Linear equations
A linear equation is a statement that two expressions are equal —
both built from numbers and a variable, with no powers above 1. Your
job is to find the value of the variable that makes the equation true.
In grade 7 you meet four flavours:
| form | example | solution |
| `px + q = r` | `3x + 5 = 14` | `x = 3` |
| `p(x + q) = r` | `3(x + 2) = 18` | `x = 4` |
| `ax + b = cx + d` | `2x + 5 = x + 11` | `x = 6` |
| word problem → equation | "age", "perimeter", "budget" | depends |
Plus a tiny relative — one-step inequalities like `x + 3 > 7`.
The core principle
Do the same thing to both sides of the equation. The equality is
preserved.
If `A = B`, then `A + k = B + k`, `A − k = B − k`, `A · k = B · k`,
`A ÷ k = B ÷ k` (with k ≠ 0). The values may change, but the equality
itself doesn't break.
That's all you need. Every solving technique below is just **a clever
choice of what to add, subtract, multiply or divide on both sides**.