Expressions & Order of Operations

A letter stands for a number you have not been told yet, and the order of operations settles what a written expression means.

A club charges $5 a person plus a $2 booking fee.

Three people pay 3 × 5 + 2. Seven people pay 7 × 5 + 2. Twelve pay 12 × 5 + 2.

one rule, written once the long way 3 people: 3 × 5 + 27 people: 7 × 5 + 212 people: 12 × 5 + 240 people: … 5n + 2 any n at all the letter stands for the number you have not been told yet

The same sentence, over and over, with one number changing. Writing it out for every group is tedious.

So write it once: 5n + 2, where n is however many people turn up.

The letter is not mysterious. It is a placeholder for the number you have not been told, and the expression is the rule that will handle any of them.

The parts of an algebraic expression reading an expression is half of algebra 3 x 2 + 5 x - 8 coefficient variable exponent term term constant term 3 terms degree 2 one variable + and - signs cut an expression into terms; the sign in front belongs to the term after it so the third term is -8, not 8

Writing a rule with a letter

Look for what stays the same and what changes. The changing thing gets the letter.

"Five dollars each plus two" becomes 5n + 2, whatever n turns out to be.

5n means 5 × n; the multiplication sign is left out to save writing.

What does the letter stand for

Say it in words: "n is the number of people". A letter without a meaning attached is where word problems go wrong.

The order of operations

Brackets first, then exponents, then multiplication and division, then addition and subtraction.

Multiplication and division rank equally and are done left to right; so do addition and subtraction.

2 + 3 × 4 is 14, not 20, because the multiplication goes first.

Equal ranks go left to right

12 ÷ 3 × 2 is 8, not 2. Division does not automatically wait for multiplication — whichever comes first goes first.

Exponents

53 means 5 × 5 × 5 = 125. The small number counts the factors.

It is not 5 × 3. That confusion is the most common exponent error at this level.

24 = 16, while 2 × 4 = 8 — the two answers drift apart quickly.

Say it aloud

"Five to the third" means three fives multiplied. Reading it as a count of factors keeps it straight.

Evaluating

Substitute the number for the letter, putting it in brackets, then follow the order of operations.

For 5n + 2 with n = 7: 5(7) + 2 = 37.

Brackets matter when the value is negative: 3n2 with n = -2 is 3(-2)2 = 12, not -12.

Brackets around the substitution

Writing 3 · -22 invites squaring before the minus is applied. Brackets remove the ambiguity entirely.

Equivalent expressions

3(x + 4) and 3x + 12 give the same answer for every x, so they are equivalent.

Expanding a bracket multiplies each term inside by what is outside.

Collecting like terms goes the other way: 4x + 3x is 7x, because seven xs is seven xs.

Test with a number

Unsure whether two expressions match? Put x = 2 into both. If they disagree they are not equivalent; if they agree, try one more.

Worked examples

Order of operations

Work out 2 + 3 × 42.

  1. Exponent first: 42 = 16. Before any multiplying.
  2. Then multiply: 3 × 16 = 48. Multiplication outranks addition.
  3. Then add: 2 + 48 = 50. Working left to right would have given 400.

50

Writing an expression

A taxi charges $3 to start and $2 a mile. Write the cost for m miles.

  1. The $3 is paid once, whatever happens. A constant term.
  2. The $2 is paid for each mile, so that is 2m. The changing part gets the letter.
  3. Total: 2m + 3. Check with m=4: $11, which matches $3 + $8.

2m + 3

Evaluating

Find the value of 4n2 - 5 when n = 3.

  1. Substitute with brackets: 4(3)2 - 5. Brackets keep the order clear.
  2. Exponent first: (3)2 = 9. Before multiplying by 4.
  3. 4 × 9 - 5 = 36 - 5 = 31. Multiplication, then subtraction.

31

Words to know

Variable
A letter standing for a number.
Coefficient
The number multiplying a variable — the 5 in 5n.
Term
A piece of an expression separated by + or -.
Evaluate
To work out the value once you know what the variable is.
Exponent
A small raised number counting how many times to multiply.

Mistakes to watch for

Working strictly left to right. 2 + 3 × 4 is 14, not 20 — multiplication goes before addition whatever the order on the page.

Reading 53 as 5 × 3. It means 5 × 5 × 5 = 125, not 15. The exponent counts the factors.

Doing multiplication before division out of habit. 12 ÷ 3 × 2 is 8: they rank equally, so the leftmost goes first.

Squaring without brackets after substituting. For n = -2, 3n2 is 3(-2)2 = 12; writing 3 · -22 makes it look like -12.

Multiplying only the first term in a bracket. 3(x+4) is 3x + 12, not 3x + 4 — everything inside gets multiplied.

Using letters for unknowns is due to Viète in the 1590s. Before that, algebra was written out in sentences and was far harder to work with.

The convention that multiplication comes before addition is just that — a convention — but it is universal, and every calculator and programming language obeys it.

Spreadsheets are expressions with letters: a formula in a cell is a rule waiting for whatever numbers arrive.

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