To determine the equation of line $q$, which is parallel to line $p$ and passes through the point $(5, -8)$, follow these steps:

1. Since line $q$ is parallel to $p$, they have the same slope. From the equation $y = -\frac{5}{4}x - 6$, the slope is $m = -\frac{5}{4}$.

2. Use the point-slope form of a line: $y - y_1 = m(x - x_1),$ where $(x_1, y_1) = (5, -8)$ and $m = -\frac{5}{4}$.

3. Substitute the values: $y - (-8) = -\frac{5}{4}(x - 5).$ Simplify: $y + 8 = -\frac{5}{4}x + \frac{25}{4}.$

4. Rearrange to slope-intercept form: $y = -\frac{5}{4}x + \frac{25}{4} - 8.$ Convert $8$ to a fraction: $y = -\frac{5}{4}x + \frac{25}{4} - \frac{32}{4}.$ Combine like terms: $y = -\frac{5}{4}x - \frac{7}{4}.$

Thus, the equation of line $q$ is: $y = -\frac{5}{4}x - \frac{7}{4}.$

Would you like further clarification on any steps? Here are five related questions for further exploration:

1. How do you determine the slope of a line from an equation in slope-intercept form?
2. What happens to the equation of a line if it is perpendicular instead of parallel?
3. Can you derive the same equation for line $q$ using standard form instead of slope-intercept form?
4. How do you verify if a point lies on a specific line?
5. What is the general process to write an equation for a line passing through two points?

Tip: Always simplify fractions completely to ensure clarity in your final equation.