Variable
A variable is a letter or symbol used in algebra to represent an unknown or changing quantity.
Definition
A variable is a letter or symbol, such as $x$, $y$, or $n$, that stands in for a value we do not know yet or that can change, think of it as a blank in a fill-in-the-blank question. It represents a quantity within an expression, equation, or formula, and can act either as an unknown with one fixed value to find, or as a genuine placeholder that varies, as in a function rule like $y = 2x$. Formally, a variable is a symbol bound to a domain of values within a mathematical context: a free variable ranges over its entire domain, while a bound variable is quantified (as in "for all $x$"), and in ring theory a polynomial ring $R[x]$ treats $x$ as an indeterminate rather than a variable ranging over real numbers.
Example
In the expression $x + 5$, the letter $x$ is a variable: if $x = 3$ the expression equals $8$, and if $x = 10$ it equals $15$. In the equation $2x + 3 = 11$, solving for the variable $x$ gives $x = 4$, and in the formula $A = lw$, both $l$ and $w$ are variables representing length and width. In the polynomial $f(x) = 3x^2 - x + 7$, $x$ is an indeterminate defined for any value in the domain; substituting $x = 2$ yields $f(2) = 12 - 2 + 7 = 17$.
Key Insight
Variables let us write one rule that works for many different numbers instead of writing out every case separately, whether they act as an unknown to solve for or as an indeterminate in a ring. That distinction, solving equations versus studying polynomial structure, underpins much of algebra.