Variables & Expressions
What a letter is doing in a maths problem, and how to read one.
The explanation
A variable is a letter standing in for a number. In 3n + 5, the n could be any number, and the expression tells you what to do once you know it.
Reading the notation matters. 3n means 3 times n — multiplication is implied by writing them together. n/4 means n divided by 4. n² means n times itself.
An expression has no equals sign. It is a recipe, not a question. 3n + 5 is an expression; 3n + 5 = 20 is an equation, and only equations get solved.
An algebraic expression is a finite combination of constants, variables and operations. Its structure is a tree: in 3n + 5 the root is addition, with 3n and 5 as its two operands, and 3n itself is a product. Reading that structure correctly is what determines which operations you are allowed to move.
Vocabulary that later instructions depend on: terms are the pieces separated by + or −; the coefficient is the numeric factor of a term; a constant is a term with no variable. In 7x² − 4x + 9 there are three terms, coefficients 7 and −4, and constant 9. The sign belongs to the term that follows it, which is why −4x is the term rather than 4x.
Juxtaposition denoting multiplication is a convention adopted because multiplication is the most common operation in algebraic work; it is also why the × symbol is abandoned, since it collides visually with the variable x.
Worked example
Write an expression for: five less than twice a number, then name its terms.
- Twice a number: 2n.
- Five less than that: subtract 5 from it, so 2n − 5.
- Terms: 2n and −5.
Answer: 2n − 5, with coefficient 2 and constant −5
Common mistakes
- Writing 'five less than 2n' as 5 − 2n. 'Less than' reverses the order.
- Reading 3n as 'three and n' rather than 3 times n.