expression −0.4×(−6.3b)×2

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

− 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 Expression

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

Now we multiply the numbers separately and multiply the letters separately:

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

When simplifying expressions, fractions can be reduced in the process of solving, 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 choosing both the factor in the numerator and the denominator and reducing these factors by their greatest 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 12 and in the denominator, the factor 4 can be reduced by 4. We keep the four in mind, and dividing 12 and 4 by this four, we write the answers next to these numbers, having previously crossed them out

Now you can multiply the resulting small factors. In this case, there are not many of them and you can multiply them in your mind:

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

Example 4 Simplify Expression

So the expression simplified to

Example 5 Simplify Expression

We multiply numbers separately and letters separately:

So the expression simplified to mn.

Example 6 Simplify Expression

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

Now we multiply the numbers separately and the letters separately. For convenience of calculations, the decimal fraction −6.4 and the 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 Expression

We multiply numbers separately and letters separately. For convenience of calculation, the mixed number and decimal fractions 0.1 and 0.6 can be converted to ordinary fractions:

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

Notice how the fraction has been reduced. New multipliers, which are obtained by reducing the previous multipliers, 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 it cannot be written as follows:

This is equivalent to the fact that if we were asked to add two numbers, and we would multiply them instead of adding them.

When substituting any values ​​of variables a and b expression 5a+4b turns into a simple numeric expression. Let's assume the variables a and b have the following meanings:

a = 2 , b = 3

Then the value of the expression will be 22

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

First, the 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 x 2 x 3 = 120

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

After simplifying the 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 simplification.

With expression 5a + 4b actually nothing can be done. It doesn't get easier.

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

Example 8 Simplify 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 Expression −7.5a − 2.5b + 4a

To simplify this expression, you can add like 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, since there was nothing to fold it with.

Example 10 Simplify Expression

To simplify this expression, you can add like terms:

The coefficient was for the convenience of calculation.

So the expression simplified to

Example 11. Simplify Expression

To simplify this expression, you can add like terms:

So the expression simplified to .

In this example, it would make more sense to add the first and last coefficient first. In this case, we would get a short solution. It would look like this:

Example 12. Simplify Expression

To simplify this expression, you can add like 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 omits the steps of replacing subtraction with addition and a detailed record of how the fractions were reduced to a common denominator.

Another difference is that in the detailed solution, the answer looks like , but in short as . Actually, it's the same expression. The difference is that in the first case, subtraction is replaced by addition, because at the beginning, when we wrote down the solution in detail, we replaced subtraction with addition wherever possible, and this replacement was preserved for the answer.

Identities. Identical equal expressions

After we have simplified any expression, it becomes simpler and shorter. To check whether the expression is simplified correctly, it is enough to substitute any values ​​of the variables first into the previous expression, which was required to be simplified, and then into the new one, which was simplified. If the value in both expressions is the same, then the expression is simplified correctly.

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

2a × 7b = 2 × 7 × a × b = 14ab

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

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

a = 4 , b = 5

Substitute them in the first expression 2a × 7b

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

14ab = 14 x 4 x 5 = 280

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

2a × 7b = 2 × 4 × 7 × 5 = 280

14ab = 14 x 4 x 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 x 1 x 2 = 28

Thus, for any values ​​of the variables, the expressions 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, since they are equal to the same value.

2a × 7b = 14ab

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

And the 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.

True numerical equalities are also identities. For example:

2 + 2 = 4

3 + 3 = 5 + 1

10 = 7 + 2 + 1

When solving a complex problem, in order to facilitate the calculation, a complex expression is replaced by a simpler expression that is identically equal to the previous one. Such a replacement is called identical transformation of the expression or simply expression conversion.

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

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

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

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, the left side of the equality became equal to the right side of the equality. So we have proved 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, bring like terms, and also simplify some expressions.

But these are far from all identical transformations that exist in mathematics. There are many more identical transformations. We will see this again and again in the future.

Tasks for independent solution:

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With the help of any language, you can express the same information in different words and phrases. Mathematical language is no exception. But the same expression can be equivalently written in different ways. And in some situations, one of the entries is simpler. We will talk about simplifying expressions in this lesson.

People communicate in different languages. For us, an important comparison is the pair "Russian language - mathematical language". The same information can be reported in different languages. But, besides this, it can be pronounced differently in one language.

