Why is this so that a higher value of slope indicates a steeper incline? I can't take it into my head. What could be the reason behind that? I know that it is a fact because I've also noticed it but don't know the reason which could assist my understanding of this concept.
2026-04-24 20:56:21.1777064181
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Slope of a straight line
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Well we could take these two lines as an example.
a) $y=x+1$
b) $y= 10x+1$
In line a, we find that the slope is x which is equal to $\dfrac 11$ which means rise 1, run 1.
In line b, we find that the slope is $10x$ which is equal to $\dfrac {10}1$ which means rise 10, run 1.
Notice the difference in the rise of both lines even though the y-intercept's are the same.
Think about why $$\text{slope}=\frac{\text{rise}}{\text{run}}=\frac{\Delta y}{\Delta x},$$ where the $\Delta$ means "change in". The more the $y$ changes over an interval of a fixed length (so $\Delta x$ is fixed), the slope will get larger. Conversely, if the slope gets larger and $\Delta x$ is fixed, then $\Delta y$ must get larger.