In the prior section, we ended with a general linear equation. $$7x-2y=24 \tag{EQ 1}\label{EQ 1}$$ If $\eqref{EQ 1}$ is solved for variable $y$, we get $$y=\frac{7}{2}x-12 \tag{EQ 2}\label{EQ 2}$$ $\eqref{EQ 2}$ is called the slope intercept form of the line. In this form, the coefficient in front of the x variable is called the slope and given the Latin letter $m$. The constant on the right side turns out to be the value where the line crosses the $y$-axis $(y\;intercept)$ and is most often given the Latin letter $b$. Therefore the slope intercept form becomes $$y=mx+b. \tag{EQ 3}\label{EQ 3}$$ Using general point coordinates $(x_{1},y_{1})$ and $(x_{2},y_{2})$ then equation $\eqref{EQ 3}$ would become $$y=\left(\frac{y_{1}-y_{2}}{x_{1}-x_{2}}\right)x+\left(y_{1}-\frac{y_{1}-y_{2}}{x_{1}-x_{2}}x_{1}\right)=\frac{\Delta y}{\Delta x}x+\left(y_{1}-\frac{\Delta y}{\Delta x}x_{1}\right)$$ That is, $m$ and $b$ are defined in terms of the two known points, $(x_{1},y_{1})$ and $(x_{2},y_{2}).$ $$m=\frac{y_{1}-y_{2}}{x_{1}-x_{2}}\qquad b=y_{1}-\frac{y_{1}-y_{2}}{x_{1}-x_{2}}x_{1} \tag{EQ 4}\label{EQ 4}$$ Given how much more complicated these formulas are from the general form, I find it odd that the slope intercept form is the most common. Possibly it is popular because the two coefficients, $m$ and $b$ have some physical meaning. The slope can be described verbally as the “directional rise” divided by the “directional run”. Most often we just say “rise” over “run” but since positive and negative matter, we have to fall back on calculating by the formula of $\eqref{EQ 4}$.