How and why is Rise/Run gives the inclination of the line (or slope)?

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How and why is Rise/Run gives the inclination of the line (or slope)? How to understand the theoratical concept behind it ?

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It's a definition. We define the slope to be the rise over the run. This way a positive slope is a higher number when it's steeper, which is nice. But we could very well have defined it as "run" over "rise".