Like Terms – Same Variable, Different Coefficient
Like terms are terms that have exactly the same variable parts — the same letters raised to the same powers. You can only add or subtract like terms, just as you can only add the same kinds of objects together.
“You can’t add apples and oranges” is the classic way this idea is taught, and it is a genuinely useful analogy: 3 apples + 2 apples = 5 apples makes sense, but 3 apples + 2 oranges cannot be combined into a single count. In algebra, x and y (or x and x²) are the “apples and oranges” — collecting like terms is simply counting how many of each distinct kind of quantity you have, which is exactly the same logic used when a shop totals up separate columns for different items on a receipt.
What Makes Terms Like?
Two terms are like terms if they have identical variable parts (same letters AND same powers). The coefficients do not need to match — only the variable part matters.
Like vs Unlike Terms
| Terms | Like or Unlike? | Reason |
|---|---|---|
| 3x and 7x | Like | Both have x to the power 1 |
| 5y squared and 2y squared | Like | Both have y squared |
| 4x and 4y | Unlike | Different variables |
| 3x and 3x squared | Unlike | Same letter but different powers |
| 6ab and -2ab | Like | Both have ab |
| 5x and 5 | Unlike | One has a variable, one does not |
Collecting Like Terms
Collecting like terms means adding or subtracting them to produce a simpler expression. Only the coefficient changes — the variable part stays the same.
(3 + 5)x = 8x.
(7 - 2 + 1)y = 6y. (The invisible coefficient of y is 1.)
Group: (4a - 2a) + (3b + 5b) = 2a + 8b. Answer: 2a + 8b.
x squared terms: 5 + 2 = 7. x terms: -3 + 7 = 4. Constant: -4. Answer: 7x squared + 4x - 4.
Common Mistakes
- Treating 3x and 3x squared as like terms — they are not.
- Forgetting the sign in front of a term: in 5a - 3b, the term is -3b, not +3b.
- Adding unlike terms: 2x + 3y cannot be simplified further.
Key Takeaways
- Like terms have identical variable parts (same letters AND same powers).
- Add or subtract the coefficients; keep the variable part unchanged.
- Unlike terms cannot be combined — leave them as separate terms.