For example: “Peter is friends with Vasya”, “Vasya is friends with Petya”, “Peter and Vasya are friends”. Said differently, but one and the same. By any of these phrases, we would understand what is at stake.

Let's look at this phrase: "The boy Petya and the boy Vasya are friends." We understand what is at stake. However, we don't like how this phrase sounds. Can't we simplify it, say the same, but simpler? “Boy and boy” - you can say once: “Boys Petya and Vasya are friends.”

"Boys" ... Isn't it clear from their names that they are not girls. We remove the "boys": "Petya and Vasya are friends." And the word "friends" can be replaced with "friends": "Petya and Vasya are friends." As a result, the first, long, ugly phrase was replaced with an equivalent statement that is easier to say and easier to understand. We have simplified this phrase. To simplify means to say it easier, but not to lose, not to distort the meaning.

The same thing happens in mathematical language. The same thing can be said differently. What does it mean to simplify an expression? This means that for the original expression there are many equivalent expressions, that is, those that mean the same thing. And from all this multitude, we must choose the simplest, in our opinion, or the most suitable for our further purposes.

For example, consider a numeric expression. It will be equivalent to .

It will also be equivalent to the first two: .

It turns out that we have simplified our expressions and found the shortest equivalent expression.

For numeric expressions, you always need to do all the work and get the equivalent expression as a single number.

Consider an example of a literal expression . Obviously, it will be simpler.

When simplifying literal expressions, you must perform all the actions that are possible.

Is it always necessary to simplify an expression? No, sometimes an equivalent but longer notation will be more convenient for us.

Example: Subtract the number from the number.

It is possible to calculate, but if the first number were represented by its equivalent notation: , then the calculations would be instantaneous: .

That is, a simplified expression is not always beneficial for us for further calculations.

Nevertheless, very often we are faced with a task that just sounds like "simplify the expression."

Simplify the expression: .

Solution

1) Perform actions in the first and second brackets: .

2) Calculate the products: .

Obviously, the last expression has a simpler form than the initial one. We have simplified it.

In order to simplify the expression, it must be replaced with an equivalent (equal).

To determine the equivalent expression, you must:

1) perform all possible actions,

2) use the properties of addition, subtraction, multiplication and division to simplify calculations.

Properties of addition and subtraction:

1. Commutative property of addition: the sum does not change from the rearrangement of the terms.

2. Associative property of addition: in order to add a third number to the sum of two numbers, you can add the sum of the second and third numbers to the first number.

3. The property of subtracting a sum from a number: to subtract the sum from a number, you can subtract each term individually.

Properties of multiplication and division

1. The commutative property of multiplication: the product does not change from a permutation of factors.

2. Associative property: to multiply a number by the product of two numbers, you can first multiply it by the first factor, and then multiply the resulting product by the second factor.

3. The distributive property of multiplication: in order to multiply a number by a sum, you need to multiply it by each term separately.

Let's see how we actually do mental calculations.

Calculate:

Solution

1) Imagine how

2) Let's represent the first multiplier as the sum of bit terms and perform the multiplication:

3) you can imagine how and perform multiplication:

4) Replace the first factor with an equivalent sum:

The distributive law can also be used in the opposite direction: .

Follow these steps:

1) 2)

Solution

1) For convenience, you can use the distribution law, just use it in the opposite direction - take the common factor out of brackets.

2) Let's take the common factor out of brackets

It is necessary to buy linoleum in the kitchen and hallway. Kitchen area - hallway -. There are three types of linoleums: for, and rubles for. How much will each of the three types of linoleum cost? (Fig. 1)

Rice. 1. Illustration for the condition of the problem

Solution

Method 1. You can separately find how much money it will take to buy linoleum in the kitchen, and then add it to the hallway and add up the resulting works.

Remark 1

A logical function can be written using a logical expression, and then you can go to the logical circuit. It is necessary to simplify logical expressions in order to obtain the simplest possible (and therefore cheaper) logical circuit. In fact, a logical function, a logical expression, and a logical circuit are three different languages ​​that talk about the same entity.

To simplify logical expressions, use laws of the algebra of logic.

Some transformations are similar to the transformations of formulas in classical algebra (bracketing the common factor, using commutative and associative laws, etc.), while other transformations are based on properties that classical algebra operations do not have (using the distribution law for conjunction, laws of absorption, gluing, de Morgan's rules, etc.).

