Section 1.2 Combining Like Terms
ΒΆObjectives: PCC Course Content and Outcome Guide
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.
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
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.
Example 1.2.4.
List the terms in the expression
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
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
permalinkIn the examples above, you may have wanted to combine terms in some cases. For example, if you have 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 we have somethings and then we have more of the same thing. All together, we have of those things. So is the same as
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 which simplfies to
Sometimes terms are like terms because they have the same units, like with which simplfies to 5 cm.
Sometimes terms are like terms because they have something else in common, like with which simplfies to
Example 1.2.7.
In the expressions below, look for like terms and then simplify where possible by adding or subtracting.
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.
\(5\,\text{in}+20\,\text{in}=25\,\text{in}\)
\(16\,\text{ft}^2+4\,\text{ft}\) 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
\(20\,\text{m}-6\,\text{m}=14\,\text{m}\)
permalinkOne of the examples from Example 1.2.7 was 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.
Checkpoint 1.2.9.
Example 1.2.10.
Simplify the expression 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 \(y\) are different.
Example 1.2.11.
Simplify the expression 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 1.2.12.
Simplify the expression if possible, by combining like terms.
This expression cannot be simplified because there are not any like terms.
Example 1.2.13.
Simplify the expression if possible, by combining like terms.
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:
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.
Example 1.2.14.
Simplify the expression if possible, by combining like terms.
Note that this expression can be rewritten as \(1.00x-0.15x\text{,}\) and combined like this:
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 and in These are different from factors, which are parts that are multiplied together. For example, the term has two factors: and (with the multiplication symbol implied between them). The term has three factors: and
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 βlike terms.β
3.
Describe the difference between βtermsβ and βfactorsβ in an algebraic expression. Give examples.
Exercises 1.2.4 Exercises
Review and Warmup
Counting, Identifying, and Combining Terms
21.
Simplify each expression, if possible, by combining like terms.
22.
Simplify each expression, if possible, by combining like terms.
23.
Simplify each expression, if possible, by combining like terms.
24.
Simplify each expression, if possible, by combining like terms.
25.
Simplify each expression, if possible, by combining like terms.
26.
Simplify each expression, if possible, by combining like terms.
27.
Simplify each expression, if possible, by combining like terms.
28.
Simplify each expression, if possible, by combining like terms.
29.
Simplify each expression, if possible, by combining like terms.
30.
Simplify each expression, if possible, by combining like terms.
31.
Simplify each expression, if possible, by combining like terms.
32.
Simplify each expression, if possible, by combining like terms.