Two linear equations in x, y have a unique solution when:

A system of two linear equations in two variables, $x$ and $y$, represents two straight lines in a Cartesian plane. The solution to the system corresponds to the point(s) where the two lines share the same $(x, y)$ coordinates. There are three possible geometric relationships between two lines: 1. **Intersecting Lines**: If the lines have different slopes, they cross at exactly one point. This point $(x, y)$ represents the **unique solution**. 2. **Parallel Lines**: If the lines have the same slope but different $y$-intercepts, they never meet. This means the system has **no solution**. 3. **Coincident Lines**: If the lines have the same slope and the same $y$-intercept, they lie exactly on top of each other. This means they share every point, resulting in **infinitely many solutions**. Therefore, a unique solution exists if and only if the lines intersect at exactly one point.