Simplification is the art of recognising what belongs together.
Section 01
What Are Like Terms?
Two or more terms are like terms when they have exactly the same variable part — the same letters raised to the same powers. The coefficient (the number in front) is irrelevant to whether terms are “like”.
The Like Terms Rule
Terms are like if and only if their variable parts are identical — same letters, same exponents. Constants are like terms with each other.
Visual Identification — Colour-Coded Expression
In the expression below, each colour marks a group of like terms:
5x² − 3x + 7 + 2x² + 4y − x − 3
x² termsx termsy termsconstants
Group
Like Terms Found
Can Combine?
x²
5x², 2x²
✓ Yes
x
−3x, −x
✓ Yes
y
4y
— Only one term
constant
7, −3
✓ Yes
Like terms are like coins of the same denomination. You can count ten-pence pieces together, and pound coins together — but you cannot merge a ten-pence and a pound into a single coin. 3x and 3x² are different denominations; they may not be combined.
Section 02
How to Collect Like Terms
To simplify an expression, add or subtract the coefficients of like terms. The variable part remains unchanged.
ax + bx = (a + b)x
Three-Step Method
1
Identify — scan the expression and group like terms (same variable part)
2
Add/subtract coefficients — treat the variable as a label, work only with the numbers
3
Write the result — attach the combined coefficient back to the variable part. If the coefficient is 1, it stays invisible.
Worked Example A — Simplify 5x² − 3x + 7 + 2x² − x − 3
1
Group: 5x² + 2x²− 3x − x+ 7 − 3
2
Combine coefficients: (5+2)x²(−3−1)x(7−3)
3
Result: 7x² − 4x + 4
Worked Example B — Simplify 6a − 2b + 3a + 5b − a
1
Group a-terms: 6a + 3a − a Group b-terms: −2b + 5b
One of the most common errors in collecting like terms is losing track of signs. The golden rule: the sign (+ or −) directly in front of a term belongs to that term, and travels with it when you reorder.
Sign Tracking — The Critical Habit
Rewrite 4x − 7 − 3x + 2 by grouping:
4x − 3x and −7 + 2
The minus sign before 7 belongs to 7. The plus sign before 2 belongs to 2.
Coefficients: (4−3)x + (−7+2) = x − 5 ✓
Wrong: treating 7 as positive gives x + 9 ✗
Worked Example C — Simplify 3 − 2x² + x − 5 + 4x² − 3x
1
Identify signs carefully: +3 −2x² +x −5 +4x² −3x
2
x² group: −2x² + 4x² = 2x²
3
x group: +x − 3x = −2x
4
Constants: +3 − 5 = −2
5
Result: 2x² − 2x − 2
Section 04
Unlike Terms — What Cannot Be Combined
Unlike terms have different variable parts. They cannot be merged — they must remain as separate terms in the simplified expression.
Cannot Combine
✗ x + x² ≠ x³
✗ 3x + 3y ≠ 6xy
✗ 2ab + 2a ≠ 4a²b
✗ 5 + 5x ≠ 10x
Can Combine
✓ 4x + 7x = 11x
✓ 3ab − ab = 2ab
✓ x² + 5x² = 6x²
✓ 8 − 3 = 5
Section 05
Multi-Variable Expressions
Expressions can contain multiple different variables and multiple powers. The method is identical — group each family of like terms, combine coefficients, write the result.
Worked Example D — Simplify 4x²y − 3xy² + 2x²y + 5xy² − xy²
Note: x²y and xy² are unlike — the exponent positions differ.
4
Result: 6x²y + xy²
Invisible Coefficient of 1
When the combined coefficient is exactly 1 or −1, we do not write the 1 — only the sign (if negative) and the variable: 1xy² = xy² and −1xy² = −xy². The coefficient 1 is always invisible in final answers.
False. 3x and 2y are unlike terms — they have different variable parts (x vs y). They cannot be combined. The expression 3x + 2y is already fully simplified.
7. Simplify: ½x + ¾x − ¼x
All x-terms. Coefficients: ½ + ¾ − ¼ = 2/4 + 3/4 − 1/4 = 4/4 = 1. Answer: x
8. Challenge: A rectangle has length (3x + 5) and width (x − 2). Write a simplified expression for the perimeter.
Perimeter = 2(length + width) = 2[(3x + 5) + (x − 2)] = 2[4x + 3] = 8x + 6. (Collecting like terms inside the brackets first: 3x + x = 4x and 5 − 2 = 3.)
Summary
Lesson Checklist
You Can Now
Identify like terms by comparing variable parts (letters and exponents)
Collect like terms by adding or subtracting their coefficients
Track signs correctly when reordering or grouping terms
Recognise unlike terms and leave them unsimplified
Handle expressions with multiple variables and mixed powers
Write results with invisible coefficients of 1 correctly
Up Next → Lesson 12
The Distributive Property
Learn to expand brackets — and discover why collecting like terms becomes twice as powerful once brackets are in play.