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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to combine like terms using addition and subtraction. By the end of the module students should be able to combine like terms in an algebraic expression.

Section overview

  • Combining Like Terms

Combining like terms

From our examination of terms in [link] , we know that like terms are terms in which the variable parts are identical. Like terms is an appropriate name since terms with identical variable parts and different numerical coefficients represent different amounts of the same quantity. When we are dealing with quantities of the same type, we may combine them using addition and subtraction.

Simplifying an algebraic expression

An algebraic expression may be simplified by combining like terms.

This concept is illustrated in the following examples.

  1. 8 records + 5 records = 13 records . size 12{"8 records "+" 5 records "=" 13 records" "." } {}

    Eight and 5 of the same type give 13 of that type. We have combined quantities of the same type.

  2. 8 records + 5 records + 3 tapes = 13 records + 3 tapes . size 12{"8 records "+" 5 records "+" 3 tapes "=" 13 records "+" 3 tapes" "." } {}

    Eight and 5 of the same type give 13 of that type. Thus, we have 13 of one type and 3 of another type. We have combined only quantities of the same type.

  3. Suppose we let the letter x represent "record." Then, 8 x + 5 x = 13 x size 12{"8x "+" 5x "=" 13x"} {} . The terms 8 x size 12{"8x"} {} and 5 x size 12{"5x"} {} are like terms. So, 8 and 5 of the same type give 13 of that type. We have combined like terms.
  4. Suppose we let the letter x represent "record" and y represent "tape." Then,

    8 x + 5 x + 3 y = 13 x + 5 y size 12{"8x "+" 5x "+" 6y "=" 13x "+" 5y"} {}

    We have combined only the like terms.

After observing the problems in these examples, we can suggest a method for simplifying an algebraic expression by combining like terms.

Combining like terms

Like terms may be combined by adding or subtracting their coefficients and affixing the result to the common variable.

Sample set a

Simplify each expression by combining like terms.

2 m + 6 m - 4 m . All three terms are alike. Combine their coefficients and affix this result to m : 2 + 6 - 4 = 4 .

Thus, 2 m + 6 m - 4 m = 4 m .

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5 x + 2 y 9 y . size 12{5x+2y-9y "." } {} The terms 2 y size 12{2y} {} and 9 y size 12{-9y} {} are like terms. Combine their coefficients: 2 9 = - 7 size 12{2-9"=-"7} {} .

Thus, 5 x + 2 y 9 y = 5 x 7 y size 12{5x+2y-9y=5x-7y} {} .

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- 3 a + 2 b 5 a + a + 6 b . size 12{-3a+2b-5a+a+6b "." } {} The like terms are

- 3 a , - 5 a , a - 3 - 5 + 1 = - 7 - 7 a 2 b , 6 b 2 + 6 = 8 8 b

Thus, 3 a + 2 b 5 a + a + 6 b = - 7 a + 8 b . size 12{-3a+2b-5a+a+6b"=-"7a+8b "." } {}

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r 2 s + 7 s + 3 r 4 r 5 s size 12{r-2s+7s+3r-4r-5s} {} . The like terms are

Two bracketed lists. The first list is r, 3r, and -4r. Below this is the equation, 1+3-4=0. Below this is the expression, 0r. The second list is -2s, 7s, and -5s. Below this is the equation -2+7-5=0. Below this is the expression, 0s. The results of the two lists can be simplified to 0r + 0s = 0.

Thus, r 2 s + 7 s + 3 r 4 r 5 s = 0 size 12{r-2s+7s+3r-4r-5s=0} {} .

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Practice set a

Simplify each expression by combining like terms.

4 x + 3 x + 6 x size 12{4x+3x+6x} {}

13 x size 12{"13"x} {}

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5 a + 8 b + 6 a 2 b size 12{5a+8b+6a-2b} {}

11 a + 6 b size 12{"11"a+6b} {}

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10 m 6 n 2 n m + n size 12{"10"m-6n-2n-m+n} {}

9 m 7 n size 12{9m-7n} {}

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16 a + 6 m + 2 r 3 r 18 a + m 7 m size 12{"16"a+6m+2r-3r-"18"a+m-7m} {}

- 2 a r size 12{-2a-r} {}

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5 h 8 k + 2 h 7 h + 3 k + 5 k size 12{5h-8k+2h-7h+3k+5k} {}

0

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Exercises

Simplify each expression by combining like terms.

