Reducing expressions. How to simplify algebraic expression

In the fifth century BC, the ancient Greek philosopher Zenon Elayky formulated his famous apiorials, the most famous of which is Achilles and Turtle Aritia. This is how it sounds:

Suppose Achilles runs ten times faster than the turtle, and is behind it at a distance of a thousand steps. For the time, for which Achilles is running through this distance, a hundred steps will crash in the same side. When Achilles runs a hundred steps, the turtle will crawl about ten steps, and so on. The process will continue to infinity, Achilles will never catch up to the turtle.

This reasoning has become a logical shock for all subsequent generations. Aristotle, Diogen, Kant, Hegel, Hilbert ... All of them somehow considered the Apriology of Zenon. Shock turned out to be so strong that " ... Discussions continue and at present, to come to the general opinion on the essence of paradoxes to the scientific community has not yet been possible ... A mathematical analysis, the theory of sets, new physical and philosophical approaches was involved in the study of the issue; None of them became a generally accepted issue of the issue ..."[Wikipedia," Yenon Apriya "]. Everyone understands that they are blocked, but no one understands what deception is.

From the point of view of mathematics, Zeno in his Aproria clearly demonstrated the transition from the value to. This transition implies application instead of constant. As far as I understand, the mathematical apparatus of the use of variables of units of measurement is either yet not yet developed, or it was not applied to the Aporition of Zenon. The use of our ordinary logic leads us to a trap. We, by inertia of thinking, use permanent time measurement units to the inverter. From a physical point of view, it looks like a slowdown in time to its complete stop at the moment when Achilles is stuffed with a turtle. If time stops, Achilles can no longer overtake the turtle.

If you turn the logic usually, everything becomes in place. Achilles runs at a constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on its overcoming, ten times less than the previous one. If you apply the concept of "infinity" in this situation, it will correctly say "Achilles infinitely will quickly catch up the turtle."

How to avoid this logical trap? Stay in permanent time measurement units and do not move to reverse values. In the language of Zenon, it looks like this:

For that time, for which Achilles runs a thousand steps, a hundred steps will crack the turtle to the same side. For the next time interval, equal to the first, Achilles will run another thousand steps, and the turtle will crack a hundred steps. Now Achilles is an eight hundred steps ahead of the turtle.

This approach adequately describes reality without any logical paradoxes. But it is not complete solution Problems. On the Zenonian Agrac of Achilles and Turtle is very similar to the statement of Einstein on the irresistibility of the speed of light. We still have to study this problem, rethink and solve. And the decision should be sought not in infinitely large numbers, but in units of measurement.

Another interesting Yenon Aproria tells about the flying arrows:

The flying arrow is still, since at every moment she rests, and since it rests at every moment of time, it always rests.

In this manor, the logical paradox is very simple - it is enough to clarify that at each moment the flying arrow is resting at different points of space, which, in fact, is the movement. Here you need to note another moment. According to one photo of the car on the road, it is impossible to determine the fact of its movement, nor the distance to it. To determine the fact of the car's motion, you need two photos made from one point at different points in time, but it is impossible to determine the distance. To determine the distance to the car you need two photos made from different points Spaces at one point in time, but it is impossible to determine the fact of movement (naturally, additional data is still needed for calculations, trigonometry to help you). What I want to pay special attention is that two points in time and two points in space are different things that do not be confused, because they provide different opportunities for research.

wednesday, July 4, 2018

Very good differences between many and multiset are described in Wikipedia. We look.

As you can see, "there cannot be two identical elements in a set", but if identical elements are in the set there are, such a set is called "Mix". A similar logic of absurd reasonable beings never understand. This is the level of speaking parrots and trained monkeys, which are missing from the word "at all." Mathematics act as ordinary trainers, preaching our absurd ideas.

Once the engineers who built the bridge during the tests of the bridge were in the boat under the bridge. If the bridge collapsed, the talentless engineer died under the wreckage of his creation. If the bridge has withstood the load, a talented engineer built other bridges.

As math did not hide behind the phrase "Chur, I am in a house", more precisely, "mathematics studies abstract concepts," there is one umbilical cord, which inextricably binds them with reality. This umbilical cord is money. Apply mathematical theory of sets to mathematics themselves.

We taught mathematics very well and now we sit at the checkout, we issue a salary. That comes to us the mathematician for your money. We count on it the entire amount and lay out on your table on different stacks, in which we add bills of one dignity. Then we take from each stack on one bill and hand the mathematics of his "mathematical set of salary". Explain the mathematics that the rest of the bills will receive only when it proves that the set without the same elements is not equal to the set with the same elements. Here the most interesting will begin.

First of all, the logic of deputies will work: "It is possible to apply it to others, to me - low!". There will be further assurances of us that there are different numbers on bills of equal dignity, which means that they cannot be considered the same elements. Well, count the salary with coins - there are no numbers on the coins. Here the mathematician will begin to dislike physics: on different coins there is a different amount of dirt, the crystal structure and the location of atoms each coin is unique ...

And now I have the most interesting question: where is the line, behind which the elements of the multisament turn into elements of the set and vice versa? Such a face does not exist - everyone solves the shamans, the science here and not lying close.

Here are looking. We take football stadiums with the same field area. The field area is the same - it means we have a multipart. But if we consider the names of the same stadiums - we have many, because the names are different. As you can see, the same set of elements is both set and multiset. How correct? And here the mathematician-shaman-shuller pulls out the trump ace from the sleeve and begins to tell us either about the set or about the multiset. In any case, he will convince us of her right.

To understand how modern shamans operate the theory of sets, tie it to reality, it is enough to answer one question: how are the elements of one set differ from the elements of another set? I will show you, without any "imaginable as not a single whole" or "not thoughtful as a whole."

sunday, March 18, 2018

The amount of numbers is a dance of shamans with a tambourine, which does not have any relation to mathematics. Yes, in the lessons of mathematics, we are taught to find the amount of numbers of numbers and use it, but they are shamans to train your descendants to their skills and wisdoms, otherwise the shamans will simply be cleaned.

Do you need evidence? Open Wikipedia and try to find the number of numbers page. It does not exist. There is no formula in mathematics at which you can find the amount of numbers of any number. After all, the numbers are graphic symbols, with which we write numbers and in mathematics language, the task sounds like this: "Find the sum of graphic characters depicting any number". Mathematics can not solve this task, but shamans are elementary.

