Geometric picture of a 2-by-2 linear system

A linear equation in 3 unknowns $ax + by + cz = d$ represents a plane in $\mathbb{R}^3$.

• If one coefficient is zero (e.g., $c = 0$), the plane is parallel to the corresponding axis
• If two coefficients are zero, the plane is parallel to a coordinate plane

Special cases:

• $ax + by + cz = 0$ — the plane passes through the origin
• $0x + 0y + 0z = 0$ — the entire space $\mathbb{R}^3$
• $0x + 0y + 0z = d$ (with $d \neq 0$) — the empty set

Example: $2x + 3y + z = 6$ is a plane in $\mathbb{R}^3$ passing through $(3, 0, 0)$, $(0, 2, 0)$, and $(0, 0, 6)$.