The given problem asks for the equation of line $r$, which passes through the point $(9, -3)$ and is parallel to line $q$, whose equation is:

$y = -\frac{10}{9}x + 2$

Step-by-Step Solution:

1. Find the slope of line $r$:
   Since line $r$ is parallel to line $q$, the two lines share the same slope. The slope of line $q$ is $-\frac{10}{9}$ (from the equation $y = mx + b$, where $m$ is the slope).

   Therefore, the slope of line $r$ is also: $m = -\frac{10}{9}$

2. Use the point-slope form of the equation:
   The point-slope form of a line is: $y - y_1 = m(x - x_1)$ where $m$ is the slope, and $(x_1, y_1)$ is a point on the line.

   Substituting $m = -\frac{10}{9}$ and the point $(9, -3)$, we get: $y - (-3) = -\frac{10}{9}(x - 9)$ Simplifying: $y + 3 = -\frac{10}{9}(x - 9)$

3. Simplify the equation: Now, distribute the slope on the right-hand side: $y + 3 = -\frac{10}{9}x + 10$

   Next, subtract 3 from both sides to get $y$ by itself: $y = -\frac{10}{9}x + 10 - 3$ $y = -\frac{10}{9}x + 7$

Final Answer:

The equation of line $r$ in slope-intercept form is: $y = -\frac{10}{9}x + 7$

Would you like further details or explanations? Here are five questions to expand on this topic:

1. What is the difference between parallel and perpendicular lines in terms of slope?
2. How would the process change if the lines were perpendicular instead of parallel?
3. Can you find the equation of a line given two points on the line?
4. How do you convert from point-slope form to standard form?
5. Why is the slope of parallel lines always the same?

Tip: When solving problems involving parallel or perpendicular lines, always pay attention to the slope, as it's a key factor in determining the relationship between the lines.