Let's deal with what and how we do in order to find the amount of the numbers of the specified number. And so, let us have a number of 12345. What should be done in order to find the amount of numbers of this number? Consider all the steps in order.

1. Record the number on the piece of paper. What did we do? We transformed the number in the graphic symbol of the number. This is not a mathematical action.

2. We cut one image obtained into several pictures containing individual numbers. Cutting pictures is not a mathematical action.

3. We convert individual graphic characters in numbers. This is not a mathematical action.

4. We fold the numbers. This is already mathematics.

The amount of numbers of 12345 is 15. These are the "cutters and sewing courses" from the shamans apply mathematicians. But that's not all.

From the point of view of mathematics, it does not matter in which number system we write the number. So, in different systems Number of numbers of numbers of the same number will be different. In mathematics, the number system is indicated in the form of the lower index to the right of the number. FROM large number 12345 I do not want to fool my head, consider the number 26 of the article about. We write this number in binary, octal, decimal and hexadecimal number systems. We will not consider every step under the microscope, we have already done. Let's look at the result.

As you can see, in different number systems, the sum of the numbers of the same number is obtained different. This result for mathematics has nothing to do. It is like determining the area of \u200b\u200bthe rectangle in meters and centimeters you would get completely different results.

Zero in all surge systems looks the same and the amount of numbers does not have. This is another argument in favor of what. Question to mathematicians: how in mathematics is indicated that is not a number? What, for mathematicians, nothing but numbers does not exist? For shamans, I can be allowed, but for scientists - no. Reality consists not only of numbers.

The result obtained should be considered as proof that the number systems are units of numbers. After all, we can not compare the numbers with different units Measurements. If the same action with different units of measurement of the same value lead to different results after their comparison, it means that it has nothing to do with mathematics.

What is real mathematics? This is when the result of mathematical action does not depend on the value of the number used by the unit of measurement and on who performs this action.

Plate on doors Opens the door and says:

Oh! Isn't that a female toilet?
- Girl! This is a laboratory for the study of the Indefile Holiness of the Souls in Ascension to Heaven! Nimbi from above and arrow up. What else toilet?

Female ... Nimbi from above and arrogant down - it's a male.

If you in front of your eyes several times a day flashes this is the work of designer art,

Then it is not surprising that in your car you suddenly find a strange icon:

Personally, I am doing an effort on myself to be in a cuffing person (one picture), to see a minus four degrees (a composition of several pictures: a minus sign, a number four, designation of degrees). And I do not think this girl is a fool who does not know physics. It is simply an arc stereotype of the perception of graphic images. And mathematics we are constantly taught. Here is an example.

1A is not "minus four degrees" or "One A". This is a "cuffing person" or the number of "twenty-six" in a hexadecimal number system. Those people who constantly work in this number system automatically perceive the figure and letter as one graphic symbol.

An alpoint expression (or expression with variables) is mathematical expressionwhich consists of numbers, letters and signs of mathematical operations. For example, next expression is alphabetic:

a + b + 4

With the help of letter expressions, you can record laws, formulas, equations and functions. The ability to manipulate letterproof expressions is the key to good knowledge of algebra and higher mathematics.

Any serious task in mathematics is reduced to solving equations. And to be able to solve equations, you need to be able to work with the letter expressions.

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

Design of lesson

Variables

Letters that are contained in alphabet expressions are called variables. For example, in expression a + b + 4 Changes are letters a. and b.. If instead of these variables, substitute any numbers, then the letter expression a + b + 4 Contact the numeric expression whose value can be found.

Numbers that are substituted instead of variables call values \u200b\u200bof variables. For example, change the values \u200b\u200bof variables a. and b.. To change the values, an equal sign is used

a \u003d 2, b \u003d 3

We changed the values \u200b\u200bof variables a. and b.. Variable a. Assigned importance 2 , variable b. Assigned importance 3 . As a result, an alphabetic expression a + b + 4 appeals to the usual numerical expression 2+3+4 whose value can be found:

2 + 3 + 4 = 9

When multiplication of variables occurs, they are recorded together. For example, writing aB means the same as recording a × B.. If we substitute instead of variables a.and B. numbers 2 and 3 then we will get 6

2 × 3 \u003d 6

It can also be supposed to comply with the multiplication of the number on the expression in brackets. For example, instead a × (B + C) can be recorded a (B + C). Applying the distribution law of multiplication, we get a (B + C) \u003d AB + AC.

Factors

In letter expressions, you can often find a record in which the number and variable are recorded together, for example 3A . In fact, this is a short recording of multiplication of the number 3 to the variable a. And this entry looks like 3 × A. .

In other words, expression 3A It is a product of the number 3 and 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 but", Or" increase the value of the variable a. three times ", but most often read as" three a.«

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

3 × 5 \u003d 15

Speaking simple language, The coefficient is a number that is facing the letter (before the variable).

Letters can be somewhat, 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 "either" increase the expression value aBC five times "or" five aBC«.

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

5 × 2 × 3 × 4 \u003d 120

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

The coefficient mark applies only to the coefficient, and does not relate to variables.

Consider expression -6B.. Minus factor 6 , applies only to the coefficient 6 and does not apply to the variable b.. Understanding this fact will not make mistakes in the future with signs.

Find the value of the expression -6B. for b \u003d 3..

-6B. -6 × B.. For clarity, write the expression -6B. in deployment and substitute the value of the variable b.

-6b \u003d -6 × b \u003d -6 × 3 \u003d -18

Example 2. Find an expression value -6B. for b \u003d -5.

We write expression -6B. in deployed video

-6b \u003d -6 × b \u003d -6 × (-5) \u003d 30

Example 3. Find an expression value -5A + B. for a \u003d 3.and B \u003d 2.

-5A + B. This is a short form of recording from -5 × a + b , so for clarity we write the expression -5 × a + b in deployment and substitute the values \u200b\u200bof variables a. and b.

-5A + b \u003d -5 × a + b \u003d -5 × 3 + 2 \u003d -15 + 2 \u003d -13

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

but the unit according to tradition is not recorded, so they just write a. or aB

If there is a minus before the letter, then the coefficient is the number −1 . For example, expression -A. actually looks like -1A.. This is a product minus units and variable a.It was as follows:

-1 × a \u003d -1a

Here lies a little catch. In expression -A. minus facing variable a. in fact refers to the "invisible unit", and not to the variable a. . Therefore, when solving tasks should be attentive.

