Let's solve the problem step-by-step.

Step 1: Determine the slope of the given line

The given equation of the line is: $7y - 1 = -2(4 - 2x)$

First, expand and simplify the equation: $7y - 1 = -8 + 4x$

Rearrange to get the equation in slope-intercept form ($y = mx + b$): $7y = 4x + 7$ $y = \frac{4}{7}x + 1$

From this equation, the slope ($m$) of the given line is $\frac{4}{7}$.

Step 2: Write the equation of the parallel line

Lines that are parallel have the same slope. Thus, the slope of the line that passes through the point $(-8, -6)$ is also $\frac{4}{7}$.

Using the point-slope form of a line equation, which is $y - y_1 = m(x - x_1)$ where $(x_1, y_1)$ is a point on the line and $m$ is the slope, we can plug in the values: $y + 6 = \frac{4}{7}(x + 8)$

Now simplify this to get the equation in slope-intercept form: $y + 6 = \frac{4}{7}x + \frac{32}{7}$ $y = \frac{4}{7}x + \frac{32}{7} - 6$ $y = \frac{4}{7}x + \frac{32}{7} - \frac{42}{7}$ $y = \frac{4}{7}x - \frac{10}{7}$

So, the equation of the line in slope-intercept form is: $y = \frac{4}{7}x - \frac{10}{7}$

This equation represents the line parallel to the given line and passing through the point $(-8, -6)$.

Would you like more details on any of these steps or have any other questions?

Here are a few related questions you might find interesting:

1. How would you find the perpendicular slope to this line?
2. What would be the equation of a line parallel to the given line but passing through the origin?
3. How can we determine the y-intercept of a line if we know a point and the slope?
4. What is the significance of the slope in real-life applications?
5. Can you find the intersection point between the original line and the y-axis?

Tip: When you need to find a line parallel to a given line, always use the same slope for the new line as the given line's slope.