The problem asks for the equation of a line that passes through the point $(-1, -4)$ and has a slope of 9, expressed in the form $Ax + By = C$, where $A$, $B$, and $C$ are integers.

Step-by-Step Solution:

1. Point-Slope Form: We can start by using the point-slope form of a line: $y - y_1 = m(x - x_1)$ where $m$ is the slope and $(x_1, y_1)$ is the point on the line. Here, $m = 9$, $x_1 = -1$, and $y_1 = -4$.

   Substituting these values, we get: $y - (-4) = 9(x - (-1))$ Simplifying: $y + 4 = 9(x + 1)$ $y + 4 = 9x + 9$

2. Rearranging into Slope-Intercept Form: $y = 9x + 9 - 4$ $y = 9x + 5$ This is the slope-intercept form of the line.

3. Converting to Standard Form: Now we convert it to the form $Ax + By = C$. We need to move all terms to one side of the equation: $9x - y = -5$ This is the equation in the required form, where $A = 9$, $B = -1$, and $C = -5$.

Final Answer:

$9x - y = -5$

Would you like more details on any part of the process or have further questions?

Here are five related questions:

1. How do you convert any line equation from slope-intercept form to standard form?
2. What is the general form of a line equation, and how does it differ from the slope-intercept form?
3. How would the equation change if the slope were negative or a fraction?
4. Can you find the equation of a line parallel to this one but passing through a different point?
5. How would you find the distance between this line and a given point not on the line?

Tip: When converting from slope-intercept form to standard form, always aim to keep $A$, $B$, and $C$ as integers and $A$ positive.