4 a + 7 a size 12{4a+7a} {}

11 a size 12{"11"a} {}

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3 m + 5 m size 12{3m+5m} {}

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6 h 2 h size 12{6h-2h} {}

4 h size 12{4h} {}

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11 k 8 k size 12{"11"k-8k} {}

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5 m + 3 n 2 m size 12{5m+3n-2m} {}

3 m + 3 n size 12{3m+3n} {}

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7 x 6 x + 3 y size 12{7x-6x+3y} {}

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14 s + 3 s 8 r + 7 r size 12{"14"s+3s-8r+7r} {}

17 s r size 12{"17"s-r} {}

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5 m 3 n + 2 m + 6 n size 12{-5m-3n+2m+6n} {}

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7 h + 3 a 10 k + 6 a 2 h 5 k 3 k size 12{7h+3a-"10"k+6a-2h-5k-3k} {}

5 h + 9 a 18 k size 12{5h+9a-"18"k} {}

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4 x 8 y 3 z + x y z 3 y 2 z size 12{4x-8y-3z+x-y-z-3y-2z} {}

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11 w + 3 x 6 w 5 w + 8 x 11 x size 12{"11"w+3x-6w-5w+8x-"11"x} {}

0

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15 r 6 s + 2 r + 8 s 6 r 7 s s 2 r size 12{"15"r-6s+2r+8s-6r-7s-s-2r} {}

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- 7 m + 6 m + 3 m size 12{ lline -7 rline m+ lline 6 rline m+ lline -3 rline m} {}

16 m size 12{"16"m} {}

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2 x + 8 x + 10 x size 12{ lline -2 rline x+ lline -8 rline x+ lline "10" rline x} {}

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- 4 + 1 k + 6 3 k + 12 4 h + 5 + 2 k size 12{ left (-4+1 right )k+ left (6-3 right )k+ left ("12"-4 right )h+ left (5+2 right )k} {}

8 h + 7 k size 12{8h+7k} {}

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- 5 + 3 a 2 + 5 b 3 + 8 b size 12{ left (-5+3 right )a- left (2+5 right )b- left (3+8 right )b} {}

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5 + 2 Δ + 3 Δ - 8

5 Δ - 3

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9 + 10 - 11 - 12

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16 x 12 y + 5 x + 7 5 x 16 3 y size 12{"16"x-"12"y+5x+7-5x-"16"-3y} {}

16 x 15 y 9 size 12{"16"x-"15"y-9} {}

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3 y + 4 z 11 3 z 2 y + 5 4 8 3 size 12{-3y+4z-"11"-3z-2y+5-4 left (8-3 right )} {}

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Exercises for review

( [link] ) Convert 24 11 size 12{ { {"24"} over {"11"} } } {} to a mixed number

2 2 11 size 12{2 { {2} over {"11"} } } {}

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( [link] ) Determine the missing numerator: 3 8 = ? 64 . size 12{ { {3} over {8} } = { {?} over {"64"} } "." } {}

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( [link] ) Simplify 5 6 1 4 1 12 size 12{ { { { {5} over {6} } - { {1} over {4} } } over { { {1} over {"12"} } } } } {} .

7

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( [link] ) Convert 5 16 size 12{ { {5} over {"16"} } } {} to a percent.

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( [link] ) In the expression 6 k size 12{6k} {} , how many k ’s are there?

