Is . In this article, we will define like terms, figure out what is called the reduction of like terms, consider the rules by which this action is performed, and give examples of reducing like terms with a detailed description of the solution.
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Definition and examples of similar terms.
A conversation about such terms arises after getting acquainted with literal expressions, when it becomes necessary to carry out transformations with them. According to the textbooks of mathematics N. Ya. Vilenkin definition of like terms is given in the 6th grade, and it has the following wording:
Definition.
Similar terms are terms that have the same letter part.
It is worth considering this definition carefully. First, we are talking about terms, and, as you know, terms are constituent elements of sums. This means that such terms can only be present in expressions that are sums. Secondly, in the voiced definition of such terms there is an unfamiliar concept of “literal part”. What is meant by the letter part? When this definition is given in the sixth grade, the letter part refers to one letter (variable) or the product of several letters. Thirdly, the question remains: “What are these terms with a letter part”? These are terms that are the product of a certain number, the so-called numerical coefficient, and the letter part.
Now you can bring examples of similar terms. Consider the sum of two terms 3·a and 2·a of the form 3·a+2·a . The terms in this sum have the same letter part, which is represented by the letter a , therefore, by definition, these terms are similar. The numerical coefficients of these similar terms are the numbers 3 and 2 .
Another example: total 5 x y 3 z+12 x y 3 z+1 the terms 5·x·y 3 ·z and 12·x·y 3 ·z with the same literal part x·y 3 ·z are similar. Note that y 3 is present in the literal part, its presence does not violate the definition of the literal part given above, since it is, in fact, the product of y·y·y .
Separately, we note that the numerical coefficients 1 and −1 for such terms are often not written explicitly. For example, in the sum 3 z 5 +z 5 −z 5 all three terms 3 z 5 , z 5 and −z 5 are similar, they have the same letter part z 5 and coefficients 3 , 1 and −1 respectively, of which 1 and −1 are not clearly visible.
Proceeding from this, in the sum 5+7 x−4+2 x+y, not only 7 x and 2 x are similar terms, but also the terms without the literal part 5 and −4 .
Later, the concept of the literal part also expands - I begin to consider the literal part not only the product of letters, but an arbitrary literal expression. For example, in the textbook of algebra for grade 8 authors Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorov, edited by S. A. Telyakovsky, a sum of the form is given, and it is said that its components terms are similar. The common literal part of these similar terms is an expression with a root of the form .
Similarly, similar terms in the expression 4 (x 2 +x−1/x)−0.5 (x 2 +x−1/x)−1 we can consider the terms 4 (x 2 +x−1/x) and −0.5 (x 2 +x−1/x) , since they have the same letter part (x 2 +x−1/x) .
Summarizing all the above information, we can give the following definition of similar terms.
Definition.
Similar terms terms in a literal expression are called that have the same literal part, as well as terms that do not have a literal part, where the literal part is understood to be any literal expression.
Separately, we say that similar terms can be the same (when their numerical coefficients are equal), or they can be different (when their numerical coefficients are different).
In concluding this paragraph, we will discuss one very subtle point. Consider the expression 2 x y+3 y x . Are the terms 2 x y and 3 y x similar? This question can also be formulated as follows: “Are the literal parts x y and y x of the indicated terms the same”? The order of the literal factors in them is different, so that in fact they are not the same, therefore, the terms 2·x·y and 3·y·x in the light of the definition introduced above are not similar.
However, quite often such terms are called similar terms (but for the sake of rigor it is better not to do this). In this case, they are guided by the following: according to the permutation of factors in the product, it does not affect the result, so the original expression 2 x y+3 y x can be rewritten as 2 x y+3 x y , whose terms are similar. That is, when they talk about similar terms 2 x y and 3 y x in the expression 2 x y+3 y x , they mean the terms 2 x y and 3 x y in transformed expression of the form 2 x y+3 x y .
Reduction of similar terms, rule, examples
The transformation of expressions containing similar terms implies the addition of these terms. This action has a special name - reduction of like terms.
The reduction of similar terms is carried out in three stages:
- first, the terms are rearranged so that similar terms are next to each other;
- after that, the literal part of similar terms is taken out of brackets;
- finally, the value of the numerical expression formed in brackets is calculated.
