## Equation of a line: The derivation of y = mx + b

We have discussed in context the origin (click here and here) of the linear equation $y = ax + b$, where $a$ and $b$ are real numbers.  We have also talked about the slope of a line and many of its properties. In this post, we will discuss the generalization of the equation of a line in the coordinate plane based on its slope and y-intercept.

We have learned that to get a slope of a line, we only need two points.  We have also learned that given two points on a line, its slope is described as the rise (difference in the y-coordinates) over the run (difference in x-coordinates).  Therefore, if we have two points with coordinates $(x_1,y_1)$ and $(x_2,y_2)$, the slope $m$ is  defined the formula

$m = \displaystyle\frac{y_2 - y_1}{x_2 - x_1}$.

All the points on a vertical line have similar x-coordinates; therefore, the run ${x_2 - x_1}$ is equal to $0$ making $m$ undefined.  From here, we can conclude a vertical line has no slope. » Read more