Algebra is arithmetic with a letter standing in for a number you do not know yet. Solving an equation just means undoing whatever was done to that letter.
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An equation is a balance. Whatever you do to one side you must do to the other, and the balance holds.
⚖️ Undo in reverse order
Look at what has been done to x. In 3x + 5 = 20, x was multiplied by 3, then 5 was added.
Undo the last thing first: subtract 5 from both sides → 3x = 15.
Undo the next: divide both sides by 3 → x = 5. Then check: 3(5) + 5 = 20 ✅
A term changes sign when it crosses the equals sign. From 5x = 2x + 9, moving 2x across gives 5x − 2x = 9, so 3x = 9 and x = 3. Forgetting the sign change is the commonest error in this chapter.
Factorising is expanding run backwards: turning a sum into a product.
🔍 The three methods
| Method | Use when | Example |
|---|---|---|
| Common factor | every term shares something | 6x + 9 = 3(2x + 3) |
| Grouping | four terms | ax + ay + bx + by = (a+b)(x+y) |
| Identity | it matches a known pattern | x² − 25 = (x+5)(x−5) |
Always take out the common factor first. 2x² − 18 looks awkward until you write 2(x² − 9), and then the difference of squares gives 2(x+3)(x−3).
Turn the sentence into an equation one phrase at a time, then solve as usual.
📝 From words to algebra
Let the unknown be x and say clearly what it stands for.
Translate phrase by phrase: "5 more than twice a number" → 2x + 5.
Form the equation, solve it, and read the answer back into the question.
Always check against the story, not just the algebra. If x comes out as −3 ages or 2.5 chairs, you have set the equation up wrongly even if the arithmetic is right.
12 questions on expressions, factorising and equations. Check your answer by substituting. 🍀