Let's analyze the recorded steps with an example. We present similar terms in the expression 3 x y+1+5 x y . First, we rearrange the terms so that the like terms 3 x y and 5 x y are next to each other: 3 x y+1+5 x y=3 x y+5 x y+1. Secondly, we take out the literal part of the brackets, we get the expression x·y·(3+5)+1 . Thirdly, we calculate the value of the expression that was formed in brackets: x·y·(3+5)+1=x·y·8+1 . Since it is customary to write the numerical coefficient before the letter part, we will transfer it to this place: x·y·8+1=8·x·y+1. This completes the reduction of similar terms.
For convenience, the three steps above are combined into rule for reducing like terms: to bring similar terms, you need to add their coefficients and multiply the result by the letter part (if any).
The solution of the previous example using the rule of reduction of like terms will be shorter. Let's bring him. The coefficients of similar terms 3 x y and 5 x y in the expression 3 x y+1+5 x y are the numbers 3 and 5, their sum is 8, multiplying it by the letter part x y , we get the result of reducing these terms is 8·x·y . It remains not to forget about the term 1 in the original expression, as a result we have 3 x y+1+5 x y=8 x y+1 .
Instruction
Before bringing similar terms in a polynomial, it often becomes necessary to perform intermediate actions: open all the brackets, raise and bring the terms themselves into standard form. That is, write them as a product of a numerical factor and variables. For example, the expression 3xy(-1.5)y², reduced to standard form, will look like this: -4.5xy³.
Expand all brackets. Omit parentheses in expressions like A+B+C. If there is a plus sign in front of it, then all terms are preserved. If there is a minus sign in front of the brackets, then reverse the signs of all terms. For example, (x³–2x)–(11x²–5ax)=x³–2x–11x²+5ax.
If you need to multiply a polynomial by a polynomial, multiply all the terms together and add the resulting monomials. When raising a polynomial A+B to a power, use abbreviated multiplication. For example, (2ax–3y)(4y+5a)=2ax∙4y–3y∙4y+2ax∙5a–3y∙5a.
Bring monomials to standard form. To do this, group numbers and degrees with bases. Then multiply them together. If necessary, raise the monomial to a power. For example, 2ax∙5a–3y∙5a+(2xa)³=10a²x–15ay+8a³x³.
Find the terms in the expression that have the same letter part. Highlight them with a special underline for clarity: one straight line, one wavy line, two simple lines, etc.
Add up the coefficients of like terms. Multiply the resulting number by the literal expression. Similar terms are given. For example, x²–2x–3x+6+x²+6x–5x–30–2x²+14x–26=x²+x²–2x²–2x–3x+6x–5x+14x+6–30–26=10x–50.
Sources:
- monomial and polynomial
- Wash please: write down: a) the amount, where the first term
Even the most complex equation ceases to look intimidating if you reduce it to the form that you have already encountered. The simplest way, which helps out in any situation, is to bring polynomials to a standard form. This is the starting point from which you can move forward towards a solution.
You will need
- paper
- colored pens
Instruction
Remember the standard form so that you know what you should get as a result. Even the order of writing is significant: the first should be the terms with the largest . In addition, it is customary to first write down unknowns, indicated by letters at the beginning of the alphabet.
Write down the original polynomial and start looking for similar terms. These are the members of the equation given to you, the same letter part or (and) numeric. For greater clarity, underline the found pairs. Please note that similarity does not mean identity - the main thing is that one member of the pair contains the second. So, there will be members xy, xy2z and xyz - they have a common part in the form of the product of x and y. The same is true for the power ones.
Label different like terms in different ways. To do this, it is better to emphasize with single, double and triple lines, use color and other line shapes.
Having found all similar terms, proceed to combine them. To do this, take similar terms out of brackets in the found ones. Keep in mind that a polynomial has no like terms in standard form.
Check if you still have the same items in the entry. In some cases, you may have similar members again. Repeat the operation with their combination.
Follow the second condition required to write a polynomial in standard form: each of its members must be depicted as a monomial in standard form: in the first place - a numerical factor, in the second - a variable or variables, following in the already indicated order. In this case, it has a letter sequence specified by the alphabet. Decreasing degrees are taken into account in the second place. So, the standard form of the monomial is 7xy2, while y27x, x7y2, y2x7, 7y2x, xy27 are not required.
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The signs of the zodiac are the basic element of astrology. These are 12 sectors (according to the number of months in a year), into which the zodiac zone is divided, according to the astrological tradition of Europe. Each of them has a name, depending on the zodiac constellation located in this area. There is a version according to which the names of the signs originated from ancient Greek myths.
