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

Put content hereA 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.

# Parents

* The geometry of linear systems