equation

An equation is a claim of balance: it says the thing on the left of the equals sign weighs exactly the same as the thing on the right. The symbol = is the fulcrum of a scale, and an equation insists the scale is level. “3 + 4 = 7” is a true equation; “x + 1 = 5” is a claim that becomes true for the right choice of x.

Formally, an equation states that two expressions are equal. When it contains an unknown, solving it means finding every value that keeps the scale balanced. The golden rule of solving is that whatever you do to one side you must do to the other — add the same amount, multiply by the same nonzero number — so the balance is preserved while you simplify toward the answer.

Equations come in types worth distinguishing. A conditional equation is true only for particular values (x + 1 = 5 only when x = 4). An identity is true for every value (x + x = 2x always). A contradiction is true for no value (x = x + 1 never). Recognising which kind you have tells you whether to expect one solution, all of them, or none.