Solving linear equations
Maths · Key Stage 3 · The School
The balance rule
An equation is a set of scales: 3x + 4 = 19 says both pans weigh the same. Whatever you do to one side, do to the other, and the balance — the truth — survives. That single rule is all of equation-solving.
Undo in reverse order
The expression 3x + 4 was built by "times 3, then add 4". To find x, undo in REVERSE: subtract 4 from both sides (3x = 15), then divide both sides by 3 (x = 5). Check by substituting back: 3 × 5 + 4 = 19 ✓. Always check.
Letters on both sides
For 5x + 2 = 3x + 10, first gather the letters: subtract 3x from both sides → 2x + 2 = 10. Now it is a two-step equation: 2x = 8, so x = 4. Move the smaller letter-term to avoid negatives while you are learning.
Brackets, fractional answers, and no answer at all
Solve 3(2x − 1) = 4x + 7. Expand first: 6x − 3 = 4x + 7. Gather the letters by subtracting 4x from both sides: 2x − 3 = 7. Then undo: 2x = 10, so x = 5. Check both sides separately — 3(2×5 — 1) = 3 × 9 = 27, and 4×5 + 7 = 27. They agree, so it is right. Two things worth knowing before an exam. An answer does not have to be a whole number: 4x = 6 gives x = 1.5, and that is finished, not a mistake. And if every letter cancels out, read what is left — 2x + 3 = 2x + 5 collapses to 3 = 5, which is never true, so that equation has NO solution.
Turning a sentence into an equation
Most of the marks here are lost before any algebra starts. Take: I think of a number, multiply it by 4 and subtract 3, and the answer is 17. Write what you are told, in the order it happens: 4x − 3 = 17. Then solve as usual — 4x = 20, so x = 5 — and check against the SENTENCE rather than against your own working: 5 times 4 is 20, take away 3 is 17. Two traps do most of the damage. Writing the operations in the order the words appear rather than the order they happen, and forgetting that subtracting 3 from a number is x − 3 while subtracting a number from 3 is 3 − x, which is not the same thing at all.