Thread Academy

Variables

The letter x is a container, not a mystery. Learn what variables stand for, how they vary, and how to substitute values into expressions.

  • Explain what a variable represents
  • Substitute values into expressions
  • Distinguish variables from unknowns

A letter that stands for a number

In algebra, letters stand for numbers. The letter x in 3x + 2 isn't mysterious — it's a placeholder. Sometimes it stands for one specific number we haven't found yet (an unknown in an equation). Sometimes it stands for any number (a variable in a formula).

Definition: variable

A variable is a letter representing a number that can change or vary. In an equation like 3x + 2 = 11, x is an unknown with one specific value; in a formula like C = πd, the letters are variables that can take many values.

Mathematicians traditionally use x, y, z for unknowns and a, b, c for known-but-unspecified numbers — but any letter works. In science you'll meet t for time, m for mass, v for velocity: letters chosen for meaning.

Substitution: putting numbers in

Substituting means replacing each letter with a number, then calculating. If x = 4:

Maths
3x + 2  →  3(4) + 2  =  12 + 2  =  14
Worked example

Find the value of 2a − 5b when a = 6 and b = 2.

Maths
2a − 5b  →  2(6) − 5(2)  =  12 − 10  =  2

Write the brackets in — 2(6) not 26 — and the arithmetic takes care of itself.

Watch out

3x means 3 × x, never 3 + x. And x² means x × x. When substituting a negative number, always use brackets: if x = −3, then x² = (−3)² = 9, not −9.

Variables in formulae

A formula is a rule written with variables. The area of a trapezium:

Area of a trapezium

A = ½(a + b)h

Here a and b are the parallel sides and h is the height. Substitute any measurements and the formula does the work — that's the power of variables: one rule, infinitely many cases.

Practice

Find the value of 5x − 3 when x = 7.
Replace x with 7, then calculate.

5(7) − 3 = 35 − 3 = 32.

If y = −2, what is y²? What is −y²?
Brackets first: (−2)².

y² = (−2)² = 4. But −y² = −(4) = −4 — the minus is applied after squaring.

The formula C = 5(F − 32)/9 converts Fahrenheit to Celsius. Find C when F = 77.
Substitute F = 77, then work the brackets first.

C = 5(77 − 32)/9 = 5(45)/9 = 225/9 = 25°C.

Quick check

Variables — quick check

If x = 5, what is 2x + 1?

2(5) + 1 = 10 + 1 = 11.

If n = −4, what is n²?

(−4)² = (−4) × (−4) = 16. A negative times a negative is positive.

In the equation 3x = 15, x is best described as…

The equation pins x down to exactly one value: x = 5.
Key takeaways
  • A variable is a letter standing for a number; an unknown is a variable with one specific value to find.
  • Substitute by replacing letters with numbers in brackets, then calculate.
  • 3x means multiplication; watch signs when substituting negatives.
  • Formulae turn variables into reusable rules.
Back to Grade 9 chaptersRelated resources →