A system of 2 linear equations in 2 unknowns represents **two lines in the plane** $\mathbb{R}^2$.

$$\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}$$

There are exactly three possibilities:

1. **Unique solution** — the lines intersect at one point (different slopes)
2. **No solution** — the lines are parallel but distinct (same slope, different intercept)
3. **Infinitely many solutions** — the lines coincide (same line)

**Example (unique solution):**
$$\begin{cases} x + y = 3 \\ x - y = 1 \end{cases}$$

The lines intersect at $(2, 1)$.

**Example (no solution):**
$$\begin{cases} x + y = 2 \\ x + y = 5 \end{cases}$$

Parallel lines, no intersection.