## Section 2.3 Combining Like Terms

¶In the last section we worked with algebraic expressions. In order to simplify algebraic expressions, it is useful to identify which quantities we can combine.

### Subsection 2.3.1 Identifying Terms

In an algebraic expression, the terms are the quantities that are added. For example, the expression \(3x+2y\) has two terms, which are \(3x\) and \(2y\text{.}\) The coefficient or numerical coefficient of a term is the number attached to the variable, by multiplication.

###### Example 2.3.2

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

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

If there is any subtraction, we will rewrite the expression using addition. Here is an example of that.

###### Example 2.3.3

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

We can rewrite this expression as \(-3x^2+5x+(-4)\) to see that the terms are \(-3x^2\text{,}\) \(5x\text{,}\) and \(-4\text{.}\) The last term is negative because subtracting is the same as adding the opposite.

###### Example 2.3.4

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

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

###### Checkpoint 2.3.5

### Subsection 2.3.2 Combining Like Terms

In the examples above, you may have wanted to combine some of the terms. Look at the quantities below to see which ones you can add or subtract.

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

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

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

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

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

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

The terms that we can combine are called like terms. 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 combining their coefficients. Here are the answers:

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

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

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

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

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

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

Now let's look at some examples that have variables in them.

###### Checkpoint 2.3.6

###### Example 2.3.7

Simplify the expression \(20x-16x+4y\text{,}\) if possible, by combining like terms.

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

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

###### Example 2.3.8

Simplify the expression \(100x+100x^2\text{,}\) if possible, by combining like terms.

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 2.3.9

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

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

###### Example 2.3.10

Simplify the expression \(y+5y\text{,}\) if possible, by combining like terms.

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

So far we have combined terms with whole numbers and integers, but we can also combine like terms when the coefficients are decimals (or fractions).

###### Example 2.3.11

Simplify the expression \(x-0.15x\text{,}\) if possible, by combining like terms.

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

###### Checkpoint 2.3.12

###### Remark 2.3.13 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 than 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 \(r\text{.}\)

### Subsection 2.3.3 Exercises

###### Review and Warmup

###### 1

Add the following.

\(\displaystyle{ 2+(-6)= }\)

\(\displaystyle{ 8+(-3)= }\)

\(\displaystyle{ 9+(-9)= }\)

###### 2

Add the following.

\(\displaystyle{ 2+(-7)= }\)

\(\displaystyle{ 10+(-4)= }\)

\(\displaystyle{ 9+(-9)= }\)

###### 3

Add the following.

\(\displaystyle{ -9+3= }\)

\(\displaystyle{ -1+7= }\)

\(\displaystyle{ -2+2= }\)

###### 4

Add the following.

\(\displaystyle{ -6+3= }\)

\(\displaystyle{ -2+9= }\)

\(\displaystyle{ -2+2= }\)

###### 5

Subtract the following.

\(\displaystyle{ 4-8= }\)

\(\displaystyle{ 6-2= }\)

\(\displaystyle{ 6-12= }\)

###### 6

Subtract the following.

\(\displaystyle{ 4-7= }\)

\(\displaystyle{ 8-1= }\)

\(\displaystyle{ 6-17= }\)

###### 7

Subtract the following.

\(\displaystyle{ -1-5= }\)

\(\displaystyle{ -10-5= }\)

\(\displaystyle{ -4-4= }\)

###### 8

Subtract the following.

\(\displaystyle{ -1-3= }\)

\(\displaystyle{ -8-3= }\)

\(\displaystyle{ -4-4= }\)

###### 9

Subtract the following.

\(\displaystyle{ -1-(-6)= }\)

\(\displaystyle{ -10-(-1)= }\)

\(\displaystyle{ -5-(-5)= }\)

###### 10

Subtract the following.

\(\displaystyle{ -2-(-8)= }\)

\(\displaystyle{ -6-(-2)= }\)

\(\displaystyle{ -5-(-5)= }\)

###### Identifying Terms

###### 11

Count the number of terms in each expression.

\({8t^{2}-6t}\)

\({4x-4-9y}\)

\({-8z^{2}-z^{2}+2y^{2}+6t}\)

\({-3s+2t+2z}\)

###### 12

Count the number of terms in each expression.

\({-7t-5x}\)

\({-4s^{2}}\)

\({2x^{2}-6x^{2}}\)

\({-2s^{2}+8z+4t^{2}}\)

###### 13

Count the number of terms in each expression.

\({1.8t+0.7z^{2}+x^{2}}\)

\({8.1s-8.9s+5.2+0.6y}\)

\({3s^{2}+5.9z}\)

\({0.1s}\)

###### 14

Count the number of terms in each expression.