For example, if the expression is given -A. and we are asked to find its meaning when a \u003d 2. then in school we substituted a two instead of a variable a. and received the answer −2 , not particularly documented on how it turned out. In fact, there was a multiplication of minus units on a positive number 2

-A \u003d -1 × a

-1 × a \u003d -1 × 2 \u003d -2

If the expression is given -A. and it is required to find its meaning when a \u003d -2. then we substitute −2 Instead of a variable a.

-A \u003d -1 × a

-1 × a \u003d -1 × (-2) \u003d 2

To avoid errors, the first time invisible units can be written explicitly.

Example 4. Find an expression value aBC for a \u003d 2. , b \u003d 3. and c \u003d 4.

Expression aBC 1 × a × b × c. For clarity, write the expression aBC a, B. and c.

1 × A × B × C \u003d 1 × 2 × 3 × 4 \u003d 24

Example 5. Find an expression value aBC for a \u003d -2, b \u003d -3and C \u003d -4.

We write expression aBC in deployment and substitute the values \u200b\u200bof variables a, B.and C.

1 × a × b × c \u003d 1 × (-2) × (-3) × (-4) \u003d -24

Example 6. Find an expression value − aBC for a \u003d 3, b \u003d 5 and c \u003d 7

Expression − aBC This is a short form of recording from -1 × a × b × c. For clarity, write the expression − aBC in deployment and substitute the values \u200b\u200bof variables a, B. and c.

-Abc \u003d -1 × A × B × C \u003d -1 × 3 × 5 × 7 \u003d -105

Example 7. Find an expression value − aBC for a \u003d -2, b \u003d4 and c \u003d -3

We write expression − aBC In an expanded form:

-Abc \u003d -1 × a × b × c

We substitute the value of variables a. , b. and c.

-Abc \u003d -1 × a × b × c \u003d -1 × (-2) × (-4) × (-3) \u003d 24

How to determine the coefficient

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

To determine the coefficient in the expression, it is necessary to multiply the numbers included in this expression, and multiply multiply the letters. The resulting numerical factory and will be a coefficient.

Example 1. 7m × 5a × (-3) × n

The expression consists of several factors. It can be clearly seen if you write an expression in the deployment. That is, works 7m. and 5a Record in the form 7 × M. and 5 × A.

7 × m × 5 × a × (-3) × n

We apply a combination of multiplication law, which allows multipliers to multiply in any order. Namely, separately change the numbers and separately with the letters (variables):

-3 × 7 × 5 × m × a × n \u003d -105man

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

-105amn.

Example 2. Determine the coefficient in expression: -A × (-3) × 2

-A × (-3) × 2 \u003d -3 × 2 × (-a) \u003d -6 × (-a) \u003d 6A

The coefficient is 6.

Example 3. Determine the coefficient in expression:

Move separately numbers and letters:

The coefficient is -1. Note that the unit is not recorded because the coefficient 1 is not recorded not to record.

These seemingly the simplest tasks can play with us a very evil joke. It often finds out that the coefficient's sign is incorrect: either missed minus or vice versa it is in vain. To avoid these annoying errors, should be studied at a good level.

Consciousness in alphabetic expressions

When adding several numbers, the sum of these numbers is obtained. The numbers that are called called the terms. The components can be several, for example:

1 + 2 + 3 + 4 + 5

When the expression consists of the components, it is much easier to calculate it, because it is easier to add than to deduct. But in the expression there may be not only addition, but also subtraction, for example:

1 + 2 − 3 + 4 − 5

In this expression, the number 3 and 5 are subtracted, and not the terms. But it does not interfere with us, replace subtraction by adding. Then we again get an expression consisting of the terms:

1 + 2 + (−3) + 4 + (−5)

Do not escape that numbers -3 and -5 now with a minus sign. The main thing is that all numbers in this expression are connected by the addition mark, that is, the expression is the amount.

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

1 + 2 − 3 + 4 − 5 = −1

1 + 2 + (−3) + 4 + (−5) = −1

Thus, the value of the expression does not suffer from the fact that we will replace subtracting by adding.

Replacing subtraction by adding can also be in alphabone expressions. For example, consider the following expression:

7A + 6B - 3C + 2D - 4S

7A + 6B + (-3c) + 2D + (-4S)

With any values \u200b\u200bof 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 the teacher at school or the teacher at the institute can call the alignments even those numbers (or variables) that they are not.

For example, if a difference will be recorded on the board a - B. then the teacher will not say that a. - this is diminished, and b. - subtracted. Both variables, he will call one general word - composition . And all because the expression of the form a - B. Mathematics sees how the amount a + (-b) . In this case, the expression becomes the amount, and variables a. and (-B) become the terms.

Similar terms

Similar terms - These are the terms that have the same alphabetic part. For example, consider expression 7A + 6B + 2A . Composition 7A. and 2a. have the same alphabet part - variable a.. So the components 7A. and 2a.are similar.

Usually, the similar components are folded to simplify the expression or solve some equation. This operation is called by bringing similar terms.

To bring similar terms, you need to fold the coefficients of these terms, and the resulting result is multiplied by the overall letter.

For example, we give similar terms in the expression 3A + 4A + 5A . In this case, these are all the terms. Moving their coefficients and the result to multiply on the general lettering part - to the variable a.

3A + 4A + 5A \u003d (3 + 4 + 5) × a \u003d 12a

Similar terms usually lead in mind and the result is recorded immediately:

3A + 4A + 5A \u003d 12A

Also, one can argue as follows:

There were 3 variables A, they added another 4 variables A and 5 more variables a. As a result, 12 variables were obtained

Consider several examples of bringing similar terms. Considering that this topic It is very important, at first we will write in detail every little thing. Despite the fact that everything is very simple here, most people allow many mistakes. Basically intensifying, and not for ignorance.

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

Moving the coefficients in this expression and the result obtained to multiply on the general lettering part:

3A + 2A + 6A + 8A \u003d (3 + 2 + 6 + 8) × a \u003d 19A

Design (3 + 2 + 6 + 8) × a You can not record, so you will immediately write the answer

3A + 2A + 6A + 8A \u003d 19A

Example 2. Create similar components in expression 2A + A.

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

2A + 1A.

