Why does the slope of a vertical line not exist? Can you provide a geometric interpretation?

Vertical lines have an undefined slope because the horizontal change is 0 — you cannot divide a number by 0. The slope of a line is a measure of its steepness. Vertical lines have NO SLOPE.

Vertical lines cross the x-axis where y = 0. Can this also be a reason why vertical lines have no slope? I understand the steepness explanation.

Considering lines making angle $\theta$ with the positive x-axis. Its slope is $tan(\theta)$. The limit, as $\theta$ goes to $\frac{\pi}{2}%, $tan(\theta)$ goes to infinity.

(Above image scraped from Purplemath. To see it in context, try [here].) No. The line \(y = x\) also crosses the \(x\)-axis at the origin (that is, where \(y = 0\)), but this line has a slope of \(m = 1\), not zero. P.S. Hi, Sologuitar!