Mathématiques
What a Letter Is Doing in an Algebra Question
A letter in algebra does one of three jobs: an unknown to find, a quantity that changes, or any number at all in a general rule. How to tell which is which.
Variable
A variable is a letter used in place of a number — one that is unknown, one that changes, or any number at all in a general statement.
- Où les élèves le rencontrent
- Grade 6 mathematics, when letters replace the empty boxes used in primary arithmetic, and in every algebra topic afterwards.
La réponse en bref
A variable is a letter standing in for a number, and it does one of three jobs: an unknown with one value to find, as in 3x + 2 = 11; a quantity that varies, as t does in d = 60t; or any number at all, as in a + b = b + a.
Un exemple
3x + 2 = 11 d = 60t a + b = b + a
In the first, x has exactly one value, 3, and the job is to find it. In the second, t can be any time and each value of t produces a distance — the letters vary together. In the third, a and b stand for every number at once; the statement is not something to solve but something that is always true.
Three jobs, one letter
'What does x mean' has no single answer, and that is the actual difficulty students hit rather than the notation. The letter means something different depending on what the question is doing with it.
Naming which job is in play tells you what you are allowed to do next. You solve an unknown. You substitute into a varying quantity. You do not try to solve a general rule at all — attempting it gives you 0 = 0, which is true and useless.
- Unknown — 3x + 2 = 11. One value satisfies it, and the task is to find x = 3.
- Varying quantity — d = 60t. Every value of t gives a value of d, and the pair can be plotted.
- Generality — a + b = b + a. True for all numbers, and the letters are there to say 'any'.
The same letter can change jobs mid-question
In y = 2x + 1, x is an input you choose and y is whatever comes out. Then the next part of the question says 'find x when y = 9', and x has quietly become an unknown to be solved for.
The letter did not change and the equation did not change; the job did. Recognising that shift is what makes a multi-part algebra question feel coherent instead of arbitrary, and it is usually the thing a student who says 'I can do the working but I don't know what I'm doing' is missing.
Conventions and one trap
Letters carry habits. n usually means a whole number or a position in a sequence; x and y are coordinates and general variables; t is time. Within a single question a letter keeps one meaning throughout, and 2x always means 2 × x while xy means x × y.
The trap worth knowing is that a variable stands for a number, never for an object. In a school with six times as many students as teachers, the correct statement is s = 6t. Writing 6s = t reads the letters as labels — 'six students, one teacher' — instead of as quantities, and it is a mistake that survives well into GCSE if nobody names it.
Questions fréquentes
What is the difference between a variable and a constant?
A variable can take different values; a constant is fixed. In y = 2x + 5, x and y vary while 2 and 5 stay put. Letters can be constants too: in y = mx + c, m and c are fixed for any one straight line even though they change from line to line.
Is a variable the same as an unknown?
Not quite. An unknown is a variable being used for one particular job — standing for a specific value you are trying to find. The x in 3x + 2 = 11 is an unknown. The x in y = 2x + 1, which takes every value along the axis, is a variable but not an unknown.
Can a variable stand for more than one number at a time?
Depending on the job, yes. In a general rule like a + b = b + a, the letters stand for all numbers at once. In an inequality such as x > 4, x stands for a whole range. Only when it is functioning as an unknown in an equation does it usually pin down to one value.
What does a question mean by 'in terms of x'?
It means your answer should be an expression containing x rather than a number. 'Write the perimeter in terms of x' expects something like 6x + 4. It is a signal that x will not be given a value in this part of the question, so there is nothing to calculate to.
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