Instruction
Aries is a ram with golden wool. The name of this sign is associated with the myth of the Golden Fleece. People born under the sign of Aries are seemingly meek, like this animal, but at the decisive moment they are capable of bold deeds.
Taurus is a kind and at the same time violent animal. The origin of the name of this sign is associated with the legend of Jupiter and Europe. The loving god fell in love with a beautiful girl, in order to conquer her he turned into a beautiful snow-white bull. Europe began to caress the animal, climbed onto its back. And the insidious Jupiter took her to the island of Crete.
The twins are the personification of the myth of the brotherly love of Pollux and Castor, who were ready to die for each other. According to legend, during the battle, Castor was wounded and died in the arms of his brother, Pollux was immortal and turned to his father Zeus to let him die with his brother.
A giant crayfish dug its claws into Hercules' leg during his battle with the Hydra. He crushed the cancer and continued the battle with the snake, but Juno (it was on her orders that the cancer attacked Hercules) was grateful to him and placed the image of the cancer along with other heroes.
The Nemean lion is a terrible and formidable animal that has been attacking people for a long time in the name of keeping peace of power. Heracles defeated him. From the point of view of mythology, the lion is an attribute of power. People born under this sign have a sense of pride and great self-respect.
The virgin is mentioned in the ancient Greek myth of the creation of the world. The legend says that Pandora (the first woman) brought to earth a box that she was forbidden to open, but she could not resist the temptation and opened the lid. All misfortunes, hardships, grief and human vices scattered from the box. After that, the Gods left the earth, the last to fly away was the goddess of innocence and purity, Astrea (Virgo), and the constellation was named after her.
The name of the zodiac sign Libra is associated with the myth of the goddess of justice Themis, who had a daughter, Dika. The girl weighed the actions of people, and her scales became the symbol of the sign.
The scorpion, according to one of the legends, stung Orion, who was trying to rape the goddess Diana. After the death of Orion, Jupiter placed him and among the stars.
Sagittarius is a centaur. According to ancient Greek myths, this is a half-horse, half-man. In the myth of the centaur Chiron, the protagonist knew everything and everything, taught the gods sports, the art of healing and other knowledge and skills that they were supposed to possess.
Capricorn is an animal with powerful hooves, which is able to climb mountain steeps, clinging to ledges. In ancient Greece, it was associated with Pan (the god of nature), who was half man, half goat.
The sign Aquarius is named after a young man named Ganymede, who worked as a cupbearer and treated earthly people at holidays and celebrations. The young man had excellent human qualities, was a great friend, conversationalist and just a decent person. For this, Zeus made him the butler of the gods.
The last sign of the zodiac is Pisces. The appearance of its name is associated with the myth of Eros and Aphrodite. The goddess was walking with her son along the coast and they were attacked by the monster Typhon. To save them, Jupiter turned Eros and Aphrodite into fish, which jumped into the water and disappeared into the sea.
Casting fractions to the least denominator called differently by abbreviation fractions. If as a result of mathematical operations you get a fraction with large numbers in the numerator and denominator, check if it can be reduced.
Examples:
monomials \(2\) \(x\) and \(5\) \(x\)- are similar, since both there and there the letters are the same: x;
the monomials \(x^2y\) and \(-2x^2y\) are similar, since the letters are the same both there and there: x squared multiplied by y. The fact that there is a minus sign in front of the second monomial does not matter, it just has a negative numerical factor ();
the monomials \(3xy\) and \(5x\) are not similar, since in the first monomial the literal factors x and y are, and in the second only x;
the monomials \(xy3yz\) and \(y^2 z7x\) are similar. However, to see this, it is necessary to bring the monomials to . Then the first monomial will look like \(3xy^2z\), and the second like \(7xy^2z\) - and their similarity will become obvious;
the monomials \(7x^2\) and \(2x\) are not similar, since in the first monomial the literal factors x are squared (that is, \(x x\)) , and in the second there is just one x.
How such terms are defined does not need to be memorized, it is better to simply understand. Why are \(2x\) and \(5x\) called similar? But think about it: \(2x\) is the same as \(x+x\), and \(5x\) is the same as \(x+x+x+x+x\). That is, \(2x\) is "two x", and \(5x\) is "five x". And there, and there in the basis - the same (similar): x. Just a different "number" of these Xs.
Another thing, for example, \(5x\) and \(3xy\). Here, the first monomial is essentially "five x's", but the second one is "three x\(·\)games" (\(3xy=xy+xy+xy\)). Basically, it's not the same, it's not the same.
