Rules for solving examples with similar terms. Educational and methodological material on algebra (grade 6) on the topic: Similar terms
Example 1. Let's open the brackets in the expression - 3*(a - 2b).
Solution. Let's multiply - 3 by each of the terms a and - 2b. We get - 3*(a - 2b)= - 3*a + (- 3)*(- 2b)= - 3a + 6b.
Example 2. Let's simplify the expression 2m - 7m + 3m.
Solution. In this expression, all terms have a common factor m. This means, according to the distribution property of multiplication, 2m - 7m + Зm = m (2 - 7 + 3). The amount is written in parentheses 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 in coefficients.
To add (or say: bring) similar terms, you need to add their coefficients and multiply the result by the common letter part.
Example 3. Let us 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.
Which terms are called similar? How can similar terms differ from each other? Based on what property of multiplication is reduction (addition) performed? similar terms?
1265. Open the brackets:
a) (a-b+c)*8; e) (3m-2k + 1)*(-3);
b) -5*(m - n - k); e) - 2a*(b+2c-3m);
c) a*(b - m + n); g) (-2a + 3b+5c)*4m;
d) - a*(6b - Зс + 4); h) - a*(3m + k - n).
1266. Do the steps by applying the distributive property multiplication:
1267. Add similar terms:
Expressions of the form 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 similar terms:
1269. Open the brackets and give similar terms:
1270. Find the meaning 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. They bought 3 kg more potatoes than cabbage. They paid 1 ruble for everything. 62 k. How many kilograms of potatoes and how much cabbage did you buy?
1273. The tourist walked for 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 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 equal to -1?
1276. Find the value of the expression
1-3 + 5-7 + 9-11+ ... + 97-99.
1277. Solve the equation orally:
a) x + 4=0; c) m + m + m = 3m;
b) a+3=a -1; d) (y-3)(y + 1)=0.
1278. Perform multiplication: 
1279. What is the coefficient in each of the expressions:
1280. Distance from Moscow to Nizhny Novgorod 440 km. What scale should the map be for this distance to be 8.8 cm long?
1285. Solve the problem:
1) The combine operator exceeded the plan by 15% and harvested grain on an area of 230 hectares. How many hectares is the combine harvester expected to harvest?
2) A team of carpenters used 4.2 m3 of boards to repair the building. At the same time, she saved 16% of the boards allocated for repair. How many cubic meters boards were allocated for the renovation of the building?
1286. Find the meaning 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. Using the graph, solve the problem: “Marina, Larisa, Zhanna and Katya can play on different instruments(piano, cello, guitar, violin), but each only on one. They know foreign languages (English, French, German, Spanish), but each only one. Known:
1) the girl who plays the guitar speaks Spanish;
2) Larisa doesn’t play the violin or cello and doesn’t know in English;
3) Marina does not play the violin or 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 which instrument and which one? foreign language knows?
1288. Open the brackets:
a) (x+y-z)*3; d) (2x-y+3)*(-2);
b) 4*(m-n-р); 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 similar terms:
1291. Open the brackets and give similar terms:
1292. Solve the equation:
1293. Bought one table and 6 chairs for 67 rubles. A chair is 18 rubles cheaper than a table. How much does a chair cost and how much does a table cost?
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 students 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 it is 25 mm.
1296. How long is the distance 6.5 km shown on the map if the map scale is 1: 25,000?
1297. On the map, the 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. Shvartsburd, V.I. Zhokhov, Mathematics for grade 6, Textbook for high school
Mathematics for 6th grade free download, lesson plans, preparing for school online
Examples:
monomials \(2\) \(x\) and \(5\) \(x\)- are similar, since both there and there the letters are the same: x;
monomials \(x^2y\) and \(-2x^2y\) are similar, since in both cases the letters are the same: 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 there are letter factors x and y, and in the second there are only x;
monomials \(xy3yz\) and \(y^2 z7x\) are similar. However, to see this, it is necessary to reduce 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 are x squared (that is, \(x·x\)), and in the second there is simply one x.
There is no need to memorize how such terms are defined; it is better to simply understand. Why are \(2x\) and \(5x\) called similar? Just 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’s”, and \(5x\) is “five x’s”. Both there and there are basically the same (similar): x. Just a different “quantity” of these same X’s.
Another thing is, for example, \(5x\) and \(3xy\). Here the first monomial is essentially “five X’s”, but the second is “three X\(·\)games” (\(3xy=xy+xy+xy\)). At the core – not the same, not similar.
Reducing similar terms
The process of replacing the sum or difference of similar terms with one monomial is called “ reduction of similar terms».
Let us note that if the terms are not similar, then it will not be possible to bring them. For example, adding \(2x^2\) and \(3x\) is impossible, they are different!
Understand, fold Not Such terms are the same as adding rubles and kilograms: it turns out to be complete nonsense.
Bringing similar terms is a very common step in simplifying the expressions and , as well as when solving and . Let's see specific example application of acquired knowledge.
Example. Solve the equation \(7x^2+3x-7x^2-x=6\)
Answer: \(3\)
It is not at all necessary to rewrite the equation every time so that similar ones stand next to each other; you can present them at once. This was done here for clarity of further transformations.
Is . In this article we will give a definition of similar terms, understand what is called reducing similar terms, consider the rules by which this action is performed, and give examples of reducing similar terms with detailed description solutions.
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Definition and examples of similar terms.
A conversation about such terms arises after getting acquainted with literal expressions when there is a need to carry out transformations with them. Based on mathematics textbooks by N. Ya. Vilenkin definition of similar terms is given in 6th grade, and it has the following wording:
Definition.
Similar terms- these are terms that have the same letter part.
