Why slope of line defined as height to length?

Question

I want to understand: why slope defined as ΔY/ΔX but not ΔX/ΔY ?

Answer 1

A higher slope should correspond to a steeper line. Thus, if $\Delta Y$ increases while $\Delta X$ stays the same, the line gets steeper (visualizing this on the $XY$-plane), so we should expect the slope to increase as well. If the slope was $\Delta X/\Delta Y$, the slope would decrease, and thus a word like "flatness" might be more adequate to describe this ratio.

Answer 2

When you're graphing a function like $y=f(x)$, the slope of the function $f$ means "when you change the input $x$ by an amount, how much does the output $y=f(x)$ change?"

To measure that, you take the ratio of the change in output $\Delta y$ and divide it by the change in the input $\Delta x$.

For example, $f(x)$ might measure the temperature of the air at a certain distance $x$ from a heater. The slope $\Delta y / \Delta x$ asks "How much does the temperature change as a result of moving a short distance?"