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Section 1.2 Combining Like Terms

In 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.

Figure 1.2.1. Alternative Video Lesson

Subsection 1.2.1 Identifying Terms

Definition 1.2.2.

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

Example 1.2.3.

List the terms in the expression \(2\ell+2w\text{.}\)

Explanation

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

If 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.

Example 1.2.4.

List the terms in the expression \(-3x^2+5x-4\text{.}\)

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.

Example 1.2.5.

List the terms in the expression \(3\,\text{cm}+2\,\text{cm}-3\,\text{cm}+2\,\text{cm}\text{.}\)

Explanation

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

Checkpoint 1.2.6.

Subsection 1.2.2 Combining Like Terms

In the examples above, you may have wanted to combine terms in some cases. For example, if you have \(3\,\text{cm}+2\,\text{cm}\text{,}\) 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 \(2x+3x\text{,}\) 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 \(5x\text{.}\)

Terms 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 \(2x+3x\text{,}\) which simplfies to \(5x\text{.}\)

  • Sometimes terms are like terms because they have the same units, like with \(3\,\text{cm}+2\,\text{cm}\text{,}\) which simplfies to 5 cm.

  • Sometimes terms are like terms because they have something else in common, like with \(3\sqrt{7}+2\sqrt{7}\text{,}\) which simplfies to \(5\sqrt{7}\text{.}\)

Example 1.2.7.

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

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

  2. \(16\,\text{ft}^2+4\,\text{ft}\)

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

  4. \(5\,\text{min}+50\,\text{ft}\)

  5. \(5\,\dog-2\,\cat\)

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

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

One of the examples from Example 1.2.7 was \(16\,\text{ft}^2+4\,\text{ft}\text{.}\) 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.

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

Simplify the expression \(20x-16x+4y\text{,}\) 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.

Example 1.2.11.

Simplify the expression \(100x+100x^2\text{,}\) 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).

Example 1.2.12.

Simplify the expression \(-10r+2s-5t\text{,}\) if possible, by combining like terms.

Explanation

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

Example 1.2.13.

Simplify the expression \(y+5y\text{,}\) 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*}

So 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.

Example 1.2.14.

Simplify the expression \(x-0.15x\text{,}\) 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*}
Checkpoint 1.2.15.
Remark 1.2.16. The Difference Between Terms and Factors.

We have learned that terms are quantities that are added, such as \(3x\) and \(-2x\) in \(3x-2x\text{.}\) 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 \(2\pi r\) has three factors: \(2\text{,}\) \(\pi\text{,}\) and \(r\text{.}\)

Reading Questions 1.2.3 Reading Questions

1.

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

2.

Describe at least two different ways in which a pair of terms are considered to be “like terms.”

3.

Describe the difference between “terms” and “factors” in an algebraic expression. Give examples.

Exercises 1.2.4 Exercises

Review and Warmup
1.

Add the following.

  1. \(4+(-6)\)

  2. \(6+(-2)\)

  3. \(9+(-9)\)

2.

Add the following.

  1. \(4+(-7)\)

  2. \(9+(-3)\)

  3. \(9+(-9)\)

3.

Add the following.

  1. \(-9+5\)

  2. \(-4+5\)

  3. \(-2+2\)

4.

Add the following.

  1. \(-6+5\)

  2. \(-2+8\)

  3. \(-2+2\)

5.

Subtract the following.

  1. \(1-9\)

  2. \(10-2\)

  3. \(6-17\)

6.

Subtract the following.

  1. \(1-7\)

  2. \(7-1\)

  3. \(6-12\)

7.

Subtract the following.

  1. \(-4-5\)

  2. \(-7-5\)

  3. \(-4-4\)

8.

Subtract the following.

  1. \(-3-1\)

  2. \(-7-4\)

  3. \(-9-9\)

9.

Subtract the following.

  1. \(-3-(-6)\)

  2. \(-8-(-1)\)

  3. \(-5-(-5)\)

10.

Subtract the following.

  1. \(-4-(-8)\)

  2. \(-5-(-1)\)

  3. \(-5-(-5)\)

Counting, Identifying, and Combining Terms
11.

Count the number of terms in each expression.

  1. \({-5t-9y-5x-5x^{2}}\)

  2. \({5x^{2}+8-8s^{2}}\)

  3. \({3y-8x}\)

  4. \({4z+2s^{2}+6y^{2}-7s}\)

12.

Count the number of terms in each expression.

  1. \({3t+5s-4y+6t^{2}}\)

  2. \({-4z^{2}-9y^{2}}\)

  3. \({3t^{2}+5y^{2}+4y+7}\)

  4. \({-2y-6t^{2}-y}\)

13.

Count the number of terms in each expression.

  1. \({-2t+8.3x-3.6y+3y}\)

  2. \({-y^{2}-2s^{2}+4.5x}\)

  3. \({6.6y-2.1y}\)

  4. \({-4.3x}\)

14.

Count the number of terms in each expression.

  1. \({-t}\)

  2. \({-8.9z-1.8y+8.1s}\)

  3. \({-1.8x-5.3+6.2y+8.4}\)

  4. \({-8.5z-8.2+4.9s+3.4}\)

15.