Now we give similar terms. That is, lay the coefficients and the result to multiply on the general lettering:

2a + 1a \u003d (2 + 1) × a \u003d 3a

Write the decision shorter:

2a + a \u003d 3a

2A + A., it can be reasoned and different:

Example 3. Create similar components in expression 2a - A.

Replace subtraction by adding:

2a + (-a)

Second term (-A) written without a coefficient, but it looks like (-1A).Coefficient −1 Again, invisible due to the fact that it is not written. Therefore, the expression looks like this:

2a + (-1a)

Now we give similar terms. Mix the coefficients and the result to multiply on the overall letter:

2a + (-1a) \u003d (2 + (-1)) × a \u003d 1a \u003d a

Usually recorded in short:

2a - a \u003d a

Leading similar components in expression 2a-A. It can be reasoned in a different way:

There were 2 variables a, detected one variable A, as a result, one single variable remained

Example 4. Create similar components in expression 6A - 3A + 4A - 8A

6A - 3A + 4A - 8A \u003d 6A + (-3A) + 4A + (-8A)

Now we give similar terms. Mix the coefficients and the result to multiply on the overall letter

(6 + (-3) + 4 + (-8)) × a \u003d -1a \u003d -a

Write the decision shorter:

6A - 3A + 4A - 8A \u003d -A

There are expressions that contain several different groups of similar terms. For example, 3A + 3B + 7A + 2B . For such expressions, the same rules are valid as for the rest, namely the folding of the coefficients and the multiplication of the result obtained on the overall letter. But in order to prevent errors, it is convenient to emphasize different lines of components.

For example, in expression 3A + 3B + 7A + 2B those terms that contain a variable a., you can emphasize with one line, and those components that contain a variable b., you can emphasize two lines:

Now you can bring similar terms. That is, fold the coefficients and the resulting result is multiplied by the general letter. This is necessary for both groups of terms: for terms containing a variable a. And for the components containing the variable b..

3A + 3B + 7A + 2B \u003d (3 + 7) × A + (3 + 2) × B \u003d 10A + 5B

Again, we repeat, the expression is simple, and the like components can be given in mind:

3A + 3B + 7A + 2B \u003d 10A + 5B

Example 5. Create similar components in expression 5A - 6A -7B + B

Replace subtraction addition where it can be:

5a - 6A -7B + B \u003d 5A + (-6A) + (-7B) + B

We emphasize the similar terms of different lines. Condivables containing variables a. We emphasize one line, and the components of the variables b. , we emphasize the two lines:

Now you can bring similar terms. That is, fold the coefficients and the result obtained multiplied by the general letter:

5a + (-6a) + (-7b) + b \u003d (5 + (-6)) × a + ((-7) + 1) × b \u003d -a + (-6b)

If the expression contains ordinary numbers without alleged factors, they add up separately.

Example 6. Create similar components in expression 4A + 3A - 5 + 2B + 7

Replace subtraction by adding where it can be:

4A + 3A - 5 + 2B + 7 \u003d 4A + 3A + (-5) + 2B + 7

We give similar terms. Numbers −5 and 7 do not have alphabets, but they are similar terms - they need to just fold. And the foundation 2b. will remain unchanged because it is the only thing in this expression that has an alphabet b, And nothing to fold it with:

4a + 3a + (-5) + 2b + 7 \u003d (4 + 3) × A + 2B + (-5) + 7 \u003d 7A + 2B + 2

Write the decision shorter:

4A + 3A - 5 + 2B + 7 \u003d 7A + 2B + 2

The components can be organized so that those terms that have the same alphabetic part are located in one part of the expression.

Example 7. Create similar components in expression 5T + 2X + 3X + 5T + X

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

5T + 5T + 2x + 3x + x

Now you can bring similar terms:

5t + 5t + 2x + 3x + x \u003d (5 + 5) × t + (2 + 3 + 1) × x \u003d 10t + 6x

Write the decision shorter:

5T + 2X + 3X + 5T + X \u003d 10T + 6X

The sum of the opposite numbers is zero. This rule works for alphabetic expressions. If the expression will meet the same terms, but with opposite signs, then they can be rid of them at the stage of bringing similar terms. In other words, simply shut out them from the expression, since their sum is zero.

Example 8. Create similar components in expression 3T - 4T - 3T + 2T

Replace subtraction by adding where it can be:

3T - 4T - 3T + 2T \u003d 3T + (-4t) + (-3t) + 2t

Composition 3T and (-3t) are opposite. The sum of the opposite terms is zero. If you remove this zero from the expression, the value of the expression does not change, so we will remove it. And we will remove it with the usual strikeout of the terms 3T and (-3t)

As a result, we will have an expression (-4t) + 2t. In this expression, such a component can be given and get the final answer:

(-4t) + 2t \u003d ((-4) + 2) × t \u003d -2t

Write the decision shorter:

Simplification of expressions

"Similarize the expression" And then the expression is given to simplify. Simplify expression So make it easier and shorter.

In fact, we have already been simplified expressions when the fractions have shrink. After cutting, the fraction became shorter and easier for perception.

Consider the following example. Simplify the expression.

This task can literally understand this way: "Apply any permissible actions to this expression, but make it easier." .

In this case, the fraction can be reduced, namely split the numerator and denominator of the fraction 2:

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

As a result, the fraction simply simplified to 0.5.

The first question that needs to be asked to solve such tasks should be "What can I do?" . Because there are actions that can be done, and there are actions that can not be done.

Another important momentWhat needs to be remembered is that the expression value should not change after simplifying the expression. Let's return to expression. This expression is a division that can be performed. By doing this division, we obtain the value of this expression, which is 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 the expression we also tried to simplify, calculate it. As a result, they received the final answer 0.5.

Thus, no matter how we simplify the expression, the value of the expressions obtained is still 0.5. So simplification was performed correctly at each stage. It is for this that it is necessary to strive when simplifying expressions - the value of the expression should not suffer from our actions.

Often it is necessary to simplify letter expressions. For them, the same simplification rules are valid as for numerical expressions. You can perform any permissible actions, just not to change the value of the expression.

Consider several examples.

Example 1. Simplify expression 5,21s × t × 2.5

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

5,21s × t × 2,5 \u003d 5.21 × 2.5 × s × t \u003d 13,025 × ST \u003d 13,025ST

So the expression 5,21s × t × 2.5 Simplified before 13,025ST.

