You know from arithmetic that you can combine quantities of the same type. For example, if you have three cats and your roommate has two cats, then together you have (3 + 2) cats, or 5 cats.
Activity5.1.1.Like Terms.
Suppose tuition at your college is $20 per unit. If you are enrolled in eight units this semester and plan to enroll in ten units next semester, then you can figure your tuition for the year in two ways.
Method 1: Calculate the tuition for each semester, then add them up:
Either calculation gives your total tuition for the year.
Next year the tuition at your college is going up to some unknown figure, \(x\) dollars per unit. If you take the same number of units next year, your tuition will be
Now as soon as you learn the value of , you can find your tuition for the year.
In Activity 5.1.1, \(8x\) and \(10x\) are called like terms, because the variable part of each term, \(x\text{,}\) is the same. (Recall that terms are expressions that are added or subtracted, and factors are expressions that are multiplied together.)
The numbers multiplied by the variable are called the coefficients of the variable. In Activity 5.1.1 the coefficients are 8 and 10.
To add or subtract like terms.
Add or subtract the coefficients.
Do not change the variable part of the terms.
Example5.1.1.
Add or subtract like terms.
\(\displaystyle 9m-4m\)
\(\displaystyle 6st + 8st\)
Solution.
We add or subtract the coefficients of the terms. The variable part of the terms remains the same.
Because \(x = 1 \cdot x\text{,}\) the coefficient of \(x\) is \(1\text{.}\) Also, because 0 times any number is 0, we have
\begin{equation*}
0 \cdot x = 0
\end{equation*}
Example5.1.4.
Combine like terms.
\(\displaystyle 3x + x = 4x ~~~~~~~ \blert{\text{Add coefficients, keep the variable factor.}}\)
\(\displaystyle 3w - w = 2w ~~~~~~~ \blert{\text{Subtract coefficients, keep the variable factor.}}\)
Checkpoint5.1.5.
Combine like terms: \(~pq + pq\)
Answer.
\(2pq\)
Subsection5.1.2Equivalent Expressions
Combining like terms gives us a simpler expression that is equivalent to the original. Equivalent expressions have the same value when we substitute a number for the variable. It is always a good idea to replace complicated expressions with simpler ones if we can.
Note5.1.6.
We cannot combine unlike terms. If you have three cats and your roommate has two canaries, and you try to combine them, you will still have three cats and two canaries, if you’re lucky.
If the variable parts of the terms are not identical, they are not like terms, and they cannot be combined. Here are some examples of unlike terms.
\begin{align*}
3x ~~ \text{and} ~~ 5y \hphantom{0000} \amp ~~ \text{are not like terms}\\
6a ~~ \text{and} ~~ 2ab \hphantom{0000} \amp ~~ \text{are not like terms}\\
4z ~~ \text{and} ~~ 8 \hphantom{0000} \amp ~~ \text{are not like terms}
\end{align*}
So, expressions such as \(3x + 5y ~~ \text{or} ~~ 6a-2ab ~~ \text{or} ~~ 4z - 8\) cannot be simplified; they just stay the way they are.
A Quick Refresher.
Remember the order of operations: we perform all multiplications (and divisions) before additions and subtractions. For example, to evaluate
\begin{equation*}
6x ~~ \text{and} ~~ 3x^2 ~~ \text{are not like terms}
\end{equation*}
This is because \(x\) and \(x^2\) are usually not the same number (unless \(x\) happens to be 0 or 1), so we cannot combine them into one group (like cats and canaries).
Example5.1.9.
Combine like terms.
We add the coefficients, \(4 + 3\text{,}\) and leave the variable part, \(x\text{,}\) unchanged.
\(7a^2\) and \(5a\) are not like terms, because the variable, \(a\text{,}\) does not have the same exponent. So we cannot combine the terms; this expression cannot be simplified.
Note5.1.10.
When combining like terms, we do not add the exponents. For example,
What about multiplying variable expressions? We will need one fact from arithmetic: We can rearrange factors in any order and get the same product. Try it yourself. For example, multiply from left to right:
