Algebra Precalculus – Derivation of Slope of Line Formula

Question

The formula for slope of a line as we know:

$y_2 – y_1/x_2 – x_1$ or just rise / run

What is the derivation for this formula? E.g. Why is it not rise times run for example?

Answer 1

The equation of the line is $$y=ax+b$$ Assume we know that $(x_1,y_1)$ and $(x_2,y_2)$ are two points on the line. Then the following system holds $$\begin{cases}y_1=ax_1+b\\y_2=ax_2+b\end{cases}$$ Now, this is a system of two equations in two unknowns ($a$ and $b$). Solving it (by subtracting the first equation from the second), yields $$a=\frac{y_2-y_1}{x_2-x_1}$$

Graphically