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Section 1.2 Combining Like Terms

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permalinkIn Section 1.1, we worked with algebraic expressions. Algebraic expressions can be large and complicated, and anything we can do to write the same expression in a simplified form is helpful. The most basic skill for simplfying an algebraic expression is finding parts of the expression that have a certain something in common that allows them to be combined into one. Combining like terms is the topic of this section.

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Figure 1.2.1. Alternative Video Lesson
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Subsection 1.2.1 Identifying Terms

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Definition 1.2.2.

In an algebraic expression, the terms are quantities being added together.

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Example 1.2.3.

List the terms in the expression 2β„“+2w.

Explanation

The expression has two terms that are being added, \(2\ell\) and \(2w\text{.}\)

permalinkIf there is any subtraction, we can rewrite the expression using addition to make it easier to see exactly what the terms are and what sign each term has.

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Example 1.2.4.

List the terms in the expression βˆ’3x2+5xβˆ’4.

Explanation

We can rewrite this expression as \(-3x^2+5x+(-4)\) to see that the terms are \(-3x^2\), \(5x\), and \(-4\text{.}\)

Once you learn to recognize that subtraction represents a negative term, you don't need to rewrite subtraction as addition.

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Example 1.2.5.

List the terms in the expression 3cm+2cmβˆ’3cm+2cm.

Explanation

This expression has four terms: 3 cm, 2 cm, -3 cm, and 2 cm.

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Checkpoint 1.2.6.
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Subsection 1.2.2 Combining Like Terms

permalinkIn the examples above, you may have wanted to combine terms in some cases. For example, if you have 3cm+2cm, it is natural to add those together to get 5 cm. That works because their units (cm) are the same. This idea applies to some other kinds of terms that don't have units. For example, with 2x+3x, we have 2 somethings and then we have 3 more of the same thing. All together, we have 5 of those things. So 2x+3x is the same as 5x.

permalinkTerms in an algebraic expression that can be combined like these last examples are called like terms.

  • Sometimes terms are like terms because they have the same variable, like with 2x+3x, which simplfies to 5x.

  • Sometimes terms are like terms because they have the same units, like with 3cm+2cm, which simplfies to 5 cm.

  • Sometimes terms are like terms because they have something else in common, like with 37+27, which simplfies to 57.

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Example 1.2.7.

In the expressions below, look for like terms and then simplify where possible by adding or subtracting.

  1. 5in+20in

  2. 16ft2+4ft

  3. 2🍎+5🍎

  4. 5min+50ft

  5. 5πŸΆβˆ’2🐱

  6. 20mβˆ’6m

Explanation

We can combine terms with the same units, but we cannot combine units such as minutes and feet, or cats and dogs. We can combine the like terms by adding or subtracting their numerical parts.

  1. \(5\,\text{in}+20\,\text{in}=25\,\text{in}\)

  2. \(16\,\text{ft}^2+4\,\text{ft}\) cannot be simplified

  3. \(2\,\apple+5\,\apple=7\,\apple\)

  4. \(5\,\text{min}+50\,\text{ft}\) cannot be simplified

  5. \(5\,\dog-2\,\cat\) cannot be simplified

  6. \(20\,\text{m}-6\,\text{m}=14\,\text{m}\)

permalinkOne of the examples from Example 1.2.7 was 16ft2+4ft. The units on these two terms may look similar, but they are very different. 16 ft2 is a measurement of how much area something has. 4 ft is a measurement of how long something is. Figure 1.2.8 illustrates this.

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Figure 1.2.8. There is no way to add 16 ft2 to 4 ft.
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Checkpoint 1.2.9.
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Example 1.2.10.

Simplify the expression 20xβˆ’16x+4y, if possible, by combining like terms.

Explanation

This expression has two like terms, \(20x\) and \(-16x\text{,}\) which we can combine.

\begin{equation*} \highlight{20x-16x}+4y=\highlight{4x}+4y \end{equation*}

Note that we cannot combine \(4x\) and \(4y\) because \(x\) and \(y\) are different.

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Example 1.2.11.

Simplify the expression 100x+100x2, if possible, by combining like terms.

Explanation

This expression cannot be simplified because the variable parts are not the same. We cannot add \(x\) and \(x^2\) just like we cannot add feet ( a measure of length) and square feet (a measure of area).

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Example 1.2.12.

Simplify the expression βˆ’10r+2sβˆ’5t, if possible, by combining like terms.

