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Why the slope of a line isn't the change in x divided by the change in y?

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2 Answers

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I'd say it's to make the definition consistent with the normal English meaning of the word "slope". When we say that a landscape has "no slope" we mean that the ground is flat. Assuming that $y$ measures vertical height, this means that there is no change in $y$. When we say a "large slope" or a "steep slope", we mean a large change in height, specifically a large change in $y$ over a small horizontal distance.

People understand this sort of language even if they don't know the mathematical definition of "slope". It would be very confusing if the mathematical definition were inconsistent with everyday speech.

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What you are describing is an equally valid way to represent slope, however it is customary to treat the $x$ as an independent variable, and $y$ as a function of $x$.

With that in mind, drawing a line in the $xy$-plane will give you an equation of the form:

$$y(x) = mx + b$$

Using the test points $(x_1, y_1), (x_2,y_2)$:

$$y_1 = mx_1 + b \\ y_2 = mx_2 + b$$ Subtracting the first from the second, $$y_2 - y_1 = m(x_2 - x_1) \\ \implies m = \frac{y_2 - y_1}{x_2 - x_1}$$

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