Reduction of similar terms
The process of replacing the sum or difference of similar terms with one monomial is called " reduction of like terms».
At the same time, we note that if the terms are not similar, then it will not be possible to reduce them. For example, you cannot add \(2x^2\) and \(3x\) in, they are different!
Understand, fold not such terms are the same as adding rubles to kilograms: it will turn out to be complete nonsense.
Reducing like terms is a very common step in simplifying the expressions and , as well as in solving and . Let's see a specific example of applying the acquired knowledge.
Example. Solve the equation \(7x^2+3x-7x^2-x=6\)
Answer: \(3\)
Each time it is not necessary to rewrite the equation so that similar ones stand side by side, you can bring them right away. Here it was done for clarity of further transformations.
Let an expression be given that is the product of a number and letters. The number in this expression is called coefficient. For example:
in the expression, the coefficient is the number 2;
in expression - number 1;
in an expression, this is the number -1;
in the expression, the coefficient is the product of the numbers 2 and 3, that is, the number 6.
Petya had 3 sweets and 5 apricots. Mom gave Petya 2 more sweets and 4 apricots (see Fig. 1). How many sweets and apricots did Petya have in total?
Rice. 1. Illustration for the problem
Solution
Let's write the condition of the problem in the following form:
1) There were 3 sweets and 5 apricots:
2) Mom gave 2 sweets and 4 apricots:
3) That is, Petya has everything:
4) We add sweets with sweets, apricots with apricots:
Therefore, there are 5 sweets and 9 apricots in total.
Answer: 5 sweets and 9 apricots.
In Problem 1, in the fourth step, we dealt with the reduction of similar terms.
Terms that have the same letter part are called similar terms. Similar terms can differ only in their numerical coefficients.
To add (reduce) like terms, you need to add their coefficients and multiply the result by the common letter part.
By reducing like terms, we simplify the expression.
They are similar terms, since they have the same letter part. Therefore, to reduce them, it is necessary to add all their coefficients - these are 5, 3 and -1 and multiply by the common letter part - this is a.
2)
This expression contains like terms. The common letter part is xy, and the coefficients are 2, 1 and -3. Here are these similar terms:
3)
In this expression, similar terms are and , let's bring them:
4)
Let's simplify this expression. To do this, we find similar terms. There are two pairs of similar terms in this expression - these are and , and .
Let's simplify this expression. To do this, open the brackets using the distribution law:
There are similar terms in the expression - this and , let's give them:
In this lesson, we got acquainted with the concept of a coefficient, learned which terms are called similar, and formulated the rule for reducing similar terms, and we also solved several examples in which we used this rule.
Bibliography
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. M.: Mnemosyne, 2012.
- Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. M.: Gymnasium, 2006.
- Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. Moscow: Education, 1989.
- Rurukin A.N., Tchaikovsky I.V. Tasks for the course of mathematics grade 5-6. M.: ZSh MEPhI, 2011.
- Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A guide for students in grade 6 of the MEPhI correspondence school. - M.: ZSh MEPhI, 2011.
- Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-interlocutor for 5-6 grades of high school. M .: Education, Mathematics Teacher Library, 1989.
Homework
- Internet portal Youtube.com ( ).
- Internet portal For6cl.uznateshe.ru ().
- Internet portal Festival.1september.ru ().
- Internet portal Cleverstudents.ru ().
Example 1 Let's open the brackets in the expression - 3 * (a - 2b).
Solution. We multiply - 3 by each of the terms a and - 2b. We get - 3 * (a - 2b) \u003d - 3 * a + (- 3) * (- 2b) \u003d - 3a + 6b.
Example 2 Let's simplify the expression 2m - 7m + 3m.
Solution. In this expression, all terms have a common factor m. Hence, by the distributive property of multiplication, 2m - 7m + Зm = m (2 - 7 + 3). The amount in brackets coefficients all terms. It is equal to -2. Therefore 2m - 7m + 3m = -2m.
In the expression 2 m - 7 m + 3m, all terms have a common letter part and differ from each other only by coefficients. Such terms are called similar.
Terms that have the same letter part are called similar terms.
Similar terms can differ only by coefficients.
To add (or say: bring) like terms, you need to add their coefficients and multiply the result by the common letter part.
Example 3 We present similar terms in the expression 5a + a -2a.
Solution. In this sum, all terms are similar, since they have the same letter part a. Let's add the coefficients: 5 + 1 - 2 = 4. So, 5a + a - 2a = 4a.
