A variable is a symbol, usually a letter like x or n, that stands in for a number without committing to what that number actually is, whether because its value is genuinely unknown and needs to be found, because it's fixed but hasn't been specified in a given problem, or because it's deliberately allowed to vary across a whole range of possible values. An algebraic expression combines one or more variables with numbers and mathematical operations into a single phrase, 3x + 5, say, that can be simplified, evaluated for a specific value of the variable, or set equal to something else and solved.
A variable lets a single expression stand for an entire family of numerical relationships at once
Writing 3x + 5 instead of working out 3 times some specific number plus 5 lets that one expression describe the same underlying relationship for every possible value x could take, rather than writing out a separate calculation for each one individually. This is exactly the power a variable adds over arithmetic with only fixed numbers: it lets a mathematician state a general pattern or rule once, algebraically, and apply that same rule to as many specific numerical cases as needed, without redoing the underlying reasoning from scratch each time.
Simplifying, evaluating and solving are three genuinely different things to do with an expression
Simplifying an algebraic expression means rewriting it in an equivalent but more compact form without changing its value, combining like terms, for instance. Evaluating an expression means substituting a specific number in for the variable and calculating the resulting numerical value. Solving, which only applies once an expression is set equal to something in an equation, means finding exactly which value or values of the variable make that equation true. Keeping these three genuinely distinct operations straight is exactly where a lot of early algebra confusion actually comes from, since they use overlapping vocabulary and notation but do meaningfully different mathematical jobs.
What we're still unsure about
That variables let algebraic expressions state general numerical relationships once rather than case by case, and that simplifying, evaluating and solving are genuinely distinct operations on those expressions, are well established, foundational mathematics taught consistently for centuries. What's more genuinely a matter of ongoing pedagogical research is exactly why so many students, even ones fluent in arithmetic, initially struggle specifically with the shift to variables, since that transition asks students to reason about a quantity's general behaviour rather than any one specific number, and maths education researchers continue to study exactly which teaching approaches most reliably help that conceptual shift actually take hold.
This sits inside Variables & Algebraic Expressions, one of eight topics in Algebra, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.