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# The geometry of linear systems

Put content hereA system of 2 linear equations in 2 unknowns represents **two lines in the plane** $\mathbb{R}^2$.
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$$\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}$$
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There are exactly three possibilities:
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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)
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**Example (unique solution):**
$$\begin{cases} x + y = 3 \\ x - y = 1 \end{cases}$$
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The lines intersect at $(2, 1)$.
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**Example (no solution):**
$$\begin{cases} x + y = 2 \\ x + y = 5 \end{cases}$$
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Parallel lines, no intersection.

# Parents

* Linear systems of equations