Question

Why does a horizontal line have a slope of zero, but a vertical line have an undefined slope?

Answer 1

Refer to the diagram shown below.
Define two positions on each line as A (x₁, y₁) and B(x₂, y₂).
Then the slope is
$m= \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Horizontal line:
Define A (0, 1) and B (1,1).
The slope is
$m= \frac{1-1}{1-0}=0$

Vertical line:
Define A( (1,0) and B(1,1).
The slope is
$m= \frac{1-0}{1-1}= \frac{1}{0}$  (undefined)