solution of a system

Think of each equation in a system as a club with its own membership rule. A solution of the system is a set of values admitted to every club at once — a number-tuple that passes the test of each equation without exception. Belonging to most of the clubs is not enough; it has to be all of them.

Concretely, for a system in the variables x and y, a solution is an ordered pair (x, y) that makes each equation a true statement when substituted in. For three variables it is an ordered triple, and so on. You can always check a candidate by plugging it back into every original equation — if even one fails, it is not a solution.

A system may have exactly one solution, no solution, or infinitely many. When there is more than one, the full collection of them is called the solution set. Geometrically, a solution is a point lying on every graph in the system simultaneously — the common intersection of all the curves or surfaces involved.