Expressions, formulae and equations
Constructing and solving equations
An equation is a balance. Whatever you do to one side, do to the other — and the unknown is soon left on its own.
Solving by doing the opposite
To solve an equation, undo whatever has been done to the letter — and do the same to BOTH sides. x + 6 = 10 x + 6 − 6 = 10 − 6 x = 4 x − 6 = 10 x − 6 + 6 = 10 + 6 x = 16 2x = 10 2x ÷ 2 = 10 ÷ 2 x = 5
When the letter is on the right
If the unknown is on the right-hand side, just turn the equation round first. 12 = y + 3 Write it as y + 3 = 12 Then y + 3 − 3 = 12 − 3 y = 9
Building an equation from words
'I think of a number and add 3. The answer is 18.' Let the number be n. Then n + 3 = 18, so n = 15. 'I think of a number and multiply it by 4. The answer is 24.' That gives 4n = 24, so n = 6. Always CHECK by putting your answer back in: 15 + 3 = 18 ✓
Worked examples
Solve x + 6 = 10
- Undo the + 6 by subtracting 6.
- Do it to BOTH sides.
- x = 10 − 6.
Answer: x = 4
Solve x − 6 = 10
- Undo the − 6 by adding 6 to both sides.
- x = 10 + 6.
Answer: x = 16
Solve 2x = 10
- 2x means 2 × x, so undo it by dividing.
- Divide both sides by 2.
Answer: x = 5
Solve 12 = y + 3
- Turn it round first: y + 3 = 12.
- Subtract 3 from both sides.
Answer: y = 9
'I think of a number and add 3. The answer is 18.'
- Let the number be n.
- n + 3 = 18.
- Subtract 3 from both sides.
Answer: n = 15 — check: 15 + 3 = 18 ✓
'I think of a number and multiply by 4. The answer is 24.'
- 4n = 24.
- Divide both sides by 4.
Answer: n = 6
Where you see this in real life
- You paid £20 and got £6 change for 2 identical books: 20 − 2b = 6, so each book cost £7.
- Checking your answer by substituting it back is the fastest way to catch a slip — and it takes seconds.
Key words
- Equation
- a statement that two expressions are equal
- Solve
- find the value of the unknown
- Solution
- the value that makes the equation true
- Inverse operation
- the operation that undoes another
- Balance
- the idea that both sides of an equation stay equal