Variables and Expressions — Mathematics Knowledge System

Overview

A variable is a symbol, usually a letter like x, y, or n, that stands in for a number whose value is unknown, unspecified, or free to vary. An expression combines variables, numbers, and operations (like +, −, ×, ÷) into a single mathematical phrase — but unlike an equation, it makes no claim of equality and cannot be 'solved,' only simplified or evaluated. Variables and expressions are the basic vocabulary of algebra: they let us write general statements about numbers instead of working with one specific number at a time.

Intuition

Think of a variable as a labeled box that can hold a number. If you write 'x + 3,' you're describing a recipe — 'take whatever number is in the box, and add 3' — without knowing yet what's in the box. The same expression can be evaluated for many different values: if x = 2, the expression becomes 5; if x = 10, it becomes 13. This is the whole power of algebra: one expression captures a pattern that would otherwise take infinitely many separate arithmetic statements to describe.

Formal Definition

Definition

A variable is a symbol representing an element of a specified set (often the real numbers). An algebraic expression is a combination of variables, constants, and operation symbols, built up according to the usual rules of arithmetic, that represents a number once every variable is assigned a value.

3x + 5

A variable x, a coefficient 3, and a constant term 5, combined by multiplication and addition

Expression

2x^2 - 7x + 1

A polynomial expression in the single variable x

Expression

Notation

Notation	Meaning
$x, y, n$	Common letters used as variables
$3x$	Coefficient 3 multiplied by the variable x (the multiplication symbol is omitted by convention)
$f(x)$	A variable expression viewed as a function of x

Properties

Like terms

ax + bx = (a+b)x

Condition: Terms with the same variable raised to the same power can be combined by adding their coefficients.

Distributive property

a(b + c) = ab + ac

Condition: Used to expand or factor expressions containing parentheses.

Substitution / evaluation

\text{Replace each variable with its given numeric value, then compute.}

Applications

Variables in programming languages are a direct descendant of algebraic variables — a named storage location whose value can change.

Worked Examples

Substitute x = 3 into the expression.
$4(3) + 7$
Multiply first, following order of operations.
$12 + 7$

Answer: 19

Practice Problems

Difficulty 1/10

Evaluate 5x - 2 when x = 4.

Difficulty 2/10

Simplify: 6a + 3b - 2a + b.

Common Mistakes

Common Mistake

Combining terms that are not actually 'like terms,' e.g. simplifying 3x + 2x² to 5x³ or 5x².

Only terms with the exact same variable part (same variable(s) raised to the same power) can be combined. 3x and 2x² have different powers of x and must stay as 3x + 2x².

Common Mistake

Forgetting to distribute a coefficient to every term inside parentheses, e.g. writing 2(x + 3) as 2x + 3.

The distributive property requires multiplying the outside factor by EVERY term inside: 2(x + 3) = 2x + 6.

Summary

A variable is a symbol that stands in for a number that can vary or is not yet known.
An expression combines variables, numbers, and operations, but is not itself a statement of equality.
Expressions are evaluated by substituting specific values for the variables.
Like terms (same variable part) can be combined by adding their coefficients.
The distributive property, a(b+c) = ab + ac, is essential for expanding and simplifying expressions.

References

WebsiteWikipedia — Variable (mathematics)
WebsiteWikipedia — Expression (mathematics)