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.

Now, it is clear that every non-vertical line has a slope (can you see why?). It is also clear that every non-vertical line will always pass through the y-axis — in fact, both axes. If we let y = b be the y-intercept of line l (see blue line below), then the coordinates of f that point are (0,b).  If we choose any point on line l with coordinates (x,y) , then we can get the slope m of the line passing through the two points as shown in the figure.

Therefore, the slope m of any line passing through (0,b) and (x,y) is y = mx + b. This is the general equation of a line given a point (on the line) and its y-intercept.  This equation is called the slope-intercept form.

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  1. Pingback: March 2011 Top Posts « Mathematics and Multimedia

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