Example 2. Simplify expression -0.4 × (-6,3b) × 2

Second work (-6,3b) can be translated to understandable to us, namely, write in the form ( -6.3) × b,then send the numbers separately and multiply the letters separately:

− 0,4 × (-6,3b) × 2 = − 0,4 × (-6.3) × b × 2 \u003d 5,04b

So the expression -0.4 × (-6,3b) × 2 Simplified before 5,04b

Example 3. Simplify expression

Cut this expression in more detail to see well where numbers, and where letters:

Now separately alternate numbers and separately alternate the letters:

So the expression Simplified before -ABC.This solution can be written shorter:

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

The fraction can be reduced by choosing in a multiplier in a numerator and in the denominator and cut these factors to their largest common divisor. In other words, to use in which we do not paint in detail what the numerator and denominator were divided.

For example, in the numerator multiplier 12 and in the denominator, the multiplier 4 can be reduced by 4. The fourth is stored in the mind, and dividing 12 and 4 to this fourth, the answers are recorded next to these numbers, after after following them

Now you can multiply the resulting small multipliers. In this case, they are a bit and can multiply in the mind:

Over time, it can be found that solving one or another task, expressions begin to "fat", so it is desirable to learn to rapid calculations. What can be calculated in the mind must be calculated in the mind. What you can quickly cut, you need to quickly cut.

Example 4. Simplify expression

So the expression Simplified before

Example 5. Simplify expression

Move separately the numbers and separate letters:

So the expression Simplified before mn.

Example 6. Simplify expression

We write this expression in more detail to see well where numbers, and where letters:

Now separately alternate the number and separate letters. For the convenience of computing decimal fraction -6.4 and mixed number You can translate into ordinary fractions:

So the expression Simplified before

The solution for this example can be recorded significantly shorter. It will look like this:

Example 7. Simplify expression

Move separately the numbers and separate letters. For convenience of calculating a mixed number and decimal fractions 0.1 and 0.6 can be translated into ordinary fractions:

So the expression Simplified before abcd.. If you skip the details, this decision can be recorded significantly in short:

Pay attention to how the fraction has decreased. New multipliers that are obtained as a result of the reduction of previous multipliers are also allowed to reduce.

Now let's talk about what you can not do. When simplifying expressions, it is categorically impossible to multiply the numbers and letters, if the expression is the sum, and not by the work.

For example, if you need to simplify expression 5A + 4B.You can not write as follows:

It is equivalent to the fact that if we were asked to fold two numbers, and we would multiply them instead of folding.

When substituting any values \u200b\u200bof variables a. and b. expression 5A + 4B. refers to an ordinary numerical expression. Suppose that variables a. and b. have the following values:

a \u003d 2, b \u003d 3

Then the expression value will be equal to 22

5A + 4B \u003d 5 × 2 + 4 × 3 \u003d 10 + 12 \u003d 22

First, multiplication is performed, and then the results are folded. And if we tried to simplify this expression, moving the numbers and letters, it would have happened:

5A + 4B \u003d 5 × 4 × a × b \u003d 20ab

20ab \u003d 20 × 2 × 3 \u003d 120

It turns out a completely different value of the expression. In the first case it turned out 22 in the second case 120 . This means that simplification of expression 5A + 4B. It was incorrect.

After simplifying the expression, its value should not be changed at the same values \u200b\u200bof variables. If during substitution to the initial expression of any values \u200b\u200bof variables, one value is obtained, then after simplifying the expression, the same value should be obtained as before simplification.

With an expression 5A + 4B. In fact, you can not do anything. It is not simplified.

If the expression contains similar components, they can be folded if our goal is to simplify expression.

Example 8. Simplify expression 0,3A-0,4A + A

0,3A - 0,4A + a \u003d 0,3a + (-0.4a) + a \u003d (0.3 + (-0.4) + 1) × a \u003d 0,9A

or shorter: 0,3A - 0,4A + A = 0.9A.

So the expression 0,3A-0,4A + A Simplified before 0.9A.

Example 9. Simplify expression -7,5A - 2.5B + \u200b\u200b4A

To simplify this expression, you can bring similar terms:

-7,5A - 2.5B + \u200b\u200b4a \u003d -7,5A + (-2,5b) + 4a \u003d ((-7.5) + 4) × A + (-2,5B) \u003d -3,5A + (-2,5B)

or shorter -7,5A - 2.5B + \u200b\u200b4A \u003d -3,5A + (-2,5B)

Speed (-2,5B) It remains unchanged, because it has nothing to be folded.

Example 10. Simplify expression

To simplify this expression, you can bring similar terms:

The coefficient was for the convenience of calculating.

So the expression Simplified before

Example 11. Simplify expression

To simplify this expression, you can bring similar terms:

So the expression Simplified before.

IN this example It would be more expedient to fold the first and last coefficient in the first place. In this case, we would get a short decision. It looked as follows:

Example 12. Simplify expression

To simplify this expression, you can bring similar terms:

So the expression Simplified before .

The term remained unchanged, because it has nothing to be folded.

This solution can be recorded significantly shorter. It will look like this:

IN short decision The stages of replacement of subtraction by addition and the detailed entry, as the fraction was brought to a common denominator.

Another distinction is that detailed decision The answer looks like , and in short as. In fact, this is the same expression. The difference is that in the first case, subtraction is replaced by adding, because at the beginning when we recorded the decision in detailedWe are everywhere where you can replace subtraction by adding, and this replacement has been preserved for answering.

Identities. Identically equal expressions

After we simplified any expression, it becomes easier and shorter. To check whether the expression is simplified, it is enough to substitute any values \u200b\u200bof variables first into the previous expression that was required to simplify, and then to the new one that was simplified. If the value in both expressions is the same, the expression is simplified correctly.

Consider the simplest example. Let it take to simplify the expression 2a × 7b. . To simplify this expression, you can multiply numbers and letters separately:

2a × 7b \u003d 2 × 7 × a × b \u003d 14ab

Check whether we simplified the expression. To do this we will substitute any values \u200b\u200bof variables a. and b. First, in the first expression that was required to simplify, and then second, which was simplified.

Let the values \u200b\u200bof the variables a. , b. will be as follows:

a \u003d 4, b \u003d 5

Substitute them in the first expression 2a × 7b.

