Geometric picture of the solution set of a linear equation in 3 unknowns

A system of 3 linear equations in 3 unknowns represents three planes in $\mathbb{R}^3$.

The solution is the intersection of all three planes. Possible outcomes:

1. Unique solution — three planes intersect at a single point
2. No solution — planes have no common intersection (e.g., two parallel, or triangular prism configuration)
3. Infinitely many solutions — planes intersect along a line, or all three coincide

Example (unique solution):
$$\begin{cases} x + y + z = 6 \\ x - y + z = 2 \\ x + y - z = 0 \end{cases}$$

The three planes intersect at the point $(2, 2, 2)$.

Geometric insight: Each additional equation adds a constraint, reducing the dimension of the solution set by at most 1.