It is worth looking carefully at this definition. Firstly, we are talking about terms, and, as you know, terms are constituent elements sums This means that such terms can only be present in expressions that represent sums. Secondly, in the stated definition of such terms there is an unfamiliar concept of “letter 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 the 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, according to the definition, these terms are similar. The numerical coefficients of these similar terms are the numbers 3 and 2.
Another example: in 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 letter part x·y 3 ·z are similar. Note that y 3 is present in the letter part; its presence does not violate the definition of the letter 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 down 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.
Based on this, in the sum 5+7·x−4+2·x+y similar terms are not only 7·x and 2·x, but also the terms without the letter part 5 and −4.
Later, the concept of a letter part expands - I begin to consider not only a product of letters, but an arbitrary letter expression as a letter part. For example, in an algebra textbook for grade 8 by the 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 the terms are similar. The common letter part of these similar terms is the expression with the 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 information presented, we can give the following definition of similar terms.
Definition.
Similar terms are called terms in literal expression, having the same letter part, as well as terms that do not have a letter part, where the letter part is understood as any letter expression.
Separately, we will 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).
At the end of 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 this way: “Are the letter parts x·y and y·x of the indicated terms the same”? The order of the letter 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 (but for the sake of rigor it is better not to do this). In this case, they are guided by this: according to the rearrangement of factors in the product does not affect the result, therefore the original expression 2·x·y+3·y·x can be rewritten as 2·x·y+3·x·y, the terms of which 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.
Bringing similar terms, rules, examples
Converting expressions containing similar terms implies performing the addition of these terms. This action received a special name - reduction of similar terms.
Reduction of similar terms is carried out in three stages:
- first carried out rearrangement of terms so that similar terms are next to each other;
- after that is taken out of brackets letter part of similar terms;
- finally, the value of a numeric expression is calculated, formed in brackets.
Let's look at the recorded steps using an example. Let us present similar terms in the expression 3·x·y+1+5·x·y. First, we rearrange the terms so that similar terms 3 x y and 5 x x y are next to each other: 3 x y+1+5 x y=3 x y+5 x y+1. Secondly, we take the literal part out of brackets and 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 move it to this place: x·y·8+1=8·x·y+1. This completes the reduction of similar terms.
For convenience, the three steps listed above are combined into rule for reducing similar terms: to bring similar terms, you need to add their coefficients and multiply the resulting result by the letter part (if there is one).
The solution to the previous example using the rule for reducing similar 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 bringing these terms 8·x·y. It remains not to forget about term 1 in the original expression, as a result we have 3 x x y+1+5 x x y=8 x x y+1.
Let a expression be given, which appears as a result of numbers and letters. The number in this form is called co-ef-fi-tsi-en-tom. For example:
in the expression of the coefficient, the number 2 appears;
in the expression - number 1;
in the expression, this is the number -1;
in the calculation of the coefficient, it is the result of the numbers 2 and 3, that is, the number 6.
Problem 1
Petya had 3 con-fe-ty and 5 ab-ri-ko-sov. Mom po-da-ri-la Petya 2 more kon-fe-ty and 4 ab-ri-ko-sa (see Fig. 1). How many candies and ab-ri-ko-sovs does Petya have in total?
Rice. 1. Illu-strat-tion to za-da-che
Solution
We write the condition for the problem in this form:
1) There were 3 conf-fe-you and 5 ab-ri-ko-sov:
2) Mom po-da-ri-la 2 con-fe-you and 4 ab-ri-ko-sa:
3) That is, Petya’s total:
4) Warehouses-va-em kon-fe-you with kon-fe-ta-mi, ab-ri-ko-sy with ab-ri-ko-sa-mi:
Next, in total there were 5 candies and 9 ab-ri-ko-sovs.
Answer: 5 candies and 9 ab-ri-ko-sov.
Reducing similar terms
In the fourth act, we for-we-were-at-no-sweetnesses.
Sla-ga-e-my, having the same letter-vein part, are called-by-sla-ga-e-we -mi. Such weak people can only emanate from their own numbers.
In order to add up (pre-ve-sti) similar weaknesses, you need to add up their coefficients and multiply the result by common letter-vein part.
When we eat the same slacks, we simplify you.
Examples of reducing similar terms
They are additionally weak, since they have the same letter part. Next, for their admission it is necessary to add up all their coefficients - these are 5, 3 and -1 and multiplying by the common letter part is a.
2)
In this case you are very weak. The common letter-vein part is xy, and the coefficients are 2, 1 and -3. Let's take these sweet-sweet ones:
3)
In the given you-are-the-extra-we-we-are-we-are and, let's bring them:
4)
Let's simplify this expression. To do this, we need some special slacks. In this expression there are two pairs of similar slurs - these are and , and .
Let's simplify this expression. To do this, we cut out the brackets, using the pre-de-li-tel-law:
There are similar syllables in you - these are and, let's introduce them:
Lesson summary
In this lesson, we got acquainted with the co-ef-fi-tsi-ent, and found out what the weak ones are called -sya in addition to us, and for-mu-li-ro-va-li pra-vi-lo pri-ve-de-niya of the-additional sla-ga-e-my, and also we we decided on several examples, in which the given rule was used.
source of abstract - http://interneturok.ru/ru/school/matematika/6-klass/undefined/privedenie-podobnyh-slagaemyh
video source - http://www.youtube.com/watch?v=GdRqwj5sXzE
video source - http://www.youtube.com/watch?v=z2_XZDtGr3o
video source - http://www.youtube.com/watch?v=qagWrAOPxGI
video source - http://www.youtube.com/watch?v=Ty5DBUIGB5I
video source - http://www.youtube.com/watch?v=t0mOyseNddg
video source - http://www.youtube.com/watch?v=S8DoWa5wrfA
presentation source - http://ppt4web.ru/matematika/podobnye-slagaemye2.html