What terms are called similar terms? How can similar terms differ from each other? Based on what property of multiplication is the reduction (addition) of like terms performed?
1265. Expand the brackets:
a) (a-b + c) * 8; e) (3m-2k + 1)*(-3);
b) -5*(m - n - k); f) - 2a*(b+2c-3m);
c) a*(b - m + n); g) (-2a + 3b + 5c) * 4m;
d) - a*(6b - 3c + 4); h) - a*(3m + k - n).
1266. Perform actions by applying the distribution property multiplication:
1267. Add like terms:
Expressions like 7x-3x+6x-4x read like this:
- the sum of seven x, minus three x, six x and minus four x
- seven x minus three x plus six x minus four x
1268. Reduce like terms:
1269. Open the brackets and give like terms:
1270. Find the value of the expression:
1271. Decide the equation:
a) 3*(2x + 8)-(5x+2)=0; c) 8*(3-2x)+5*(3x + 5)=9.
b) - 3*(3y + 4)+4*(2y -1)=0;
1272. A kilogram of potatoes costs 20 kopecks, and a kilogram of cabbage costs 14 kopecks. Potatoes were bought 3 kg more than cabbages. They paid 1 for everything. 62 k. How many kilos of potatoes and how many cabbages did they buy?
1273. A tourist walked 3 hours and rode a bicycle for 4 hours. In total, he traveled 62 km. At what speed did he walk if he walked 5 km/h slower on foot than he rode a bicycle?
1274. Calculate orally:
1275. What is the sum of a thousand terms, each of which is equal to -1? What is the product of a thousand factors, each of which is -1?
1276. Find the value of the expression
1-3 + 5-7 + 9-11+ ... + 97-99.
1277. Orally solve the equation:
a) x + 4=0; c) m + m + m = 3m;
b) a+3=a -1; d) (y-3)(y + 1)=0.
1278. Multiply: 
1279. What is the coefficient in each of the expressions:
1280. The distance from Moscow to Nizhny Novgorod is 440 km. What should be the scale of the map so that on it this distance has a length of 8.8 cm?
1285. Solve the problem:
1) The combine operator overfulfilled the plan by 15% and harvested grain on an area of 230 hectares. How many hectares, according to the plan, should the combine harvester harvest?
2) A team of carpenters spent 4.2 m3 of planks to renovate the building. At the same time, she saved 16% of the boards allocated for repair. How many cubic meters of boards were allocated for the renovation of the building?
1286. Find the value of the expression:
1) - 3,4 7,1 - 3,6 6,8 + 9,7 8,6; 2) -4,1 8,34+2,5 7,9-3,9 4,2.
1287. Use the graph to solve the problem: “Marina, Larisa, Zhanna and Katya can play on different instruments (piano, cello, guitar, violin), but each only on one. They also know foreign languages (English, French, German, Spanish), but each only one. Known:
1) the girl who plays the guitar speaks Spanish;
2) Larisa plays neither the violin nor the cello and does not know English;
3) Marina does not play the violin or the cello and does not know either German or English;
4) a girl who speaks German does not play the cello;
5) Jeanne knows French, but does not play the violin. Who plays what instrument and what foreign language does he know?”
1288. Expand the brackets:
a) (x+y-z)*3; d) (2x-y+3)*(-2);
b) 4*(m-n-p); e) (8m-2n+p)*(-1);
c) - 8 * (a - b-c); e) (a + 5- b-c) * m.
1289. Find the value of the expression by applying the distributive property of multiplication:
1290. Give like terms:
1291. Open the brackets and give like terms:
1292. Solve the equation:
1293. Bought one table and 6 chairs for 67 rubles. The chair is cheaper than the table by 18 rubles. How much is a chair and how much is a table?
1294. There are 119 students in three classes. There are 4 more students in the first grade than in the second grade and 3 fewer than in the third grade. How many students are in each class?
1295. Determine the scale of the map if the distance between two points on the ground is 750 m, and on the map 25 mm.
1296. What is the length of the segment shown on the map at a distance of 6.5 km, if the scale of the map is 1:25,000?
1297. On the map, a segment has a length of 12.6 cm. What is the length of this segment on the ground if the map scale is 1: 150,000?
N.Ya.Vilenkin, A.S. Chesnokov, S.I. Schwarzburd, V.I. Zhokhov, Mathematics for grade 6, Textbook for high school
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