Variables & Expressions
“A letter is a container for 'a number not yet decided'.”
The formula
2x + 3How to read it: some number x doubled, then with 3 added
- x
- — a number not yet fixed, free to vary
- 2x
- — twice x (2 × x, with the × sign dropped)
- 3
- — the constant term, always added regardless of x
The hook
x is just a 'slot' for a number that hasn't shown its face yet — nothing to fear.
In plain words
'Double some number and add 3' written compactly with the letter x as 2x+3.
The intuition
It's shorthand: what you'd spell out in a long sentence, squeezed into one letter. Whatever number x is, the rule 'double it and add 3' stays the same — the letter is the container for that rule.
How it's built
x is the changing number, the 2 in front is the coefficient it's multiplied by (the × sign is dropped), and the 3 at the end is the constant term, independent of x. Together they make one expression.
Example
Substitute x=5: 2·5+3 = 13. Drop a number into the x slot and the expression becomes a single value.
Common mistake
2x is not 2 'plus' x — it's 2 'times' x. And x² is completely different from 2x — one multiplies x by itself, the other adds x twice.
Where it's used
Area, perimeter, fares, speed — any situation where 'a number varies' gets captured in one line, and later you just swap in the value.
Where it came from
Writing unknowns as letters was shaped by Viète in the 1500s and locked in by Descartes, who fixed x, y, z as the unknowns we still use.
Prerequisites
Quick check
What is 3x − 1 when x = 4?
- 9
- 11✓
- 12
- 13
Practice
Find the value of 2x − 5 when x = 6.
Answer: 7
- 2·6 − 5 = 12 − 5
- = 7
Takeaway: Drop a number into the letter's slot and the expression becomes one value.
What is 3a + b when a = 4 and b = −2?
Answer: 10
- 3·4 + (−2) = 12 − 2
- = 10
Takeaway: When substituting a negative, wrap it in parentheses to keep the sign.
Write, in terms of x, the cost of 5 pencils at x won each plus one 500-won eraser.
Answer: undefined
- 5 pencils: 5 × x = 5x
- eraser: 500
- total: 5x + 500
Takeaway: 'Varying amounts' become letters; 'fixed amounts' stay constants.