The laws of the algebra of logic are formulated for basic logical operations - “NOT” - inversion (negation), “AND” - conjunction (logical multiplication) and “OR” - disjunction (logical addition).

The law of double negation means that the “NOT” operation is reversible: if you apply it twice, then in the end the logical value will not change.

The law of the excluded middle states that any logical expression is either true or false (“there is no third”). Therefore, if $A=1$, then $\bar(A)=0$ (and vice versa), which means that the conjunction of these quantities is always equal to zero, and the disjunction is equal to one.

$((A + B) → C) \cdot (B → C \cdot D) \cdot C.$

Let's simplify this formula:

Figure 3

This implies that $A = 0$, $B = 1$, $C = 1$, $D = 1$.

Answer: students $B$, $C$ and $D$ are playing chess, but student $A$ is not playing.

When simplifying logical expressions, you can perform the following sequence of actions:

1. Replace all “non-basic” operations (equivalence, implication, exclusive OR, etc.) with their expressions through the basic operations of inversion, conjunction, and disjunction.
2. Expand inversions of complex expressions according to de Morgan's rules in such a way that only individual variables have negation operations.
3. Then simplify the expression using parentheses expansion, bracketing common factors, and other laws of the algebra of logic.

Example 2

Here, de Morgan's rule, the distributive law, the law of the excluded middle, the commutative law, the law of repetition, the again commutative law, and the law of absorption are used in succession.

An algebraic expression in the record of which, along with the operations of addition, subtraction and multiplication, also uses division into literal expressions, is called a fractional algebraic expression. Such are, for example, the expressions

We call an algebraic fraction an algebraic expression that has the form of a quotient of division of two integer algebraic expressions (for example, monomials or polynomials). Such are, for example, the expressions

the third of the expressions).

Identity transformations of fractional algebraic expressions are for the most part intended to represent them as an algebraic fraction. To find a common denominator, the factorization of the denominators of fractions - terms is used in order to find their least common multiple. When reducing algebraic fractions, the strict identity of expressions can be violated: it is necessary to exclude the values ​​of quantities at which the factor by which the reduction is made vanishes.

Let us give examples of identical transformations of fractional algebraic expressions.

Example 1: Simplify an expression

All terms can be reduced to a common denominator (it is convenient to change the sign in the denominator of the last term and the sign in front of it):

Our expression is equal to one for all values ​​except these values, it is not defined and fraction reduction is illegal).

Example 2. Represent expression as an algebraic fraction

Solution. The expression can be taken as a common denominator. We find successively:

Exercises

1. Find the values ​​of algebraic expressions for the specified values ​​of the parameters:

2. Factorize.

Section 5 EXPRESSIONS AND EQUATIONS

In the section you will learn:

ü o expressions and their simplifications;

ü what are the properties of equalities;

ü how to solve equations based on the properties of equalities;

ü what types of problems are solved with the help of equations; what are perpendicular lines and how to build them;

ü what lines are called parallel and how to build them;

ü what is a coordinate plane;

ü how to determine the coordinates of a point on a plane;

ü what is a dependency graph between quantities and how to build it;

ü how to apply the learned material in practice

§ 30. EXPRESSIONS AND THEIR SIMPLIFICATION

You already know what literal expressions are and know how to simplify them using the laws of addition and multiplication. For example, 2a ∙ (-4 b) = -8 ab . In the resulting expression, the number -8 is called the coefficient of the expression.

Does the expression cd coefficient? So. It is equal to 1 because cd - 1 ∙ cd .

Recall that converting an expression with parentheses to an expression without parentheses is called parenthesis expansion. For example: 5(2x + 4) = 10x + 20.

The reverse action in this example is to put the common factor out of brackets.

Terms containing the same literal factors are called similar terms. By taking the common factor out of brackets, similar terms are erected:

5x + y + 4 - 2x + 6 y - 9 =

= (5x - 2x) + (y + 6y )+ (4 - 9) = = (5-2)* + (1 + 6)* y-5=

B x + 7y - 5.

Bracket expansion rules

1. If there is a “+” sign in front of the brackets, then when opening the brackets, the signs of the terms in brackets are preserved;

2. If there is a “-” sign in front of the brackets, then when the brackets are opened, the signs of the terms in brackets are reversed.

