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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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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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