When you first start learning algebra, it can feel like you’re looking at a different language. However, once you understand the basic “grammar” of algebraic expressions, everything becomes much simpler. Today, we are focusing on one of the most important building blocks: Like and Unlike Terms.
In algebra, terms are called Like Terms (or similar terms) if they contain the exact same literal factors. This simply means the letter part (variable) must be identical.
The Rule: If the variables are the same, they are like terms. If they are different, they are unlike.
$a$ and $2a$: These are Like Terms because they both share the literal factor ‘$a$‘.
$2a$ and $3a$: These are Like Terms for the same reason.
$2a$ and $3b$: These are Unlike Terms. Even though they both have numbers, the letters ($a$ and $b$) are different.
$2$ and $a$: These are Unlike Terms. One is a constant (just a number) and the other is a variable.
The reason we identify like terms is so we can simplify expressions. You can add like terms together, but you cannot add unlike terms into a single unit.
Think of it like fruit: 3 apples + 2 apples = 5 apples. But 3 apples + 2 bananas? You still just have 3 apples and 2 bananas.
To add like terms, you simply add the numerical coefficients (the numbers in front) and keep the literal factor the same.
Example 1:
Example 2:
In many algebra problems, you will see a mix of different variables and constants. The key is to group the terms that match before doing any math.
Mixed Variables: If you have both “$a$” terms and “$b$” terms, solve them separately. For example, $3a + 5a + 3b + 4b$ simplifies to $8a + 7b$.
Variables and Constants: If you have variables mixed with plain numbers (constants), keep them separate. For example, $3a + 2 + 5a + 7$ becomes $8a + 9$.
Pro Tip: Adding a variable $a$ to itself $n$ times can be written as $na$ (e.g., $a + a + a = 3a$).
Subtracting terms follows the same logic as addition: you only subtract the numerical coefficients of terms that have the exact same literal factor.
When the first number is larger, it’s straightforward:
$5a – 3a = (5 – 3)a = 2a$
$12x – 3x = 9x$
Sometimes, subtracting results in a negative coefficient:
Smaller minus Larger: $5a – 9a = (5 – 9)a = -4a$.
Negative minus Positive: $-3a – 2a = -5a$.
Mixed Signs: $-3a + 2a = -a$.
For longer expressions, work from left to right or group the terms first:
Example: $9a + 2a – 6a$
First, $9a + 2a = 11a$.
Then, $11a – 6a = 5a$.
In complex expressions, you might have to add and subtract different types of terms at the same time. Always remember to “match” your terms first.
Complex Example:
Watch Out for “Hidden” Ones:
In the problem $3a + a – 5a – 1$, remember that $a$ is the same as $1a$.
$(3 + 1 – 5)a – 1 = -a – 1$. (Note: There is a small typo in the original image’s final result for item 11; based on standard math, it should be $-a – 1$).