Now we will substitute the same values \u200b\u200bof variables in the expression that happened as a result of simplification 2a × 7b., namely, the expression 14ab

14ab \u003d 14 × 4 × 5 \u003d 280

We see that when a \u003d 4. and b \u003d 5. The value of the first expression 2a × 7b. and the value of the second expression 14ab equal

2a × 7b \u003d 2 × 4 × 7 × 5 \u003d 280

14ab \u003d 14 × 4 × 5 \u003d 280

The same thing will happen for any other values. For example, let it a \u003d 1. and b \u003d 2.

2a × 7b \u003d 2 × 1 × 7 × 2 \u003d 28

14ab \u003d 14 × 1 × 2 \u003d 28

Thus, with any values \u200b\u200bof variable expression 2a × 7b. and 14ab equal to the same meaning. Such expressions are called identically equal.

We conclude that between expressions 2a × 7b. and 14ab You can put a sign of equality, since they are equal to the same value.

2a × 7b \u003d 14ab

The equality is called any expression that is connected by the sign of equality (\u003d).

A equality of type 2a × 7b \u003d 14ab Call identity.

The identity is called equality that is true for any values \u200b\u200bof variables.

Other examples of identities:

a + b \u003d b + a

a (B + C) \u003d AB + AC

a (BC) \u003d (AB) C

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

Faithful numeric equality are also identities. For example:

2 + 2 = 4

3 + 3 = 5 + 1

10 = 7 + 2 + 1

Solving a complex task to facilitate the calculation, the complex expression is replaced by a simpler expression, identically equal to the previous one. Such a replacement is called identical transformation of expression or simply transformation of the expression.

For example, we have simplified expression 2a × 7b. and got a simpler expression 14ab . This simplification can be called identical conversion.

Often you can meet the task in which it is said "Prove that equality is the identity" And then the equality that needs to be proved is given. Typically, this equality consists of two parts: the left and right part of equality. Our task is to perform identical conversions with one of the parts of equality and get another part. Either perform identical transformations with both parts of equality and make such an equal expression in both parts of equality.

For example, we prove that equality 0,5A × 5B \u003d 2,5ab Is identity.

We simplify the left part of this equality. To do this, change the number and letters separately:

0.5 × 5 × a × b \u003d 2,5ab

2,5ab \u003d 2,5ab

As a result of a small identical transformation, the left side of the equality has become equal to the right part of equality. So we have proven equality 0,5A × 5B \u003d 2,5ab Is identity.

Of the identical transformations, we learned to fold, deduct, multiply and divide the numbers, cut the fractions, bring such components, and simplify some expressions.

But this is not all identical transformations that exist in mathematics. Identical transformations a lot more. In the future, we will be convinced more than once.

Tasks for self solutions:

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

Simplification of algebraic expressions is one of key moments Studying algebra and extremely useful skill for all mathematicians. Simplification allows you to bring a complex or long expression to a simple expression, with which it is easy to work. Basic simplification skills are well given even to those who are not delighted with mathematics. Observing several simple rules, It is possible to simplify many of the most common types of algebraic expressions without any special mathematical knowledge.

Steps

Important definitions

1. Similar members. These are members with a variable of one order, members with the same variables or free members (members not containing a variable). In other words, such members include one variable in the same extent, include several identical variables or do not include a variable at all. The procedure for members in expression does not matter.

   • For example, 3x 2 and 4x 2 are similar members, as they contain a variable "x" of the second order (second degree). However, X and X 2 are not similar members, as they contain the variable "x" of different orders (first and second). Similarly, -3yx and 5xz are not similar members, as they contain different variables.

2. Decomposition of multipliers. This is the finding of such numbers whose product leads to the initial number. Any initial number may have several factors. For example, the number 12 may be decomposed on the following range of multipliers: 1 × 12, 2 × 6 and 3 × 4, so we can say that numbers 1, 2, 3, 4, 6 and 12 are multipliers of the number 12. Multiplers coincide with divisors , Well, the numbers to which the initial number is divided.

   • For example, if you want to decompose the number 20 on multipliers, write it up like this: 4 × 5.
   • Please note that during decomposition of multipliers, the variable is taken into account. For example, 20x \u003d 4 (5x).
   • Simple numbers cannot be laid out for multipliers, because they are only divided into ourselves and 1.

3. Remember and follow the procedure for performing operations to avoid errors.

   • Brackets
   • Power
   • Multiplication
   • Division
   • Addition
   • Subtraction

   Bringing similar members

   1. Write down the expression. The simplest algebraic expressions (which do not contain fractions, roots and so on) can be solved (simplify) in just a few steps.

      • For example, simplify the expression 1 + 2x - 3 + 4x.

   2. Determine such members (members with a variable of one order, members with the same variables or free members).

      • Find similar members in this expression. Members 2x and 4x contain a variable of one order (first). In addition, 1 and -3 are free members (do not contain a variable). Thus, in this member member 2x and 4x are similar and members 1 and -3. Also are similar.

   3. Bring such members. This means folded or subtract them and simplify the expression.

      • 2x + 4x \u003d 6x
      • 1 - 3 = -2

   4. Rewrite the expression taking into account the above members. You will get a simple expression with a smaller number of members. The new expression is equal to the original one.

      • In our example: 1 + 2x - 3 + 4x \u003d 6x - 2., that is, the initial expression is simplified and easier to work with it.

   5. Observe the procedure for performing operations when bringing such members. In our example it was easy to bring similar members. However, in the case of complex expressions in which members are enclosed in brackets and there are fractions and roots, bring similar members is not so easy. In these cases, follow the procedure for performing operations.

      • For example, consider the expression 5 (3x - 1) + x ((2x) / (2)) + 8 - 3x. Here it would be a mistake to immediately determine 3x and 2x as such members and bring them, because you first need to reveal brackets. Therefore, perform operations according to their order.
        • 5 (3x-1) + x ((2x) / (2)) + 8 - 3x
        • 15x - 5 + x (x) + 8 - 3x
        • 15x - 5 + x 2 + 8 - 3x. NowWhen there are only additions and subtraction operations in the expression, you can cite such members.
        • x 2 + (15x - 3x) + (8 - 5)
        • x 2 + 12x + 3

   Multiplier for brackets

   1. Find the largest common divider (node) of all expression coefficients. NOD is the largest number on which all the expression coefficients are divided.

      • For example, consider equation 9x 2 + 27X - 3. In this case, the Node \u003d 3, since any coefficient of this expression is divided into 3.