Task 1 . Simplify the expression:

1) 4x+(-7x + 5);

2) 15 y -(-8 + 7 y ).

Solutions. 1. There is a “+” sign before the brackets, therefore, when opening the brackets, the signs of all terms are preserved:

4x + (-7x + 5) \u003d 4x - 7x + 5 \u003d -3x + 5.

2. There is a “-” sign in front of the brackets, therefore, during the opening of the brackets: the signs of all terms are reversed:

15 - (- 8 + 7y) \u003d 15y + 8 - 7y \u003d 8y +8.

To open brackets, use the distributive property of multiplication: a( b + c) = ab + ac. If a > 0, then the signs of the terms b and with do not change. If a< 0, то знаки слагаемых b and from are reversed.

Task 2. Simplify the expression:

1) 2(6y -8) + 7y;

2) -5 (2-5x) + 12.

Solutions. 1. The factor 2 in front of the brackets e is positive, therefore, when opening the brackets, we keep the signs of all terms: 2(6 y - 8) + 7 y = 12 y - 16 + 7 y =19 y -16.

2. The factor -5 in front of the brackets e is negative, therefore, when opening the brackets, we change the signs of all terms to the opposite ones:

5(2 - 5x) + 12 = -10 + 25x +12 = 2 + 25x.

Find out more

1. The word "sum" comes from the Latin summa , which means "total", "total".

2. The word "plus" comes from the Latin plus , which means "more", and the word "minus" - from the Latin minus , which means "less". The signs "+" and "-" are used to indicate the operations of addition and subtraction. These signs were introduced by the Czech scientist J. Vidman in 1489 in the book "A quick and pleasant account for all merchants"(Fig. 138).

Rice. 138

REMEMBER THE MAIN THINGS

1. What terms are called similar? How are like terms constructed?

2. How do you open brackets preceded by a “+” sign?

3. How do you open brackets preceded by a "-" sign?

4. How do you open brackets that are preceded by a positive factor?

5. How do you open brackets that are preceded by a negative factor?

1374". Name the coefficient of the expression:

1) 12 a; 3) -5.6 xy;

2)4 6; 4)-s.

1375". Name the terms that differ only by the coefficient:

1) 10a + 76-26 + a; 3) 5n + 5m -4n + 4;

2) bc -4d - bc + 4d; 4) 5x + 4y-x + y.

What are these terms called?

1376". Are there similar terms in the expression:

1) 11a + 10a; 3)6n + 15n; 5) 25r - 10r + 15r;

2) 14s-12; 4)12 m + m; 6) 8k +10k - n?

1377". Is it necessary to change the signs of the terms in brackets, opening the brackets in the expression:

1)4 + (a + 3b); 2)-c +(5-d ); 3) 16-(5m-8n)?

1378°. Simplify the expression and underline the coefficient:

1379°. Simplify the expression and underline the coefficient:

1380°. Reduce like terms:

1) 4a - Po + 6a - 2a; 4) 10 - 4 d - 12 + 4d;

2) 4b - 5b + 4 + 5b; 5) 5a - 12b - 7a + 5b;

3)-7ang="EN-US">c+ 5-3 c + 2; 6) 14 n - 12 m -4 n -3 m.

1381°. Reduce like terms:

1) 6a - 5a + 8a -7a; 3) 5s + 4-2s-3s;

2)9 b +12-8-46; 4) -7n + 8m - 13n - 3m.

1382°. Take the common factor out of brackets:

1) 1.2 a +1.2 b; 3) -3 n - 1.8 m; 5) -5p + 2.5k -0.5t;

2) 0.5 s + 5d; 4) 1.2 n - 1.8 m; 6) -8p - 10k - 6t.

1383°. Take the common factor out of brackets:

1) 6a-12b; 3) -1.8 n -3.6 m;

2) -0.2 s + 1 4 d; A) 3p - 0.9k + 2.7t.

1384°. Open brackets and reduce like terms;

1) 5 + (4a -4); 4) -(5 c - d) + (4 d + 5c);

2) 17x-(4x-5); 5) (n - m) - (-2 m - 3 n);

3) (76 - 4) - (46 + 2); 6) 7 (-5x + y) - (-2y + 4x) + (x - 3y).

