### Instruction: 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?### Response: I need to find the smallest slope of a line that passes through $(4,365)$ and has a $y$-intercept between $1$ and $9$.
The slope of a line is the ratio of the change in $y$ to the change in $x$ between any two points on the line.
So, to find the slope of a line that passes through $(4,365)$ and a given $y$-intercept $(0,b)$, I can use the formula $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{b - 365}{0 - 4} = \frac{b - 365}{-4}$.
To minimize the slope, I need to maximize the numerator of this fraction, which is $b - 365$.
Since $b$ is on or between $1$ and $9$, the largest possible value of $b$ is $9$.
Therefore, the minimum possible slope is $m = \frac{9 - 365}{-4} = -\frac{356}{4} = -89$.