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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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
7hours 36 min - 4hours 50 min
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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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