1385°. Open the brackets and reduce like terms:

1) 10a + (4 - 4a); 3) (s - 5 d) - (- d + 5s);

2) -(46-10) + (4-56); 4) - (5 n + m) + (-4 n + 8 m) - (2 m -5 n).

1386°. Expand the brackets and find the meaning of the expression:

1)15+(-12+ 4,5); 3) (14,2-5)-(12,2-5);

2) 23-(5,3-4,7); 4) (-2,8 + 13)-(-5,6 + 2,8) + (2,8-13).

1387°. Expand the brackets and find the meaning of the expression:

1) (14- 15,8)- (5,8 + 4);

2)-(18+22,2)+ (-12+ 22,2)-(5- 12).

1388°. Open parenthesis:

1) 0.5 ∙ (a + 4); 4) (n - m) ∙ (-2.4 p);

2)-s ∙ (2.7-1.2 d ); 5) 3 ∙ (-1.5 p + k - 0.2 t);

3) 1.6 ∙ (2n + m); 6) (4.2 p - 3.5 k -6 t) ∙ (-2a).

1389°. Open parenthesis:

1) 2.2 ∙ (x-4); 3)(4 c - d )∙(-0.5 y );

2) -2 ∙ (1.2 n - m); 4) 6- (-p + 0.3 k - 1.2 t).

1390. Simplify the expression:

1391. Simplify the expression:

1392. Reduce like terms:

1393. Reduce like terms:

1394. Simplify the expression:

1) 2.8 - (0.5 a + 4) - 2.5 ∙ (2a - 6);

2) -12 ∙ (8 - 2, by) + 4.5 ∙ (-6 y - 3.2);

4) (-12.8 m + 24.8 n) ∙ (-0.5)-(3.5 m -4.05 m) ∙ 2.

1395. Simplify the expression:

1396. Find the meaning of the expression;

1) 4-(0.2 a-3) - (5.8 a-16), if a \u003d -5;

2) 2-(7-56)+ 156-3∙(26+ 5), if = -0.8;

m = 0.25, n = 5.7.

1397. Find the value of the expression:

1) -4∙ (i-2) + 2∙(6x - 1), if x = -0.25;

1398*. Find the error in the solution:

1) 5- (a-2.4) -7 ∙ (-a + 1.2) \u003d 5a - 12-7a + 8.4 \u003d -2a-3.6;

2) -4 ∙ (2.3 a - 6) + 4.2 ∙ (-6 - 3.5 a) \u003d -9.2 a + 46 + 4.26 - 14.7 a \u003d -5.5 a + 8.26.

1399*. Expand the brackets and simplify the expression:

1) 2ab - 3(6(4a - 1) - 6(6 - 10a)) + 76;

1400*. Arrange the parentheses to get the correct equality:

1) a-6-a + 6 \u003d 2a; 2) a -2 b -2 a + b \u003d 3 a -3 b.

1401*. Prove that for any numbers a and b if a > b , then the following equality holds:

1) (a + b) + (a-b) \u003d 2a; 2) (a + b) - (a - b) \u003d 2 b.

Will this equality be correct if: a) a< b; b) a = 6?

1402*. Prove that for any natural number a, the arithmetic mean of the preceding and following numbers is equal to a.

APPLY IN PRACTICE

1403. To prepare a fruit dessert for three people, you need: 2 apples, 1 orange, 2 bananas and 1 kiwi. How to make a letter expression to determine the amount of fruit needed to prepare a dessert for guests? Help Marin to calculate how many fruits she needs to buy if she comes to visit: 1) 5 friends; 2) 8 friends.

1404. Make a literal expression to determine the time required to complete homework in mathematics, if:

1) a min was spent on solving problems; 2) simplification of expressions is 2 times more than for solving problems. How much time did Vasilko do his homework if he spent 15 minutes solving problems?

1405. Lunch in the school canteen consists of salad, borscht, cabbage rolls and compote. The cost of salad is 20%, borscht - 30%, cabbage rolls - 45%, compote - 5% of the total cost of the entire meal. Write an expression to find the cost of lunch at the school cafeteria. How much does lunch cost if the price of a salad is 2 UAH?

REPETITION TASKS

1406. Solve the equation:

1407. Tanya spent on ice creamall available money, and for sweets -the rest. How much money does Tanya have?

if sweets cost 12 UAH?