Undefined Slope
Undefined slope describes a vertical line where the horizontal change is zero, making the slope calculation a division by zero.
Formula
\text{run} = 0 \text{ (division by zero)}
Definition
A vertical line has an undefined slope: it goes straight up and down, and slope cannot be calculated because the "run" (the denominator $x_2 - x_1$) is zero, and you cannot divide by zero. Vertical lines cannot be written in $y = mx + b$ form; their equation is simply $x = c$, and algebraically they are not functions because they fail the vertical line test. In the extended real number system, the limit of $m$ as a line approaches vertical is $+\infty$ or $-\infty$ depending on the direction of approach, and in projective geometry the slope of a vertical line can be treated as the point at infinity in the direction $(0:1:0)$.
Example
$x = 3$ is vertical: every point on it, $(3, 0)$, $(3, 5)$, $(3, -2)$, has $x = 3$, so run is always $0$ even as rise differs. For points $(4, 1)$ and $(4, 9)$ on $x = 4$, the slope attempt $(9-1)/(4-4) = 8/0$ is undefined, an infinite rise for zero horizontal change.
Key Insight
An undefined slope means the line is perfectly vertical, like a wall or a cliff: it has an x-intercept but no slope you can calculate. It is easy to confuse with zero slope, zero slope is horizontal ($y = $ constant, y-intercept only), undefined slope is vertical ($x = $ constant, x-intercept only). The distinction is a reminder that slope is a property of functions, and vertical lines are not functions, connecting directly to the definition of a function and the vertical line test that runs throughout algebra.