my teacher never gave me an idea of why the formula is the way it is (which is unusual because his coverage is really, really good). So let me see if I can help clear it up for anyone else that was wondering about it.
You'll need to graph two lines but it will hopefully show you what's going on.
Let line a equal: y = 2x + 2
Let line b equal: y = 4x + 2
Notice that b's slope of 4, is larger than a's slope of 2. If you graph both lines you'll see for yourself that line b has a steeper slope.
Now we know that delta y / delta x = rise over run and would give us the slope for both lines. So in the case of line a, delta y / delta x = 2. However, if you use delta x / delta y instead, the slope is 1/2. Clearly that doesn't fit the equation of y = 2x + 2, but it does describe the same relationship, it's just the inverse. What I mean is that 1/2 is the inverse of 2. They both represent the slope of the line but 2 is in terms of y and 1/2 is in terms of x. So with that in mind, let's change both equations to be in terms of x:
a: x = (y - 2) / 2
b: x = (y - 2) / 4
If you then graph these equations, you'll see you get exactly the same lines. So why is rise over run the standard when you can calculate the slope both ways? This is why I asked you to plot two lines instead of just one: plotting both demonstrates that when a line is drawn in terms of x, as the line becomes steeper, the slope of the equation actually decreases, so it makes no intuitive sense. To see it yourself, calculate the slope for lines a and b and you'll see the following:
Slope of line a in terms of delta x / delta y will result in 1/2.
Slope of line b in terms of delta x / delta y will result in 1/4.
Like this, b's slope is smaller than a's but the line is actually steeper when you plot it and that is why rise over run is used instead. It makes far more sense for the slope of the equation to increase when a line's slope also increases.
I hope someone found that useful. :)