11.2: Combining Like Terms Using Addition and Subtraction
- Page ID
- 48900
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
Learning Objectives
- be able to combine like terms in an algebraic expression
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.
\(\text{8 records + 5 records = 13 records.}\)
Eight and 5 of the same type give 13 of that type. We have combined quantities of the same type.
\(\text{8 records + 5 records + 3 tapes = 13 records + 3 tapes.}\)
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.
Suppose we let the letter \(x\) represent "record." Then, \(8x + 5x = 13x\). The terms \(8x\) and \(5x\) are like terms. So, 8 and 5 of the same type give 13 of that type. We have combined like terms.
Suppose we let the letter \(x\) represent "record" and \(y\) represent "tape." Then,
\(8x + 5x + 3y = 13x + 5y\)
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.
\(2m + 6m - 4m\). All three terms are alike. Combine their coefficients and affix this result to \(m\): 2 + 6 - 4 = 4.
Thus, \(2m + 6m - 4m = 4m\).
Sample Set A
\(5x + 2y - 9y\). The terms \(2y\) and \(-9y\) are like terms. Combine their coefficients: 2 - 9 = -7.
Thus, \(5x + 2y - 9y = 3x - 7y\).
Sample Set A
\(-3a + 2b - 5a + a + 6b\). The like terms are
\(\underbrace{-3a, -5a, a}_{\begin{array} {c} {-3 - 5 + 1 = -7} \\ {-7a} \end{array}}\)\(\underbrace{2b,6b}_{\begin{array} {c} {2 + 6 = 8} \\ {8b} \end{array}}\)
Thus, \(-3a + 2b - 5a + a + 6b = -7a + 8b\).
Sample Set A
\(r - 2s + 7s + 3r - 4r - 5s\). The like terms are
Thus, \(r - 2s + 7s + 3r - 4r - 5s = 0\).
Practice Set A
Simplify each expression by combining like terms.
\(4x + 3x + 6x\)
- Answer
-
\(13x\)
Practice Set A
\(5a + 8b + 6a - 2b\)
- Answer
-
\(11a + 6b\)
Practice Set A
\(10m - 6n - 2n -m +n\)
- Answer
-
\(9m - 7n\)
Practice Set A
\(16a + 6m + 2r - 3r - 18a + m - 7m\)
- Answer
-
\(-2a - r\)
Practice Set A
\(5h - 8k + 2h - 7h + 3k + 5k\)
- Answer
-
0
Exercises
Simplify each expression by combining like terms.
Exercise \(\PageIndex{1}\)
\(4a + 7a\)
- Answer
-
\(11a\)
Exercise \(\PageIndex{2}\)
\(3m + 5m\)
Exercise \(\PageIndex{3}\)
\(6h - 2h\)
- Answer
-
\(4h\)
Exercise \(\PageIndex{4}\)
\(11k - 8k\)
Exercise \(\PageIndex{5}\)
\(5m + 3n - 2m\)
- Answer
-
\(3m + 3n\)
Exercise \(\PageIndex{6}\)
\(7x - 6x + 3y\)
Exercise \(\PageIndex{7}\)
\(14s + 3s - 8r + 7r\)
- Answer
-
\(17s - r\)
Exercise \(\PageIndex{8}\)
\(-5m - 3n + 2m + 6n\)
Exercise \(\PageIndex{9}\)
\(7h + 3a - 10k + 6a - 2h - 5k - 3k\)
- Answer
-
\(5h + 9a - 18k\)
Exercise \(\PageIndex{10}\)
\(4x - 8y - 3z + x - y - z - 3y - 2z\)
Exercise \(\PageIndex{11}\)
\(11 w + 3x - 6w - 5w + 8x - 11x\)
- Answer
-
0
Exercise \(\PageIndex{12}\)
\(15r - 6s + 2r + 8s - 6r - 7s - s - 2r\)
Exercise \(\PageIndex{13}\)
\(|-7|m + |6|m + |-3|m\)
- Answer
-
\(16m\)
Exercise \(\PageIndex{14}\)
\(|-2|x + |-8|x + |10|x\)
Exercise \(\PageIndex{15}\)
\((-4 + 1)k + (6 - 3)k + (12 - 4)h + (5 + 2)k\)
- Answer
-
\(8h + 7k\)
Exercise \(\PageIndex{16}\)
\((-5 + 3)a - (2 + 5) b - (3 + 8)b\)
Exercise \(\PageIndex{17}\)
\(5 \star + 2\Delta + 3\Delta - 8 \star\)
- Answer
-
\(5\Delta - 3 \star\)
Exercise \(\PageIndex{18}\)
9⊠+10⊞−11⊠−12⊞
Exercise \(\PageIndex{19}\)
\(16x - 12y + 5x + 7 - 5x - 16 -3y\)
- Answer
-
\(16x - 15y - 9\)
Exercise \(\PageIndex{20}\)
\(-3y + 4z - 11 - 3z - 2y + 5 - 4(8 - 3)\)
Exercises for Review
Exercise \(\PageIndex{21}\)
Convert \(\dfrac{24}{11}\) to a mixed number
- Answer
-
\(2 \dfrac{2}{11}\)
Exercise \(\PageIndex{22}\)
Determine the missing numerator: \(\dfrac{3}{8} = \dfrac{?}{64}\).
Exercise \(\PageIndex{23}\)
Simplify \(\dfrac{\dfrac{5}{6} - \dfrac{1}{4}}{\dfrac{1}{12}}\)
- Answer
-
7
Exercise \(\PageIndex{24}\)
Convert \(\dfrac[5}{16}\) to a percent.
Exercise \(\PageIndex{25}\)
In the expression \(6k\), how many \(k\)'s are there
- Answer
-
6