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:
- Unique solution — the lines intersect at one point (different slopes)
- No solution — the lines are parallel but distinct (same slope, different intercept)
- 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.
The geometric description assumes that each equation has at least one nonzero coefficient. An all-zero left side instead gives the whole ambient space when the constant is zero, or no solutions when it is nonzero.
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