Slope from Two Points

Slope of the line passing through (x₁, y₁) and (x₂, y₂). Undefined when x₁ = x₂.

Slope of a line

Let $P(x_1, y_1)$ and $Q(x_2, y_2)$ be two distinct points on a line, with $x_1 \neq x_2$. Draw a right triangle by dropping a horizontal from $P$ and a vertical from $Q$. The horizontal leg has length $x_2 - x_1$ and the vertical leg has length $y_2 - y_1$.

The angle $\theta$ the line makes with the positive $x$-axis satisfies:

$$\tan\theta = \frac{y_2 - y_1}{x_2 - x_1}$$

This signed ratio is the slope.

$$\boxed{m = \frac{y_2 - y_1}{x_2 - x_1}}$$

The choice of points does not matter

Take any other two points $A, B$ on the same line. The triangles formed are similar (same angle $\theta$), so the ratios of vertical to horizontal legs are equal.

Vertical lines

When $x_1 = x_2$, the horizontal leg is zero and the ratio is undefined.