Anatomy of an expression
Take the expression 5x² − 3x + 2y − 7:
| Term | Coefficient | Variable part |
|---|---|---|
| 5x² | 5 | x² |
| −3x | −3 | x |
| 2y | 2 | y |
| −7 | — | none (a constant term) |
The sign in front of a term belongs to it. In 5x² − 3x, the coefficient of x is −3, not 3.
Hidden coefficients
- x has coefficient 1.
- −y has coefficient −1.
- x/4 has coefficient 1/4.
Like and unlike terms
Like terms have exactly the same variables with the same exponents. The coefficients can differ.
| Like terms | Why |
|---|---|
| 4x and −9x | both x |
| 3a²b and a²b | both a²b |
| 7 and −2 | both constants |
| Unlike terms | Why |
|---|---|
| 3x and 3x² | different powers |
| 2a and 2b | different variables |
| 5xy and 5x | different variable parts |
Combining like terms
You can combine like terms by adding or subtracting their coefficients. The variable part stays the same.
- 4x + 9x = 13x
- 6a − 8a = −2a
- 3x² + x² = 4x²
Think of it like objects: 4 apples + 9 apples = 13 apples. But 3 apples + 2 bananas can’t become 5 of anything, and 3x + 2y stays as it is.
Simplifying an expression
- Find like terms (underline or colour them).
- Group them together.
- Combine.
Example: 5x + 3 − 2x + 8 + y
- x terms: 5x − 2x = 3x
- constants: 3 + 8 = 11
- y terms: y
- Result: 3x + y + 11
Worked examples
Example 1: Identify the coefficient of y in 6x − y + 4.
−1.
Example 2: Are 7ab and 7ba like terms?
Yes: ab and ba are the same product. Like terms.
Example 3: Simplify 2m² + 5m − m² + 3m.
m² terms: m²; m terms: 8m → m² + 8m.
Common mistakes
- Combining unlike terms: 3x + 2x² ≠ 5x³.
- Dropping negative signs: in 4 − 6x, the coefficient is −6.
- Changing the exponent when combining: x² + x² = 2x², not x⁴.
Try it yourself
- List the terms in 8a − 3b + 5.
- Coefficient of x² in 4 − x²?
- Which are like terms: 5p, 5p², −p, 3q?
- Simplify 7y − 2 + 3y + 9.
- Simplify 4ab − ab + 2a.
Answers: 1) 8a, −3b, 5 2) −1 3) 5p and −p 4) 10y + 7 5) 3ab + 2a
Quick recap
- A term has a coefficient and a variable part; signs belong to terms.
- Like terms share exactly the same variable part.
- Combine like terms by adding coefficients.