## 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 , where and 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 and , the slope is defined the formula

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