The geometry of linear systems

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.