Quick Summary
- Forming expressions from words — translate phrases like "3 more than something" or "half of something" into algebra using a letter (usually x).
- Brackets matter — the order operations happen in changes the expression completely.
- Choosing your unknown wisely — sometimes letting the *bigger* quantity be a multiple of x makes the algebra much simpler than using fractions.
- Forming equations — spot the phrase "is equal to" (or its hidden version) to know where the = sign goes.
- Forming equations from shapes — use known geometric facts (perimeter, area, angle sums, angles on parallel lines, volume of prisms) as the "equal to" statement.
- Problem solving — equations can be linear or quadratic; you must always give your final answer in context, using the words and units of the question.
- Always check your answer makes sense — e.g. a side length or age can't be negative.
1. Forming Equations from Words
1 Turning phrases into expressions
Before you can write an equation, you need to be comfortable writing expressions. An expression is just a piece of algebra with no equals sign — it doesn't "solve" to anything, it just represents a value in terms of the unknown. The whole skill here is translation: you're converting English into algebra, word by word.
Always start by letting x represent the unknown value the question is describing. Then look for keywords that tell you which operation is happening.
| Phrase | Expression |
|---|
| 2 less than "something" | x − 2 |
| Double "something" | 2x |
| 5 lots of "something" | 5x |
| 3 more than "something" | x + 3 |
| Half of "something" | x/2 |
Keyword Bank
Addition: sum, total, more than, increase
Subtraction: difference, less than, decrease
Multiplication: product, lots of, times as many, double, triple
Division: shared, split, grouped, halved, quartered
2 Why brackets matter — order changes everything
This is one of the most common places marks are lost. The order in which operations happen in a sentence must be mirrored exactly in your algebra. Compare these two very similar-sounding sentences:
Same numbers, different meaning
Sentence A: "something" add 1, then multiplied by 3
You do the addition first, so it must be trapped in brackets before the multiplication happens:
(x + 1) × 3 = 3(x + 1)
Sentence B: "something" multiplied by 3, then add 1
Here the multiplication happens first, so no brackets are needed:
x × 3 + 1 = 3x + 1
Why this trips people up
If you read too fast, "add 1 then multiply by 3" and "multiply by 3 then add 1" look almost identical on the page — but they give completely different expressions (and different final values!). Always slow down and process the sentence in the order the actions happen, not the order the numbers appear.
3 Choosing which unknown to call x
choose
x
Take this example: "Adam is 10 years younger than Barry." This is the same fact as saying "Barry is 10 years older than Adam." You have two valid options: