Maths
What is an equation and how do you solve one?
Quick answer
An equation says two things are equal, using an = sign, like 2x + 1 = 9. Solving it means finding the value of the letter that makes it true, here x = 4.
In one sentence
An equation is a maths sentence with an equals sign, and solving it means finding the missing number that makes both sides balance.
Explained
The equals sign does not mean "here comes the answer". It means "is the same as". In 3 + 5 = 2 + 6, both sides are worth 8, so the equation is true.
Think of an equation as a set of old-fashioned weighing scales. Whatever is on the left weighs exactly the same as whatever is on the right.
apple- apple
- apple
box
In algebra, one part of the equation is unknown. Take x + 4 = 11. To find x, take 4 away from both sides so the scales stay balanced: x = 11 − 4 = 7.
The golden rule: whatever you do to one side, do to the other. Each step uses the inverse (opposite) operation to peel away what has been done to the letter:
- undoes −, and − undoes +
- × undoes ÷, and ÷ undoes ×
Two-step equations. Solve 2x + 3 = 13.
- Subtract 3 from both sides: 2x = 10.
- Divide both sides by 2: x = 5.
- Check by putting it back in: 2 × 5 + 3 = 13. ✓
Undo the adding or subtracting first, then the multiplying or dividing. That is the order of operations in reverse.
Letters on both sides. For 7x − 4 = 3x + 8, first get the letters together. Subtract 3x from both sides: 4x − 4 = 8. Add 4: 4x = 12. Divide by 4: x = 3.
Real problems become equations too. "I think of a number, multiply it by 5 and subtract 6. The answer is 29." That is 5n − 6 = 29, so 5n = 35 and n = 7.
Examples
| Equation | Steps | Answer |
|---|---|---|
| x − 9 = 4 | add 9 | x = 13 |
| 6m = 54 | ÷ 6 | m = 9 |
| p/3 = 5 | × 3 | p = 15 |
| 4k + 1 = 25 | − 1, then ÷ 4 | k = 6 |
| 2(t + 3) = 18 | ÷ 2, then − 3 | t = 6 |
| 10 − y = 4 | add y, then − 4 | y = 6 |
Common mistakes
- Doing something to one side only. Fix: write the operation on both sides, every line.
- Using the wrong inverse, such as adding 3 to "undo" 3x. Multiplying is undone by dividing.
- Not checking. Substituting the answer back takes ten seconds and catches most errors.
- Getting the sign wrong in equations like 10 − y = 4. Fix: add y to both sides first.
Quick quiz
Equations
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Solve x + 7 = 15.
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Related topics
Algebra
Algebra is maths that uses letters, like x or n, to stand for numbers we don't know yet or numbers that can change. For example, if n + 3 = 10, then n = 7.
Order of operations
The order of operations is the agreed order for working out calculations: brackets, then powers, then × and ÷, then + and −. UK pupils learn it as BIDMAS; US pupils as PEMDAS.
Sequences
A sequence is a list of numbers or shapes that follows a rule, like 3, 7, 11, 15 (add 4 each time). Each item is called a term.
Subtraction
Subtraction is taking one amount away from another, or finding the difference between two numbers. The sign is −, so 9 − 4 = 5.
Composite functions
A composite function applies one function and then another. f(g(x)), also written (f ∘ g)(x) or fg(x), means do g first, then put the answer into f.
Inequalities
An inequality compares two values with <, >, ≤ or ≥, like x + 2 > 5. Solve it like an equation, but flip the sign if you multiply or divide both sides by a negative number.