   2. Divide each member of the expression on the Node. The members received will contain smaller coefficients than in the initial expression.

      • In our example, divide every member of the expression by 3.
        • 9x 2/3 \u003d 3x 2
        • 27x / 3 \u003d 9x
        • -3/3 = -1
        • Opened expression 3x 2 + 9x - 1. It is not equal to the initial expression.

   3. Record the initial expression as equal to the product of the Node on the resulting expression. That is, enter into the resulting expression in the brackets, and take a node for brackets.

      • In our example: 9x 2 + 27x - 3 \u003d 3 (3x 2 + 9x - 1)

   4. Simplify fractional expressions by making a multiplier for brackets. Why just make a multiplier for brackets, how was it previously done? Then to learn to simplify complex expressions, for example, fractional expressions. In this case, making a multiplier for parentheses can help get rid of the frarators (from the denominator).

      • For example, consider the fractional expression (9x 2 + 27x - 3) / 3. Take advantage of the multiplier for brackets to simplify this expression.
        • Take a multiplier 3 for brackets (as you did before): (3 (3x 2 + 9x - 1)) / 3
        • Note that now in the numerator, and in the denominator there is a number 3. It can be reduced, and you will receive an expression: (3x 2 + 9x - 1) / 1
        • Since any fraction in which the denominator contains the number 1 is equal to simply a numerator, the initial fractional expression is simplified to: 3x 2 + 9x - 1.

   Additional simplification methods

4. Consider a simple example: √ (90). The number 90 can be decomposed into the following factors: 9 and 10, and from 9 to extract square root (3) and make 3 from the root.
   • √(90)
   • √ (9 × 10)
   • √ (9) × √ (10)
   • 3 × √ (10)
   • 3√(10)

5. Simplification of expressions with degrees. In some expressions there are operations of multiplication or division of members with a degree. In the case of multiplication of members with one reason, they are considerable; In the case of dividing members with one reason, they are deducted.

   • For example, consider the expression 6x 3 × 8x 4 + (x 17 / x 15). In case of multiplication, fold degrees, and in the case of division - deduct them.
     • 6x 3 × 8x 4 + (x 17 / x 15)
     • (6 × 8) x 3 + 4 + (x 17 - 15)
     • 48x 7 + x 2
   • The following is an explanation of the rule of multiplication and division of members with a degree.
     • Multiplication of members with degrees is equivalent to multiplying members on themselves. For example, since x 3 \u003d x × x × x and x 5 \u003d x × x × x × x × x, then x 3 × x 5 \u003d (x × x × x) × (x × x × x × x × x), or x 8.
     • Similarly, dividing members with degrees is equivalent to the division of members on themselves. x 5 / x 3 \u003d (x × x × x × x × x) / (x × x × x). Since such members, in the numerator, and in the denominator, can be reduced, then the number of two "x", or x 2 remains in the numerator.

• Always remember signs (plus or minus) facing member of the expression, as many have difficulty choosing the right mark.
• Ask for help if necessary!
• Simplify algebraic expressions are not easy, but if you do a hand, you can use this skill all your life.

Section 5 of expressions and equations

The section will learn:

ü about expressions and their simplification;

ü what properties of equalities;

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

ü what types of tasks are solved using equations; what is perpendicular straight lines and how to build them;

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

ü what is a coordinate plane;

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

ü what is a graph of relationship between values \u200b\u200band how to build it;

ü how to apply the studied material in practice

§ 30. Expressions and their simplification

You already know what literal expressions are and know how to simplify them with the help of the laws of addition and multiplication. For example, 2a ∙ (-4b) \u003d -8 AB . In the resulting expression, the number -8 is called the expression coefficient.

Does expression havecD coefficient? So. It is equal to 1 becausecD - 1 ∙ CD.

Recall that the transformation of expressions with brackets into the expression without brackets is called disclosure, brackets. For example: 5 (2x + 4) \u003d 10x + 20.

Reverse action In this example, this is a general factor for brackets.

The components containing the same alphabetic multipliers are called similar terms. With the help of a general factor for parentheses, similar terms are built:

5x + y + 4 - 2x + 6 y - 9 \u003d

\u003d (5x - 2x) + (y + 6 y )+ (4 - 9) = = (5-2)* + (1 + 6)* y -5 \u003d.

B X + 7U - 5.

Rules for disclosure brackets

1. If the "+" sign stands in front of the brackets, then when disclosing brackets, the signs of the components in brackets retain;

2. If there is a sign "-" before brackets, then when disclosing brackets, the signs of the components are changed in brackets.

Task 1. Simplify the expression:

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

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

Solutions. 1. Before brackets there is a "+" sign, so when disclosing brackets, the signs of all terms are saved:

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

2. Before brackets there is a sign "-", so during the disclosure of the brackets: the signs of all the components change to the opposite:

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

For disclosures of brackets use the distribution property of multiplication: A (b + C) \u003d AB + AU. If a\u003e 0, then the signs of the termsb. and do not change. If A.< 0, то знаки слагаемых b. and C change to the opposite.

Task 2. Simplify the expression:

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

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

Solutions. 1. Multiplier 2 in front of brackets is positive, so when disclosing brackets, we retain all the components: 2 (6y - 8) + 7 y \u003d 12 y - 16 + 7 y \u003d 19 y -16.

2. Multiplier -5 in front of brackets e negative, so when disclosing brackets, the signs of all the terms change to the opposite:

5 (2 - 5x) + 12 \u003d -10 + 25x +12 \u003d 2 + 25x.

Find out more

1. The word "amount" comes from Latinsumma. What does "result", "total number".

2. The word "plus" comes from Latinplus which means "more", and the word "minus" - from Latinminus what does "less" mean. Signs "+" and "-" are used to designate addition and subtraction actions. These signs introduced the Czech Scientist J. Viman in 1489 in the book "Quick and pleasant account for all merchants"(Fig. 138).

Fig. 138.

Remember the main thing

1. What are the components called similar? How to build similar terms?

2. How do braces reveal, facing the "+" sign?

3. How do you open brackets, in front of which is the sign "-"?

4. How do brackets reveal, before which is a positive factor?

5. How do brackets reveal, facing a negative multiplier?

1374. Name the expression coefficient:

1) 12 A; 3) -5.6 Hu;

2) 4 6; 4) -s.

