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 may differ only in 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 distributive 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. Distance from Moscow to Nizhny Novgorod 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 boards was 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 the different instruments(piano, cello, guitar, violin), but each on only 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 does not play either the violin or the cello and does not know in 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) Zhanna knows French but does not play the violin. Who plays what instrument and what foreign language knows?"
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
Mathematics for grade 6 free download, lesson plans, getting ready for school online
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 get a look specific example application of 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.
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. Most in a simple way, which helps out in any situation, is the reduction of polynomials to the 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 participants 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 turn. So, standard view 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 for a long time attacked people in the name of keeping the 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 it's half horse, half man. In the myth of the centaur Chiron main character knew everything and about 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. V Ancient Greece associated with Pan (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.
Let an expression be given that is the product of a number and letters. The number in this expression is called coefficient. For instance:
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 ().
Simple mathematical operations - addition, subtraction, multiplication, and so on - do not cause much difficulty for students. There is simply nothing to be confused about here. However, it happens that the expression from the problem has a very long alphanumeric entry. It diverts attention, confuses the train of thought, and most importantly, most often leads a person away from the simplest solution.
It was to simplify mathematical operations that special concepts were invented - for example, like terms. What is meant by this term, and how can the principle of similarity be used?
What terms and in what expressions are considered similar?
The expression as such must consist of letters either from letters and numbers - and of course, it must have addition, because we are talking specifically about the terms. At the same time, in order to be able to talk about similarity, individual terms must have the same letter in their composition.
For example, let's analyze a small expression 2a + 3c + 4a. The first and third parts of the expression contain the same letter "a". Accordingly, according to this feature, they are similar terms.
What gives us this understanding in practice?
In order to solve the above expression, you can go in two ways:
- Find the product 2 * a, add the product 3 * c to it, add the product 4 * a to the sum. It's not that difficult - but the longer the expression, the more tedious the calculations become.
- Use the properties of similar terms and first bring the expression into a simpler and more convenient form in order to find a solution faster.
For any task, it is preferable to choose the second method - it saves time and reduces the possibility of making a mistake.
What does the term "reduction" mean for like terms?
This is a permutation of the terms in such a way that similar ones are next to each other. Of more early rules we remember that it does not matter in what order the terms of the expression are added, the sum still turns out to be the same.
Thus, our example can be transformed as follows - write it as 2a + 4a + 3c. But that's not all. For simplicity, the numerical coefficients can be taken in brackets and added separately - and the letter “a” is left out of brackets for now.
It will look like this (2 + 4)a + 3c = (6)a + 3c = 6a + 3c. We no longer need to separately calculate the product for each of these terms - we can first add them together, and only then multiply in the resulting result.