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Why is the slope of a line = rise / run, and NOT run / rise?

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Entirely convention, but making 0 steepness the number you get when you have a flat surface seems more sensible than the alternative
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Run is the change in the independent variable. Rise is the change in the dependent variable. The slope is the change in the dependent variable per unit change in the independent variable. The change in the independent variable is the "cause" and the change in the dependent variable is the "effect".

Also: a horizontal line has a slope of zero. A vertical line doesn't have a slope. But the graph of a function can't be a vertical line.
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Because we typically put the "independent variable" on the x axis and "dependent" on the y axis. Let's say time is on the x axis and distance on the y, then rise/run is equal to the speed of  whatever object you're measuring, but run/rise doesn't really have any physical meaning.
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Treat the slope as, how much y rises when x is increased by 1
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Because the larger the slope, the steeper it is, which corresponds with rise over run.

If it was run over rise, then a smaller slope would be steeper.

In the end there is no "logic" really, we just defined it to be that way.
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Because when you look at a line equation written in the form y  = ax + b, the coefficient a can be calculated as rise / run, so if you call that number the slope,  you can also read it off the equation.

Additionally, you should think of the slope as a measure of the steepness of the line, so steeper lines should have bigger slopes. Rise / run fits this intuition; run / rise does not.
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I think the steepness comment is the best way to remember this. Also the horizon line has a slope of zero helps me remember which is which. To get a zero slope, you have to have zero rise divided by a non zero run. That gets you zero for the slope. For a vertical line, which has an undefined slope, you have a non zero rise divided by zero which isn’t allowed in our math.
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Consider how speed is measured in distance over time, eg miles per hour or meters per second. Time is usually the x axis so rise over run produces exactly that.
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Typically when a quantity( anything, it could be prices value of pressure of an ideal gas etc.)  is studied, the graph of that quantity with respect to something else has the said quantity on the Y axis. This might be because we humans see up good down bad and have neutral stances of left vs right but it's a convention.

Hence when you want to look at change of the said quantity with respect to another quantity( i.e. the quantity on the X axis) the rise becomes the difference of value on the Y axis which like we discussed before is often describing the quantity which is in focus.

tldr up good down bad is an intuitive way to convey data
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Conventionally, we treat "x" as the input and "y" as the output.

What the slope really tells you is: given some change in the input, how much change do we expect to see in the output?

You might apply it to an example like driving. Your input x is the time you spend driving (in hours). Your output y is the total distance you travel (in miles). The slope of this graph is your speed in "miles per hour", (miles/hour), (y/x)
by

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