A line passes through (4,365), and the y-intercept of the line is on or between 1 and 9 . What is the minimum possible slope of the line?
+9467 Let the y-intercept be the point (0, a) where 1 ≤ a ≤ 9
So the slope = \(\frac{\text{rise}}{\text{run}}\ =\ \frac{365-a}{4-0}\ =\ \frac{365-a}{4}\)
To make the slope as small as possible, we want to make the fraction \(\frac{365-a}{4}\) as small as possible.
To make the fraction as small as possible, we want to make the numerator as small as possible (we can't change the denominator).
And to make the numerator as small as possible, we want to subtract as much as we can from 365 .
To make 365 - a as small as possible, we want to make a as large as possible.
The largest possible value of a is 9 , so the smallest possible value of the numerator is 365 - 9
And the smallest possible slope = \(\frac{365-9}{4}\ =\ \frac{356}{4}\ =\ 89\)
It's kind of hard to see the difference but here is a graph where you can see what changing the value of a does to the slope:
