Each equation in a \(3 \times 3\) linear system is a plane in \(\mathbb{R}^3\), and a solution of the system is a point lying on all three planes simultaneously. Row reducing the augmented matrix replaces the system with a new one, but each row operation preserves the solution, with perhaps “simple planes.” Geometrically, this is viewed as rotation the planes (or swapping two of them) around the common point. When there is a unique solution, we can get all three planes orthogonal (and hence, each one parallel to a single axis). This is precisely a RREF with the main diagonal all ones, which let’s us “read off” the solution.