MISSION · Algebra
Variables and Like Terms
What a letter is standing in for, and which terms can combine
LEVEL 1 · EXPLORE
Make the idea move
Change the controls and watch what happens. Follow your curiosity and see what you notice.
Loading the activity…
LEVEL 2 · LEARN
Unlock the explanation
Open the chapters in any order. Try the examples as you go.
The idea in one sentence
A letter holds a number you don’t know yet, and terms can only be added together when they hold the same kind of thing.
Reading an expression
In \(3x + 5\):
- \(x\) is the variable — an unknown, or a quantity that varies.
- \(3\) is the coefficient — how many \(x\)’s there are.
- \(5\) is a constant — a fixed number that doesn’t depend on \(x\).
- \(3x\) and \(5\) are the two terms, the pieces separated by \(+\) and \(-\).
\(3x\) means \(3 \times x\). The multiplication sign is dropped because in algebra it looks too much like another letter, and \(3x\) is quicker to read anyway.
An expression has no equals sign. It’s a quantity, not a question. \(3x + 5\) is something you can evaluate once you know \(x\); \(3x + 5 = 20\) is an equation, and that’s a different job.
Like terms
You can add 3 apples to 2 apples and get 5 apples. You cannot add 3 apples to 2 oranges and get 5 of anything. Algebra works the same way:
\[3x + 2x = 5x \qquad \text{but} \qquad 3x + 2y \text{ stays as it is}\]Like terms have exactly the same variable part. \(3x\) and \(2x\) are like. \(3x\) and \(2y\) are not. \(3x\) and \(3x^2\) are not either — \(x\) and \(x\) squared are different things, in the same way that a length and an area are.
To simplify, collect each family and add their coefficients:
\[4x + 7 + 2x - 3 = 6x + 4\]Six \(x\)’s and a spare 4. The subtraction travels with the 3 it sits in front of.
Multiplying out a bracket
\[3(x + 4) = 3x + 12\]Three lots of \((x + 4)\) means three lots of \(x\) and three lots of 4. Every term inside gets multiplied, not just the first — missing the second one is the most common slip in the whole of early algebra.
A minus sign outside flips every sign inside: \(-2(x - 5) = -2x + 10\).
Why bother
Because an expression written once can be used for any number. “A taxi charges £3 plus £2 a mile” becomes \(2m + 3\), and that single line answers the fare for every journey anyone will ever take in it. Writing the general case is the thing algebra is for.
Common mix-ups
- Combining unlike terms. \(3x + 2y\) cannot be simplified. Leaving it alone is the right answer.
- Dropping the variable. \(3x + 2x\) is \(5x\), not 5.
- Reading \(x^2\) as \(2x\). One is \(x\) times \(x\), the other is \(x + x\).
- Multiplying only the first term in a bracket. Every term inside gets it.
Check yourself
Simplify \(5a + 3 - 2a + 8\), then expand \(4(2n - 3)\).
\(3a + 11\), collecting the \(a\) terms and the constants separately. And \(8n - 12\), multiplying both terms inside by 4.
LEVEL 3 · REFLECT
Add this quest to your trail
When you feel ready, mark this idea as explored. You can always come back and try it again.
EXPLORER STRATEGIES