Explanation

This expression cannot be simplified because there are not any like terms.

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Example 1.2.13.

Simplify the expression y+5y, if possible, by combining like terms.

Explanation

This expression can be thought of as \(1y+5y\text{.}\) When we have a single \(y\text{,}\) the numerical part \(1\) is not usually written. Now we have two like terms, \(1y\) and \(5y\text{.}\) We will add those together:

\begin{align*} y+5y\amp= \highlight{1y+5y}\\ \amp=\highlight{6y} \end{align*}

permalinkSo far we have combined terms with whole numbers and integers, but we can also combine like terms when the numerical parts are decimals or fractions.

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Example 1.2.14.

Simplify the expression xβˆ’0.15x, if possible, by combining like terms.

Explanation

Note that this expression can be rewritten as \(1.00x-0.15x\text{,}\) and combined like this:

\begin{align*} x-0.15x\amp=\highlight{1.00x-0.15x}\\ \amp=\highlight{0.85x} \end{align*}
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Checkpoint 1.2.15.
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Remark 1.2.16. The Difference Between Terms and Factors.

We have learned that terms are quantities that are added, such as 3x and βˆ’2x in 3xβˆ’2x. These are different from factors, which are parts that are multiplied together. For example, the term 2x has two factors: 2 and x (with the multiplication symbol implied between them). The term 2Ο€r has three factors: 2, Ο€, and r.

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Reading Questions 1.2.3 Reading Questions

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1.

What should you be careful with when there is subtraction in an algebraic expression and you are identifying its terms?

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2.

Describe at least two different ways in which a pair of terms are considered to be β€œlike terms.”

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3.

Describe the difference between β€œterms” and β€œfactors” in an algebraic expression. Give examples.

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Exercises 1.2.4 Exercises

Review and Warmup
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1.

Add the following.

  1. 4+(βˆ’6)

  2. 6+(βˆ’2)

  3. 9+(βˆ’9)

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2.

Add the following.

  1. 4+(βˆ’7)

  2. 9+(βˆ’3)

  3. 9+(βˆ’9)

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3.

Add the following.

  1. βˆ’9+5

  2. βˆ’4+5

  3. βˆ’2+2

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4.

Add the following.

  1. βˆ’6+5

  2. βˆ’2+8

  3. βˆ’2+2

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5.

Subtract the following.

  1. 1βˆ’9

  2. 10βˆ’2

  3. 6βˆ’17

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6.

Subtract the following.

  1. 1βˆ’7

  2. 7βˆ’1

  3. 6βˆ’12

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7.

Subtract the following.

  1. βˆ’4βˆ’5

  2. βˆ’7βˆ’5

  3. βˆ’4βˆ’4

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8.

Subtract the following.

  1. βˆ’3βˆ’1

  2. βˆ’7βˆ’4

  3. βˆ’9βˆ’9

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9.

Subtract the following.

  1. βˆ’3βˆ’(βˆ’6)

  2. βˆ’8βˆ’(βˆ’1)

  3. βˆ’5βˆ’(βˆ’5)

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10.

Subtract the following.

  1. βˆ’4βˆ’(βˆ’8)

  2. βˆ’5βˆ’(βˆ’1)

  3. βˆ’5βˆ’(βˆ’5)

Counting, Identifying, and Combining Terms
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11.

Count the number of terms in each expression.

  1. βˆ’5tβˆ’9yβˆ’5xβˆ’5x2

  2. 5x2+8βˆ’8s2

  3. 3yβˆ’8x

  4. 4z+2s2+6y2βˆ’7s

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12.

Count the number of terms in each expression.

  1. 3t+5sβˆ’4y+6t2

  2. βˆ’4z2βˆ’9y2

  3. 3t2+5y2+4y+7

  4. βˆ’2yβˆ’6t2βˆ’y

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13.

Count the number of terms in each expression.

  1. βˆ’2t+8.3xβˆ’3.6y+3y

  2. βˆ’y2βˆ’2s2+4.5x

  3. 6.6yβˆ’2.1y

  4. βˆ’4.3x

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14.

Count the number of terms in each expression.

  1. βˆ’t

  2. βˆ’8.9zβˆ’1.8y+8.1s

  3. βˆ’1.8xβˆ’5.3+6.2y+8.4

  4. βˆ’8.5zβˆ’8.2+4.9s+3.4

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15.

