## Ask Professor Puzzler

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Martha asks, "In a straight line equation, why is the slope change in y over change in x, instead of change in x over change in y?"

That's a great question, Martha. One simple answer would be to just say, "because that's the way it is." But there are better answers than that. Let me give you one. Consider the equation for a line, when written in slope-intercept form. It is:

y = mx + b, and we say that m is Δy/Δx. If you're not familiar with the Greek letter delta used here (Δ), it simply means "the change in", so we would read that fraction as "the change in y over the change in x." So the equation now is:

y = (Δy/Δx)x + b.

Now that we've established that, let's talk about units. In real-world situations, the axes would both have units. For example, if the linear equation was part of a word problem about a car traveling from place to place, the horizontal axis might have a time unit, such as hours, and the vertical axis might have a distance unit like miles.

So now let's look at how those units would fit into our equation:

y is a vertical value, so its units would be miles.

Δy is a vertical change, so its units would also be miles.

Δx is a horizontal change, so its units would be hours.

x is a horizontal value, so its units would also be hours.

b represents the y-intercept (which is a y-value when x = 0) so it also has the vertical unit, miles.

Therefore, the equation, written only in terms of its units, would look like this:

miles = (miles/hour)hour + miles

Notice what happens when you multiply out (miles/hour)hour - you get just miles, leading to:

miles = miles + miles.

This is exactly what we need; when you add a two mileages, you get another mileage.

But consider what this would look like if m was Δx/Δy instead of the other way around. We would have:

miles = (hour/miles)hour + miles

miles = hour^{2}/miles + miles, which is about the most absurd conglomeration of nonsensical units I've ever seen.

So that's one way of looking at the question, "Why is the slope Δy/Δx?" Because if it was the other way around, it would produce absurd results.