6

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Questions & Answers

what are free good
Maillot Reply
how do you determine price change
Matri Reply
what is economics?
Yaya Reply
what is economic
Nana Reply
Economics is the study of how Individual consumer, institution and society as a whole uses its available finite resources to satisfy infinite needs and wants
Richard
Explain the following concepts using suitable exemple. 1) National budget. 2) National debt
Rosalie
what is international trade
BOBO Reply
other things remain constant.
Esale Reply
explain scarcity
Richard Reply
scarcity occurs when there are not enough resources to satisfy human's needs and wants therefore we need to allocate our resources using the price mechanism.
Isha
scarcity is when there is inadequate resources to catch the unlimited wants which would compel individual to make choice.
Prince
scarcity simply means when there's a shortages of resources to satisfy Hunan's need and wants in a particular time, which means the demand for it at the moment is higher than the supply
Abu
scarcity simply means when there's a shortages of resources to satisfy humans need and wants in a particular time, which means the demand for it at the moment is higher than it supply.
Abu
😂😂😂
Isha
That escalated real quick😂
Isha
scarcity is sometimes considered as the basic problems of economics resources r scarce because we live in a world of humans in which wants are infinite but the land labor and capital r required to satisfy those wants are limited
Nkwenti
scarcity means unlimited resources
Rena
resources are limited but human wants can not be limited
Himanshu
joint or complementary demand
Ryt Reply
what is demand
Qudus Reply
it maybe define as the amount or quantity of goods and services which a consumer is willing to buy with the ability to pay at a given price at a particular time
Habib
yesoo thanks dear
Gyamfua
Explain 3 reasons why the manufacturer may decide to sell directly to the consumers
Rosalie
why is economics a science
Isaac Reply
Because science is all about thinking by making models whether a computational or Mathematical. Economics is a social sciences because it effects society but to understand Economics we use maths so it is a Science
Amit
hi
Mony
hello
Amit
I hope.......Economic is social science because it makes new new currency of money,it is decided the country’s depend system and the system be repeated others benefits in our ...
Anik
yeah
Habib
so what is the disadvantages of mix economic system
Habib
Economics is regarded as a social science because it uses scientific methods to build theories that can help explain the behaviour of individuals, groups and organisations.
Ali
The question is: why is Economic a "science" and not why is economics a "social science?" Alright folks?
Xavier
In my own understanding of why economics is a science it bcz it deals mainly on human resources just like biology that deals in the human body why economics is science it also deals on the management of human resources all over the world bcz without economics there will be no human resources
Ogbonnaya
what is technology
Nkwenti
my response to the earlier question is, economics is a science but not a pure science like biology, chemistry and physics. The reason is that those pure science study inanimate object while economics study human being, their experiment are predictable.
raimi
Economics is a social science subject that shows the relationship between ends and scarce means with their alternative uses
Ajenifuja
what is Equilibrium?
Fatima Reply
it means equal price and equal quality
Arthur
thank u Arthur!
Fatima
😇🙏
Fatima
Thanks
Moses
Equilibrium is a state of balance in an economy. In as far as market forces are reasonably concerned, equilibrium means the state at which the quantity of goods supplied is equal to the quantity of goods demanded.
Xavier
what is labour
Ayham
labor can be define as a both physical and mental effort of man put forward towards production
Habib
name the types of demand and explain any two
ALIMAMYISLA
Joint demand Composite demand Competitive demand
Ajenifuja
Labourcan be defined as man mental and physical exertion
Ajenifuja
equilibrium is a state of balance especially between opposing forces or influences
Nkwenti
equilibrium means the state of point in which a person is satisfied and after that point if that person consumes more than the satisfaction level will decrease.
Himanshu
Equilibrium is a state of balance in and economy. that mean equal privé and equal quality e.t.c.
Rosalie
Equilibrium is a situation in which economic forces such as supply and demand balance and in the absence of extremal lnfiuences the values of economic variables will not change.
Gyamfua
what is elasticity
Motseoa Reply
difference between demand and supply
Adeyemi Reply
Demand- It is the desire of a buyer and his ability to pay for a particular commodity at a specific price. Supply- It is quantity of a commodity which is made available by the producers to its consumers at certain price.
Ali
yes OK thank you dear
Gyamfua
Demand can be defined as the ability a buyer is willing and able to pay at a specific price and in agiven period of time Supply can be defined as the ability the producer is willing to supply with a specific price
Ajenifuja
thanks
antony
what is labor force
Habib
demand represents the consumer while supply represents the firm
Nkwenti
restriction on international trade
Ayim Reply
formula for price elasticity of demand
Lognyuu Reply
7hours 36 min - 4hours 50 min
Tanis Reply

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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