We are tasked with finding the equation of line $s$, which is parallel to line $r$ and passes through the point $(8, 8)$. Line $r$'s equation is given as:

$y - 6 = \frac{1}{4}(x - 8)$

Step 1: Find the slope of line $r$

Line $r$'s equation is in point-slope form: $y - y_1 = m(x - x_1)$, where $m$ is the slope. From the equation:

$y - 6 = \frac{1}{4}(x - 8)$

The slope $m$ is $\frac{1}{4}$.

Step 2: Use the slope for line $s$

Since line $s$ is parallel to line $r$, it has the same slope. Therefore, the slope of line $s$ is also $\frac{1}{4}$.

Step 3: Apply the point-slope form to line $s$

We use the point-slope form of the equation again, but now for line $s$, which passes through the point $(8, 8)$:

$y - y_1 = m(x - x_1)$

Substitute $m = \frac{1}{4}$, $x_1 = 8$, and $y_1 = 8$:

$y - 8 = \frac{1}{4}(x - 8)$

Step 4: Convert to slope-intercept form

Now, simplify this equation to get it into slope-intercept form ($y = mx + b$):

$y - 8 = \frac{1}{4}(x - 8)$

Distribute $\frac{1}{4}$ on the right side:

$y - 8 = \frac{1}{4}x - 2$

Add 8 to both sides:

$y = \frac{1}{4}x + 6$

Final Equation:

The equation of line $s$ in slope-intercept form is:

$y = \frac{1}{4}x + 6$

Would you like more details or have any further questions?

Here are five related questions to explore further:

1. What is the general form of the equation of a line?
2. How can we determine if two lines are perpendicular?
3. What is the relationship between the slopes of parallel lines?
4. How do we find the equation of a line given two points?
5. What does the slope of a line represent geometrically?

Tip: When lines are parallel, they always have the same slope, no matter what point they pass through.