List the terms in each expression.

  1. t+7s2

  2. βˆ’8t

  3. z+9z

  4. 4s2βˆ’yβˆ’8s+3

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16.

List the terms in each expression.

  1. βˆ’5tβˆ’4x

  2. βˆ’7x2+5βˆ’4tβˆ’8y

  3. βˆ’5y2βˆ’8x2βˆ’9y2

  4. 3t2βˆ’5x2

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17.

List the terms in each expression.

  1. 4.5t+4.9t

  2. 0.7s2+2.4tβˆ’8.1x

  3. 7.4t+5.1y+5.1sβˆ’5x2

  4. βˆ’6.5z

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18.

List the terms in each expression.

  1. βˆ’7.8t2+0.5t+1.6x2

  2. βˆ’5.5y

  3. 1.8z+2y

  4. 3.4t+7y+0.1sβˆ’5.6s

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19.

List the terms in each expression.

  1. 3.2t+7.7βˆ’6.3y2

  2. 9t+8.9s+8.6s+7.5

  3. 2.3z2+7tβˆ’1.6s

  4. βˆ’7.5t2βˆ’6.1t2

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20.

List the terms in each expression.

  1. 5.2t2βˆ’3.2x+8.6x2+0.1s

  2. 7.5z+1.1x2

  3. βˆ’5.8t2βˆ’3.9z2

  4. 5.4z+5.9y+9z2

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21.

Simplify each expression, if possible, by combining like terms.

  1. βˆ’8t+2t

  2. 4z+7z

  3. βˆ’5z+8z

  4. 9y2+3x2

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22.

Simplify each expression, if possible, by combining like terms.

  1. 4tβˆ’9s

  2. 5xβˆ’z

  3. 6sβˆ’9s

  4. 2x+5x

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23.

Simplify each expression, if possible, by combining like terms.

  1. βˆ’4z+3z

  2. 6xβˆ’9x2βˆ’3x

  3. βˆ’7t2βˆ’6s+7t2

  4. 7z2βˆ’7s2+4z

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24.

Simplify each expression, if possible, by combining like terms.

  1. z+9s

  2. 8x2+9s+2t2βˆ’2t

  3. 9t2βˆ’6yβˆ’5z2βˆ’9x

  4. 9sβˆ’8s

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25.

Simplify each expression, if possible, by combining like terms.

  1. βˆ’8zβˆ’21s2

  2. 94s2+70s+51s

  3. βˆ’50zβˆ’51z

  4. βˆ’90yβˆ’70y+38y+44

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26.

Simplify each expression, if possible, by combining like terms.

  1. βˆ’36z+14z2+92s2

  2. 40x2+99x

  3. βˆ’18x2+91y2βˆ’70s2+59x2

  4. βˆ’5t2+46y2+99t2

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27.

Simplify each expression, if possible, by combining like terms.

  1. 2.5zβˆ’3.6z+4.6z2

  2. 3.9x2βˆ’3.5x+1.1x+8.9x

  3. 3.7z2βˆ’0.4z2

  4. βˆ’4.3z+4.4z

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28.

Simplify each expression, if possible, by combining like terms.

  1. βˆ’6.7z2+6tβˆ’8.1t

  2. 5.9yβˆ’7.3t+2t

  3. βˆ’6.5y2βˆ’2.8t2βˆ’7.5s2

  4. βˆ’8.2y2+4.5z2

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29.

Simplify each expression, if possible, by combining like terms.

  1. 6zβˆ’17z+4z

  2. 73yβˆ’12y2

  3. 37t2βˆ’12x2+65x2+27x

  4. y+2yβˆ’58y+3y

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30.

Simplify each expression, if possible, by combining like terms.

  1. βˆ’83z2βˆ’7z2+3+34z2

  2. βˆ’t+2y+13tβˆ’17x

  3. βˆ’yβˆ’2y

  4. s+13βˆ’49t

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31.

Simplify each expression, if possible, by combining like terms.

  1. βˆ’3z+23z+7zβˆ’25z

  2. 85zβˆ’83x2+32z2

  3. βˆ’x+72tβˆ’s

  4. 59zβˆ’6t

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32.

Simplify each expression, if possible, by combining like terms.

  1. 9z2+65z2

  2. 43tβˆ’43t

  3. 32x+12zβˆ’13y

  4. 58s+34s+sβˆ’32s