Variables
A variable is a letter standing in for a number — one you have not found yet, or one that is allowed to change.
Picture a closed box with a number hidden inside. You are told the box plus 3 equals 7. You do not see the number, but you can reason about it. Algebra gives that hidden number a name — a single letter — so you can write about it before you know it. In the next few minutes you will see exactly how that name works — and why it is the key move that makes all of algebra possible.
After this lesson you can say what a variable is, read a letter as a stand-in for a number, see the two jobs a variable does, substitute a value for it, and explain why the same letter must mean the same number throughout a problem.
A variable is a letter that stands for a number. Instead of leaving a blank box,
mathematics writes a letter — often x or n — to mean “some number”. That letter is
a variable. It is not a new kind of number; it is a name for a number, used when
writing about the number is needed before its exact value is known.
A variable can be an unknown you need to find. “Some number plus 3 is 7” becomes
x + 3 = 7. Here x is one fixed number — you just have not worked out which yet.
The whole job is to find it. In this case a moment’s thought gives x = 4, because
4 + 3 = 7.
| Expression | Substitute x = 4 | Result |
|---|---|---|
| x + 3 | 4 + 3 | 7 |
| 2 × x | 2 × 4 | 8 |
| 10 − x | 10 − 4 | 6 |
A variable can also be a number that is allowed to change. If apples cost 2 coins
each, the cost of n apples is 2 × n. Here n is not one hidden value — it is
whatever number of apples you choose. Buy 3 and n is 3; buy 10 and n is 10. The
same letter, used as a placeholder for any value you might plug in.
Within one problem, the same letter always means the same number. If x appears
twice in the same problem, both xs are the identical number. You cannot let x be 4
in one line and 9 in the next — that would be two different numbers wearing one name.
When you read a problem and spot the same letter used twice, treat it as one fixed value throughout — then pick a fresh letter when you mean a genuinely different number.
A classroom has n students, and every student has 2 hands. Write the total number
of hands, then find it when n is 5.
Each student brings 2 hands, and there are n students — that is n groups of 2. The
total number of hands is 2 × n.
Now substitute: replace the variable n with the actual number 5. The total
becomes 2 × 5 = 10.
So with 5 students there are 10 hands. The expression 2 × n works for any class size
— substitute a different n and it gives that class’s answer.
▸Why this works
Why use a letter at all — why not just wait until you know the number? Because a letter
lets you write a rule before you have the number. 2 × n describes the hand count
for every possible class at once. Without a variable you could only ever talk about
one specific class. The letter is what lets one statement cover infinitely many cases.
▸Common mistake
A common mistake is thinking a particular letter has one fixed, famous value — that
x “is” some special number. It does not. A variable means whatever the problem says
it means; in the next problem the same x can be something else entirely. Read each
problem fresh: the variable’s value comes from that problem alone.
If x + 2 = 9, what number is x? Type it.
The variable n stands for 4. What is the value of n + 5? Type it.
The variable n stands for 4. What is the value of 2 × n? Type it.
If x stands for 6, what is the value of x? Type it.
If 10 − x = 3, what number is x? Type it.
What is a variable?
A variable is a letter that stands for a number. It does one of two jobs: it names an unknown you need to find, or it holds a value that is allowed to change. To use a variable’s value, substitute the number in its place and compute. Within one problem the same letter always means the same number — a letter is a name, and its value comes from the problem you are working on. Now when you see a letter in a maths problem, you will reach for one question first: is this a hidden value I need to find, or a placeholder that can be anything?
Practice
Start at the top. Tasks go easiest → hardest: recall a fact, apply it to a case, then a senior-level stretch. Open one, attempt it, then reveal.
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