# Why is the slope form written how it is?

So, I was recently reviewing slope. While doing some problems with the slope formula $$m = {y_2 - y_1\over x_2 - x_1}$$, I started thinking why it was that and not $$m = {y_1 - y_2\over x_1 - x_2}$$. Because point one and two are interchangeable, this should work. Is it doors work, why did they decide to put two first, then one?

It seems like it would be more reasonable to put numbers in there natural order if you had the chance.

• It doesn't matter, you get the same answer. Oct 11, 2019 at 1:44
• Yes, but was there some reason the formula is written that way Oct 11, 2019 at 1:45
• Welcome to MSE. I don't think there's necessarily any particular reason. However, my best guess as to why this is usually used in this order is that often the curves are in the first quadrant, with $x_2 \gt x_1$ and $y_2 \gt y_1$. Thus, by using $m = \frac{y_2 - y_1}{x_2 - x_1}$, the numerator and denominator would often be positive instead of negative. At least, even if $y_2 \le y_1$, it's quite common for at least $x_2 \gt x_1$. Oct 11, 2019 at 1:47
• @John Omielan that would make sense. Post that as an answer so I can except it. Edit: Nvm Oct 11, 2019 at 1:49
• @brododragon Thanks for the offer. However, since Doug M has just posted an answer saying basically the same thing, along with another possible reason, so you may wish to accept that answer instead. Oct 11, 2019 at 1:51

## 2 Answers

They are the same. That is, $$\frac{y_2 - y_1}{x_2 - x_1} = \frac{y_1 - y_2}{x_1 - x_2}$$

So why is it convention to say $$\frac{y_2 - y_1}{x_2 - x_1}$$?

Typically, we choose the points $$(x_1,y_1), (x_2, y_2)$$ such that on the graph of the line $$(x_1,y_1)$$ is to the left of $$(x_2, y_2)$$. That is $$x_1 < x_2$$ and $$x_2 - x_1 > 0$$ and the denominator of $$\frac{y_2 - y_1}{x_2 - x_1}$$ is postive. And, if the slope is postive, the numerator is also positive.

Also, if I wanted to describe a vector from point A to point B, I would say $$B-A.$$ So, $$\frac{y_2 - y_1}{x_2 - x_1}$$ suggests to me that we are measuring from $$(x_1, y_1)$$ to $$(x_2,y_2)$$

When you start at a point like $$x$$ and introduce some increment $$h$$ you get to the point $$x+h$$ so your change in $$x$$ is $$x_2 - x_1 = (x+h)-x =h$$

As a result of the change from $$x$$ to $$x+h$$, we get some change in $$y$$ namely we go from $$f(x)$$ to $$f(x+h)$$, so our change in $$y$$ is $$f(x+h)-f(x) =y_2-y_1$$

This it is natural to define the average rate of change as $$\frac {y_2- y_1}{x_2-x_1}$$