Writing Algebraic Expressions

What is an algebraic expression?

An algebraic expression is a way of writing a calculation using letters as well as numbers. The letter is called a variable, and it stands for a number we don't know yet or a number that can change.

For example, if you don't know how many marbles are in a bag, you can call that number n. If you then add 4 more marbles, you have written your first expression:

n + 4

This single piece of algebra describes the situation for any starting number. If the bag held 10, you'd have 14; if it held 25, you'd have 29. The expression captures all of those cases at once. That is the real power of algebra, and it is the same idea you met in Algebra Basics.

Choosing a variable

The first step in writing an expression is to decide what the variable represents and give it a letter. Be specific. Don't just write "let x be the number" — write "let x be the number of students" so you remember what it means.

You can use any letter, but x, n, y, and a are the most common. Whatever you choose, use the same letter every time you mean the same quantity in one problem.

Key words and what they mean

Most word problems contain signal words that tell you which operation to use. Learning these is like learning vocabulary in a new language.

Words	Operation	Example phrase	Expression
sum, plus, more than, increased by, total	add (+)	"8 more than x"	x + 8
difference, minus, less than, decreased by, fewer	subtract (−)	"x decreased by 5"	x − 5
product, times, twice, of, multiplied by	multiply (×)	"the product of 3 and x"	3x
quotient, divided by, per, split into	divide (÷)	"x divided by 2"	x/2

Notice that "twice" means times 2, "triple" means times 3, and "half of" means divided by 2.

Worked example 1 — addition phrase

Write "the sum of a number and 12" as an expression.

1. Choose a variable. Let the number be n.
2. Find the key word. "Sum" means addition.
3. Combine the parts: the number (n) plus 12.
4. Write it: n + 12.

Read it back: "n plus 12" — yes, that is the sum of a number and 12. ✓

Worked example 2 — the tricky "less than"

Write "7 less than a number" as an expression.

This one traps a lot of students. "Less than" reverses the order you might expect.

1. Choose a variable. Let the number be x.
2. "7 less than x" means you start with x and take away 7.
3. So the x comes first: x − 7.

It is not 7 − x. To see why, imagine the number is 20. "7 less than 20" is 13, which is 20 − 7. Writing 7 − 20 would give −13, the wrong answer. Always start from the number that follows "less than."

Worked example 3 — multiplication and addition together

Write "double a number, then add 5" as an expression.

1. Choose a variable. Let the number be m.
2. "Double" means times 2, so we have 2m.
3. "Then add 5" means + 5.
4. Put it together: 2m + 5.

Be careful: 2m + 5 is different from 2(m + 5). The first doubles the number and then adds 5. The second adds 5 first and then doubles everything. The words "double a number, then add" tell us to double first, so 2m + 5 is correct.

Worked example 4 — a real situation

A taxi charges a $3 fixed fee plus $2 for every kilometre travelled. Write an expression for the total cost.

1. Decide the variable. Let k be the number of kilometres.
2. The cost for the kilometres is $2 each, so that is 2k.
3. There is also a fixed $3, no matter the distance.
4. Total cost: 2k + 3.

Check with a number: for 5 km, 2(5) + 3 = 10 + 3 = $13. That matches "five kilometres at $2 each plus $3." ✓ Building expressions like this is the first step toward solving linear equations later on.

Activity — translate these phrases

Write an expression for each. Use n as your variable.

1. Four more than a number.
2. A number multiplied by 6.
3. Ten less than a number.
4. A number divided by 3.
5. Triple a number, then subtract 2.
6. The sum of a number and itself.

Answers: 1) n + 4 2) 6n 3) n − 10 4) n/3 5) 3n − 2 6) n + n (which simplifies to 2n)

Why this matters

Almost every word problem in algebra, science, and everyday money maths begins with translating a sentence into an expression. If you can reliably turn "$3 plus $2 a kilometre" into 2k + 3, you can then evaluate it, graph it, or solve an equation with it.

Get into two habits: always state what your variable means, and always read your expression back in words to check it matches the original sentence. Those two checks catch almost every mistake.