\({-3.2t-6.6x^{2}+4.2}\)

\({-8.5t+2.5s}\)

\({-0.4x-6.2x-6.8+2s^{2}}\)

\({-3.6x^{2}+7.6t+2.7+3.6s}\)

###### 15

List the terms in each expression.

\({-2t+6x+3x-2x^{2}}\)

\({2t}\)

\({-9s+4+6x}\)

\({s^{2}-3z+2t^{2}}\)

###### 16

List the terms in each expression.

\({6t-1+4x+8t^{2}}\)

\({-7s-3z^{2}+8y^{2}}\)

\({-6s-x-3y+t^{2}}\)

\({9z^{2}-8s}\)

###### 17

List the terms in each expression.

\({-5t-8t+4.4x-3.6y^{2}}\)

\({8.3s^{2}-5.8z}\)

\({6.7t-4.9s}\)

\({3.9z}\)

###### 18

List the terms in each expression.

\({3.3t}\)

\({3.4t^{2}+5.2x}\)

\({-7.9s^{2}-4.4t^{2}+0.7-7.5x}\)

\({-2.4t+4.5y+7.5t+0.8t}\)

###### 19

List the terms in each expression.

\({-1.8t^{2}+0.3y^{2}}\)

\({7.8t+3.2t-3+8.9y}\)

\({-6.6z^{2}}\)

\({-2.2s^{2}-8.1x+5.2x}\)

###### 20

List the terms in each expression.

\({2.9t+6.9s^{2}}\)

\({-3.7x+4.1y+1.1z}\)

\({-6.6z-5.3z^{2}-4t}\)

\({-3.9t}\)

###### Combining Like Terms

###### 21

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

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

\({8s^{2}+5z}\)

\({8t^{2}+t^{2}}\)

\({-5y^{2}-5z}\)

###### 22

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

\({-9z+5z}\)

\({7y^{2}-4s^{2}}\)

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

\({-2z+5t}\)

###### 23

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

\({z+8z^{2}+8z}\)

\({-3z^{2}+7y^{2}}\)

\({-9y-y-7t^{2}+s^{2}}\)

\({3s^{2}+6y}\)

###### 24

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

\({-5z+3z^{2}+2z^{2}-3z^{2}}\)

\({-2z^{2}-9+3y^{2}+9z^{2}}\)

\({-9s-8z}\)

\({7z^{2}+y^{2}}\)

###### 25

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

\({-5z-79z-76-99z}\)

\({-79x-51t-53s}\)

\({-89x-44x-16z}\)

\({22x+38x^{2}+50x^{2}+90}\)

###### 26

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

\({-21z+14s-18z-56z}\)

\({90z-32}\)

\({30y^{2}-61t^{2}+89t^{2}+94t^{2}}\)

\({23y^{2}+53y^{2}-90y^{2}}\)

###### 27

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

\({-3.8z-2.5t}\)

\({-5.2s-6.7s-5.4y-z^{2}}\)

\({-1.5t-4.5s^{2}+1.3t^{2}}\)

\({8.8y+2.3y}\)

###### 28

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

\({1.3z-0.4z}\)

\({-5.7s^{2}-7.6s^{2}+4.5y^{2}}\)

\({0.8y-8.8-2.7t-5.1t}\)

\({5t^{2}+2.5t}\)

###### 29

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

\({4z - {\frac{5}{8}}s}\)

\({{\frac{9}{5}}y - {\frac{2}{3}}y}\)

\({{\frac{3}{4}}y-9y+y}\)

\({{\frac{4}{7}}y+z^{2}-z - {\frac{5}{2}}z^{2}}\)

###### 30

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

\({{\frac{5}{4}}z+{\frac{5}{9}}x+{\frac{4}{3}}x}\)

\({-{\frac{4}{9}}t^{2}+{\frac{7}{6}}z^{2}+1 - {\frac{6}{7}}y^{2}}\)

\({{\frac{1}{3}}y+{\frac{8}{3}}s^{2} - {\frac{8}{7}}t}\)

\({{\frac{7}{3}}x - {\frac{3}{7}}y}\)

###### 31

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

\({{\frac{8}{9}}z^{2}+{\frac{4}{7}}t^{2} - {\frac{7}{2}}z}\)

\({-{\frac{2}{5}}x^{2}+{\frac{8}{9}}y^{2}-{\frac{7}{3}}}\)

\({6t^{2} - {\frac{2}{9}}t^{2}}\)

\({-{\frac{7}{2}}t^{2}+{\frac{5}{2}}t}\)

###### 32

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

\({-{\frac{8}{3}}z+{\frac{4}{5}}z+{\frac{3}{4}}z}\)

\({-{\frac{9}{4}}y+y^{2}}\)

\({{\frac{4}{5}}y+{\frac{3}{2}}t+z^{2}}\)

\({{\frac{9}{5}}t+6t+{\frac{1}{3}}t+7x}\)