To find the equation of line $q$, we need to consider the following steps:

1. Identify the slope of line $p$: The equation of line $p$ is given as: $y = -\frac{1}{4}x - 8$ Here, the slope ($m$) of line $p$ is $-\frac{1}{4}$.

2. Slope of line $q$: Since line $q$ is parallel to line $p$, it will have the same slope. Therefore, the slope of line $q$ is also $-\frac{1}{4}$.

3. Using the point-slope form: Line $q$ passes through the point $(6, -3)$ and has a slope of $-\frac{1}{4}$. The point-slope form of a line's equation is given by: $y - y_1 = m(x - x_1)$ Substituting $m = -\frac{1}{4}$ and the point $(6, -3)$, we have: $y - (-3) = -\frac{1}{4}(x - 6)$ Simplifying: $y + 3 = -\frac{1}{4}(x - 6)$ $y + 3 = -\frac{1}{4}x + \frac{6}{4}$ $y + 3 = -\frac{1}{4}x + \frac{3}{2}$

4. Converting to slope-intercept form: To get the equation in slope-intercept form ($y = mx + b$), subtract 3 from both sides: $y = -\frac{1}{4}x + \frac{3}{2} - 3$ $y = -\frac{1}{4}x + \frac{3}{2} - \frac{6}{2}$ $y = -\frac{1}{4}x - \frac{3}{2}$

Hence, the equation of line $q$ is: $y = -\frac{1}{4}x - \frac{3}{2}$

Would you like further details or have any questions?

Related Questions:

1. How do you find the slope of a line given its equation in standard form?
2. What is the point-slope form and how is it used to write the equation of a line?
3. Can two lines be parallel if they have different slopes?
4. How do you convert between slope-intercept form and point-slope form?
5. How do you find the equation of a line perpendicular to a given line passing through a point?

Tip: Remember that parallel lines always have the same slope, but different $y$-intercepts unless they are identical lines.