List the terms in each expression.

  1. \({t+7s^{2}}\)

  2. \({-8t}\)

  3. \({z+9z}\)

  4. \({4s^{2}-y-8s+3}\)

16.

List the terms in each expression.

  1. \({-5t-4x}\)

  2. \({-7x^{2}+5-4t-8y}\)

  3. \({-5y^{2}-8x^{2}-9y^{2}}\)

  4. \({3t^{2}-5x^{2}}\)

17.

List the terms in each expression.

  1. \({4.5t+4.9t}\)

  2. \({0.7s^{2}+2.4t-8.1x}\)

  3. \({7.4t+5.1y+5.1s-5x^{2}}\)

  4. \({-6.5z}\)

18.

List the terms in each expression.

  1. \({-7.8t^{2}+0.5t+1.6x^{2}}\)

  2. \({-5.5y}\)

  3. \({1.8z+2y}\)

  4. \({3.4t+7y+0.1s-5.6s}\)

19.

List the terms in each expression.

  1. \({3.2t+7.7-6.3y^{2}}\)

  2. \({9t+8.9s+8.6s+7.5}\)

  3. \({2.3z^{2}+7t-1.6s}\)

  4. \({-7.5t^{2}-6.1t^{2}}\)

20.

List the terms in each expression.

  1. \({5.2t^{2}-3.2x+8.6x^{2}+0.1s}\)

  2. \({7.5z+1.1x^{2}}\)

  3. \({-5.8t^{2}-3.9z^{2}}\)

  4. \({5.4z+5.9y+9z^{2}}\)

21.

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

  1. \({-8t+2t}\)

  2. \({4z+7z}\)

  3. \({-5z+8z}\)

  4. \({9y^{2}+3x^{2}}\)

22.

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

  1. \({4t-9s}\)

  2. \({5x-z}\)

  3. \({6s-9s}\)

  4. \({2x+5x}\)

23.

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

  1. \({-4z+3z}\)

  2. \({6x-9x^{2}-3x}\)

  3. \({-7t^{2}-6s+7t^{2}}\)

  4. \({7z^{2}-7s^{2}+4z}\)

24.

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

  1. \({z+9s}\)

  2. \({8x^{2}+9s+2t^{2}-2t}\)

  3. \({9t^{2}-6y-5z^{2}-9x}\)

  4. \({9s-8s}\)

25.

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

  1. \({-8z-21s^{2}}\)

  2. \({94s^{2}+70s+51s}\)

  3. \({-50z-51z}\)

  4. \({-90y-70y+38y+44}\)

26.

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

  1. \({-36z+14z^{2}+92s^{2}}\)

  2. \({40x^{2}+99x}\)

  3. \({-18x^{2}+91y^{2}-70s^{2}+59x^{2}}\)

  4. \({-5t^{2}+46y^{2}+99t^{2}}\)

27.

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

  1. \({2.5z-3.6z+4.6z^{2}}\)

  2. \({3.9x^{2}-3.5x+1.1x+8.9x}\)

  3. \({3.7z^{2}-0.4z^{2}}\)

  4. \({-4.3z+4.4z}\)

28.

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

  1. \({-6.7z^{2}+6t-8.1t}\)

  2. \({5.9y-7.3t+2t}\)

  3. \({-6.5y^{2}-2.8t^{2}-7.5s^{2}}\)

  4. \({-8.2y^{2}+4.5z^{2}}\)

29.

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

  1. \({6z - {\frac{1}{7}}z+4z}\)

  2. \({{\frac{7}{3}}y - {\frac{1}{2}}y^{2}}\)

  3. \({{\frac{3}{7}}t^{2} - {\frac{1}{2}}x^{2}+{\frac{6}{5}}x^{2}+{\frac{2}{7}}x}\)

  4. \({y+2y - {\frac{5}{8}}y+3y}\)

30.

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

  1. \({-{\frac{8}{3}}z^{2}-7z^{2}+3+{\frac{3}{4}}z^{2}}\)

  2. \({-t+2y+{\frac{1}{3}}t - {\frac{1}{7}}x}\)

  3. \({-y-2y}\)

  4. \({s+{\frac{1}{3}} - {\frac{4}{9}}t}\)

31.

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

  1. \({-3z+{\frac{2}{3}}z+7z - {\frac{2}{5}}z}\)

  2. \({{\frac{8}{5}}z - {\frac{8}{3}}x^{2}+{\frac{3}{2}}z^{2}}\)

  3. \({-x+{\frac{7}{2}}t-s}\)

  4. \({{\frac{5}{9}}z-6t}\)

32.

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

  1. \({9z^{2}+{\frac{6}{5}}z^{2}}\)

  2. \({{\frac{4}{3}}t - {\frac{4}{3}}t}\)

  3. \({{\frac{3}{2}}x+{\frac{1}{2}}z - {\frac{1}{3}}y}\)

  4. \({{\frac{5}{8}}s+{\frac{3}{4}}s+s - {\frac{3}{2}}s}\)

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