Two–Point Form

Line passing through (x₁, y₁) and (x₂, y₂).

Derivation

Given two distinct points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ with $x_1 \neq x_2$ , the slope of the line joining them is:

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

Substitute into point-slope form using $P_1$ as the base point:

y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)

This can be written symmetrically as:

\boxed{\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}}

Both forms are equivalent; the symmetric form makes it clear the roles of $P_1$ and $P_2$ are interchangeable.

Note

When $x_1 = x_2$, the line is vertical: $x = x_1$. The slope is undefined and this form does not apply.