Collecting like terms means simplifying an algebraic expression by adding or subtracting terms that have the same variable and power. Only like terms can be combined — for example, 3x and 5x are like terms but 3x and 5x² are not. The result is a shorter, tidier expression.
What are like terms?
Like terms share exactly the same letter(s) and the same power(s) for each letter. The coefficient (the number in front) can be different.
| Like terms | Not like terms | Reason |
|---|---|---|
| 4x and 7x | 4x and 7y | Different letters |
| 3x² and −5x² | 3x² and −5x | Different powers of x |
| 2ab and 9ab | 2ab and 9a | Different variable combination |
| 6 and −11 | 6 and 6x | One has a letter; one doesn't |
Pure numbers (constants) are always like terms with each other — 6 and −11 can be combined to give −5.
How do you collect like terms — step by step?
- Identify each term in the expression.
- Sort the terms into groups of like terms.
- Add or subtract the coefficients within each group, keeping the variable part unchanged.
- Write the simplified expression.
Worked example 1: Simplify 3x + 7y − x + 4y
Step 1 — sort: x terms: 3x and −x; y terms: 7y and 4y. Step 2 — collect x: 3x − x = (3−1)x = 2x Step 3 — collect y: 7y + 4y = (7+4)y = 11y Step 4 — answer: 2x + 11y
How do you handle negative terms?
Negative signs belong to the term that follows them. Keep the sign with the term as you move it.
Worked example 2: Simplify 8a − 3b − 5a + 6b − 2
Step 1 — sort by type:
- a terms: 8a and −5a
- b terms: −3b and +6b
- constants: −2
Step 2 — collect:
- a: 8a − 5a = 3a
- b: −3b + 6b = 3b
- constants: −2
Step 3 — answer: 3a + 3b − 2
How do you simplify expressions with squared terms?
Remember: x² and x are not like terms. They have different powers, so treat them as separate groups.
Worked example 3: Simplify 5x² + 3x − 2x² + 7 − 4x + 1
Sort into three groups:
- x² terms: 5x² − 2x² = 3x²
- x terms: 3x − 4x = −x
- constants: 7 + 1 = 8
Answer: 3x² − x + 8
Write the terms in descending order of power (highest power first) — this is standard convention.
How do you simplify expressions with two different letters and powers?
Worked example 4: Simplify 4p²q − 3pq² + 7p²q − pq² + 2pq
Sort into groups:
- p²q terms: 4p²q + 7p²q = 11p²q
- pq² terms: −3pq² − pq² = −4pq²
- pq terms: 2pq (only one, cannot combine with others)
Answer: 11p²q − 4pq² + 2pq
Note that p²q, pq² and pq are all different terms — the letters are the same but the power distribution differs.
What mistakes should you avoid?
| Mistake | Wrong answer | Correct approach |
|---|---|---|
| Combining unlike terms | 3x + 2y = 5xy | These cannot be combined; leave as 3x + 2y |
| Dropping the sign | 5a − 3a = 8a | The −3a is negative: 5a − 3a = 2a |
| Treating x and x² as like | 4x² + 3x = 7x² | Different powers — write 4x² + 3x (cannot simplify) |
| Forgetting constants | 6 + 3x + 2 = 3x + 6 + 2 = 3x + 62 | 6 + 2 = 8; answer is 3x + 8 |
Frequently asked questions
Why can't I add 3x and 4x²?
Because they represent fundamentally different quantities. If x = 2, then 3x = 6 and 4x² = 16. Their sum is 22, which cannot be written as a single term of the form kx or kx². The only accurate way to write it is 3x + 4x². Adding different powers would give a wrong value for every value of x.
Does it matter what order I write the terms in?
In algebra, addition is commutative, so 2x + 3y and 3y + 2x are the same expression. By convention, write terms with the highest power of the first letter first, then alphabetically. This makes expressions easier to read and compare.
Can I collect like terms that include a coefficient of 1?
Yes. Remember that x = 1x and x² = 1x². So 5x − x = 5x − 1x = 4x, not 5. This is a very common slip: students write 5x − x = 5, as if the x cancels completely. It does not — 1 copy of x remains.
How does collecting like terms help with expanding brackets?
When you expand brackets (e.g. 3(2x + 1) + 4(x − 5) = 6x + 3 + 4x − 20), collecting like terms is the final step that simplifies the result. The two skills go hand in hand: expand first, then collect. Answer: 6x + 4x + 3 − 20 = 10x − 17.
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