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Combining Like Terms Using Addition and Subtraction

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: 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 (Reference), 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.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.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 x represent "record." Then, 8 x+ 5 x = 13 x8 x+ 5 x = 13 x size 12{"8x "+" 5x "=" 13x"} {}. The terms 8 x8 x size 12{"8x"} {} and 5 x5 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 x represent "record" and y y represent "tape." Then,

    8x + 5x + 3y = 13x + 5y 8x + 5x + 3y = 13x + 5y 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.

Example 1

2m+6m-4m2m+6m-4m. All three terms are alike. Combine their coefficients and affix this result to mm: 2+6-4=42+6-4=4.

Thus, 2m+6m-4m=4m2m+6m-4m=4m .

Example 2

5x+2y9y.5x+2y9y. size 12{5x+2y-9y "." } {} The terms 2y2y size 12{2y} {} and 9y9y size 12{-9y} {} are like terms. Combine their coefficients: 29= -729= -7 size 12{2-9"=-"7} {}.

Thus, 5x+2y9y=5x7y5x+2y9y=5x7y size 12{5x+2y-9y=5x-7y} {} .

Example 3

- 3a+2b5a+a+6b. - 3a+2b5a+a+6b. size 12{-3a+2b-5a+a+6b "." } {} The like terms are

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

Thus, 3a+2b5a+a+6b= - 7a+8b.3a+2b5a+a+6b= - 7a+8b. size 12{-3a+2b-5a+a+6b"=-"7a+8b "." } {}

Example 4

r2s+7s+3r4r5sr2s+7s+3r4r5s 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, r2s+7s+3r4r5s=0r2s+7s+3r4r5s=0 size 12{r-2s+7s+3r-4r-5s=0} {} .

Practice Set A

Simplify each expression by combining like terms.

Exercise 1

4x+3x+6x4x+3x+6x size 12{4x+3x+6x} {}

Solution

13x13x size 12{"13"x} {}

Exercise 2

5a+8b+6a2b5a+8b+6a2b size 12{5a+8b+6a-2b} {}

Solution

11a+6b11a+6b size 12{"11"a+6b} {}

Exercise 3

10m6n2nm+n10m6n2nm+n size 12{"10"m-6n-2n-m+n} {}

Solution

9m7n9m7n size 12{9m-7n} {}

Exercise 4

16a+6m+2r3r18a+m7m16a+6m+2r3r18a+m7m size 12{"16"a+6m+2r-3r-"18"a+m-7m} {}

Solution

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

Exercise 5

5h8k+2h7h+3k+5k5h8k+2h7h+3k+5k size 12{5h-8k+2h-7h+3k+5k} {}

Solution

0

Exercises

Simplify each expression by combining like terms.

Exercise 6

4a+7a4a+7a size 12{4a+7a} {}

Solution

11a11a size 12{"11"a} {}

Exercise 7

3m+5m3m+5m size 12{3m+5m} {}

Exercise 8

6h2h6h2h size 12{6h-2h} {}

Solution

4h4h size 12{4h} {}

Exercise 9

11k8k11k8k size 12{"11"k-8k} {}

Exercise 10

5m+3n2m5m+3n2m size 12{5m+3n-2m} {}

Solution

3m+3n3m+3n size 12{3m+3n} {}

Exercise 11

7x6x+3y7x6x+3y size 12{7x-6x+3y} {}

Exercise 12

14s+3s8r+7r14s+3s8r+7r size 12{"14"s+3s-8r+7r} {}

Solution

17sr17sr size 12{"17"s-r} {}

Exercise 13

5m3n+2m+6n5m3n+2m+6n size 12{-5m-3n+2m+6n} {}

Exercise 14

7h+3a10k+6a2h5k3k7h+3a10k+6a2h5k3k size 12{7h+3a-"10"k+6a-2h-5k-3k} {}

Solution

5h+9a18k5h+9a18k size 12{5h+9a-"18"k} {}

Exercise 15

4x8y3z+xyz3y2z4x8y3z+xyz3y2z size 12{4x-8y-3z+x-y-z-3y-2z} {}

Exercise 16

11w+3x6w5w+8x11x11w+3x6w5w+8x11x size 12{"11"w+3x-6w-5w+8x-"11"x} {}

Solution

0

Exercise 17

15r6s+2r+8s6r7ss2r15r6s+2r+8s6r7ss2r size 12{"15"r-6s+2r+8s-6r-7s-s-2r} {}

Exercise 18

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

Solution

16m16m size 12{"16"m} {}

Exercise 19

2x+8x+10x2x+8x+10x size 12{ lline -2 rline x+ lline -8 rline x+ lline "10" rline x} {}

Exercise 20

- 4+1k+63k+124h+5+2k - 4+1k+63k+124h+5+2k size 12{ left (-4+1 right )k+ left (6-3 right )k+ left ("12"-4 right )h+ left (5+2 right )k} {}

Solution

8h+7k8h+7k size 12{8h+7k} {}

Exercise 21

- 5+3a2+5b3+8b - 5+3a2+5b3+8b size 12{ left (-5+3 right )a- left (2+5 right )b- left (3+8 right )b} {}

Exercise 22

5+2Δ+3Δ-85+2Δ+3Δ-8

Solution

5Δ-35Δ-3

Exercise 23

9+10-11-129+10-11-12

Exercise 24

16x12y+5x+75x163y16x12y+5x+75x163y size 12{"16"x-"12"y+5x+7-5x-"16"-3y} {}

Solution

16x15y916x15y9 size 12{"16"x-"15"y-9} {}

Exercise 25

3y+4z113z2y+54833y+4z113z2y+5483 size 12{-3y+4z-"11"-3z-2y+5-4 left (8-3 right )} {}

Exercises for Review

Exercise 26

((Reference)) Convert 24112411 size 12{ { {"24"} over {"11"} } } {} to a mixed number

Solution

22112211 size 12{2 { {2} over {"11"} } } {}

Exercise 27

((Reference)) Determine the missing numerator: 38=?64.38=?64. size 12{ { {3} over {8} } = { {?} over {"64"} } "." } {}

Exercise 28

((Reference)) Simplify 56141125614112 size 12{ { { { {5} over {6} } - { {1} over {4} } } over { { {1} over {"12"} } } } } {}.

Solution

7

Exercise 29

((Reference)) Convert 516516 size 12{ { {5} over {"16"} } } {} to a percent.

Exercise 30

((Reference)) In the expression 6 k6 k size 12{6k} {}, how many k’s are there?

Solution

6

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