three-variable system

Two clues can usually pin down two mystery numbers; pin down three numbers and you generally need three clues. A three-variable system is a system of equations in three unknowns, commonly written with variables x, y, and z, where you seek values that satisfy all the equations at once.

Geometrically, each linear equation in three variables is not a line but a plane floating in three-dimensional space. A typical solution is the single point where three planes meet — an ordered triple (x, y, z). As with two variables, the system might instead have no solution (planes that never share a point) or infinitely many (planes meeting along a whole line or plane).

The reliable way to solve one is to peel it down step by step: eliminate one variable from a pair of equations, then another, until you reach a single equation in a single unknown — then back-substitute upward to recover the rest. This staircase of elimination is exactly Gaussian elimination, which scales gracefully to four, five, or many more variables.