Let's work through each of the steps provided in the problem:

Step 1 of 2: Express the given equation in slope-intercept form.

The given equation is:

$7y - 8 = -3(4 - x)$

First, simplify the right-hand side by distributing the $-3$ over $(4 - x)$:

$7y - 8 = -3(4) + 3x$ $7y - 8 = -12 + 3x$

Next, add 8 to both sides to isolate the term with $y$:

$7y = 3x - 12 + 8$ $7y = 3x - 4$

Now, divide both sides of the equation by 7 to solve for $y$:

$y = \frac{3x - 4}{7}$ $y = \frac{3}{7}x - \frac{4}{7}$

This is the slope-intercept form of the equation, where the slope $m = \frac{3}{7}$ and the y-intercept is $\frac{-4}{7}$.

Step 2 of 2: Find the equation of a line parallel to the given line that passes through the point (6, 4).

A line parallel to the given one will have the same slope, which we found to be $\frac{3}{7}$.

Now, using the point-slope form of the equation of a line, which is:

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

Substitute $m = \frac{3}{7}$, and the point $(x_1, y_1) = (6, 4)$:

$y - 4 = \frac{3}{7}(x - 6)$

Now, simplify:

$y - 4 = \frac{3}{7}x - \frac{18}{7}$

Add 4 (or $\frac{28}{7}$) to both sides to isolate $y$:

$y = \frac{3}{7}x - \frac{18}{7} + \frac{28}{7}$ $y = \frac{3}{7}x + \frac{10}{7}$

Thus, the equation of the line in slope-intercept form is:

$y = \frac{3}{7}x + \frac{10}{7}$

Do you need further details on any of these steps? Here are five related questions to explore:

1. What is the significance of slope-intercept form in analyzing lines?
2. How do we determine if two lines are parallel based on their equations?
3. How can we use point-slope form for solving real-world problems?
4. What happens if two lines have the same slope but different y-intercepts?
5. How can you derive the equation of a perpendicular line instead of a parallel one?

Tip: In slope-intercept form, the coefficient of $x$ is always the slope, and the constant term is the y-intercept, which helps quickly sketch or understand a line's behavior.