Cracking the Code: Algebra Expressions

Imagine a machine. You drop a number in. The machine triples it and adds 4, then spits the result out. If you drop in 2, you get 10. If you drop in 5, you get 19. If you drop in 0, you get 4. Here is the puzzle: can you write a single rule — using a letter for any number you drop in — that describes what this machine always does? That one rule is an algebraic expression, and by the end of this lesson you will write it yourself.

In maths, a variable is a letter — such as x, n, or t — that stands in for a number we do not yet know (or a number that can change). An algebraic expression is a combination of variables, numbers, and operation signs that represents a value. Crucially, an expression has no equals sign — it is not an equation. It is more like a recipe that tells you what to do with a number. Every expression is built from parts called terms. A term is a single number, a single variable, or a number multiplied by a variable. For example, in the expression 3x + 7, the terms are 3x and 7. Let's name every part precisely: • A coefficient is the number written directly in front of a variable. In 3x, the coefficient is 3. It means '3 multiplied by x'. • A constant is a term that is just a number with no variable — it stays fixed whatever x is. In 3x + 7, the constant is 7. • Variables, as noted above, are the letters. Operations work exactly as in arithmetic: 3x means 3 × x; x² means x × x; x/2 means x ÷ 2. Writing expressions from words is a key skill. 'Five more than a number n' → n + 5. 'Four times a number, minus 1' → 4n − 1. Notice that 4n is standard shorthand for 4 × n — mathematicians drop the multiplication sign when a number sits directly next to a letter. Evaluating an expression means replacing the variable with a specific number and then calculating the result. This is called substitution.