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

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

2) BC -4 D - BC + 4 D; 4) 5x + 4u-x + y.

What are these components called?

1376 ". Are there any components in the expression:

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

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

1377 ". It is necessary to change the signs of the components in brackets, revealing brackets in the expression:

1) 4 + (A + 3 B); 2) -c + (5-D); 3) 16- (5 M -8 n)?

1378 °. Simplify the expression and emphasize the coefficient:

1379 °. Simplify the expression and emphasize the coefficient:

1380 °. Two similar terms:

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

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

3) -7 ANG \u003d "EN-US"\u003e C + 5-3 C + 2; 6) 14 n - 12 m -4 n -3 m.

1381 °. Two similar terms:

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

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

1382 °. Take a common factor for brackets:

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

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

1383 °. Take a common factor for brackets:

1) 6A-12 B; 3) -1.8 N -3.6 m;

2) -0.2 C + 1 4 D; A) 3r - 0.9 k + 2.7 t.

1384 °. Open brackets and twist similar 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) - (-2u + 4x) + (x - 3ow).

1385 °. Open brackets and twist similar terms:

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

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

1386 °. Open brackets and find the value 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 °. Open brackets and find the value 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) -C ∙ (2.7-1.2 D ); 5) 3 ∙ (-1.5 P + K - 0.2t);

3) 1.6 ∙ (2 n + 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- (-r + 0.3 k - 1.2 T).

1390. Simplify the expression:

1391. Simplify the expression:

1392. Two similar terms:

1393. Two similar 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 value 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 \u003d -0.8;

m \u003d 0.25, n \u003d 5.7.

1397. Find the value of the expression:

1) -4 ∙ (I-2) + 2 ∙ (6x - 1), if x \u003d -0.25;

1398 *. Find a mistake in the decision:

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 brackets and simplify the expression:

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

1400 *. Arrange brackets so as to obtain the right 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 andb if a\u003e b , then equality is performed:

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 \u003d 6?

1402 *. Prove that for any natural Number And the arithmetic average of the previous and next numbers behind it is equal to the number a.

Apply in practice

1403. For the preparation of a fruit dessert for three people you need: 2 apples, 1 orange, 2 banana and 1 kiwi. How to make an alphabetic expression to determine the number of fruits required for the preparation of dessert I am for guests? Help Marin these calculate how much fruit you need to buy, if you come to visit: 1) 5 friends; 2) 8 friends.

1404. Make an alphabone expression to determine the time required to perform a homework in mathematics if:

1) on solving problems is spent by min; 2) Simplification of expressions 2 times more than the solution of tasks. How much time was performed homework Vasilko, if he spent 15 minutes to solve the tasks?

1405. Lunch in the school 'dining room consists of salad, borscht, cabbage rolls and compotes. The cost of salad is 20%, borscht - 30%, kaltsov - 45%, compote - 5% of the total value of the entire lunch. Make an expression to find the cost of lunch in the school canteen. How much is lunch, if the price of salad is 2 UAH?

Tasks for repetition

1406. Decide equation:

1407. Tanya spent on ice creamall the money available, and on candy -rest. How much money remains in Tanya

if candy stand 12 UAH?

Using any language, you can express the same information. different words and turns. Not exception and mathematical language. But the same expression can be recorded equivalently in different ways. And in some situations, one of the records is simpler. We will talk about simplifying expressions in this lesson.

People communicate on different languages. For us, an important comparison is a pair of "Russian language - Mathematical language". The same information can be reported in different languages. But, in addition, it can also pronounce in one language in different ways.

For example: "Petya is friends with Vasya", "Vasya is friends with Petya", "Pete with Vay Friends." Said differently, but the same thing. For any of these phrases, we would understand what we are talking about.

Let's look at this phrase: "Petya's boy and boy Vasya are friends." We understood what this is speech. Nevertheless, we do not like how this phrase sounds. Can we simplify it, say the same, but easier? "Boy and boy" - you can say once again: "Petya and Vasya boys are friends."

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

In a mathematical language, approximately the same thing happens. One thing can be said to write differently. What does it mean to simplify expression? This means that there are many equivalent expressions for the initial expression, that is, those that mean the same thing. And from all this set, we must choose the simplest, in our opinion, or the most suitable for our future goals.

For example, consider a numerical expression. It will be equivalent.

It will also be equivalent to the first two: .

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

For numerical expressions, it is always necessary to perform all actions and receive an equivalent expression in the form of one number.

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

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

Do you always need to simplify expression? No, sometimes it will be more convenient for us equivalent, but a longer recording.

Example: From the number you need to take away the number.

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

That is, a simplified expression is not always profitable for further computing.

Nevertheless, very often we face a task that it sounds "to simplify the expression".

Simplify the expression :.

Decision

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

2) Calculate the works: .

Obviously, the last expression is a simpler view than the initial one. We simplified it.

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

To determine the equivalent expression, it is necessary:

1) perform all possible actions

2) Use the properties of addition, subtraction, multiplication and divisions to simplify calculations.

Properties of addition and subtraction:

1. Move the property of addition: the amount does not change from the rearrangement of the terms.

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

3. The property of subtraction of the amount from among: To subtract the amount from the number, you can deduct each term separately.

Properties of multiplication and division

1. Movement property of multiplication: the product does not change from the permutation of multipliers.

2. Fashionable property: To multiply the number on the work of two numbers, you can first multiply it to the first factor, and then the resulting work is multiplied by the second factor.

3. The distribution property of multiplication: to multiply the number to the amount, you need to multiply it to each alone separately.

Let's see how we actually make calculations in the mind.

Calculate:

Decision

1) imagine how

2) Imagine the first factor as the sum of the discharge terms and perform multiplication:

3) You can imagine how to perform multiplication:

4) Replace the first factor of the equivalent amount:

Distribution law can be used in reverse side: .

Perform actions:

1) 2)

Decision

1) For convenience, you can use the distributional law, just to use it in the opposite direction - to make a general factor for brackets.

2) I will bring a general multiplier for brackets.

It is necessary to buy linoleum in the kitchen and an entrance hall. Square kitchen -, hallway -. There are three types of linoleums: software, and rubles for. How much will each of three species Linoleum? (Fig. 1)

Fig. 1. Illustration to the condition of the problem

Decision

Method 1. You can individually find how much money will need to buy a linoleum into the kitchen, and then add to the hallway and the obtained works.