We are given the line equation $8y = 3x + 8$ and need to find the equation of a line that is parallel to this and passes through the point $(5, 6)$.

Step 1: Determine the slope of the given line

To start, we rewrite the given equation in slope-intercept form $y = mx + b$, where $m$ represents the slope and $b$ represents the y-intercept.

The given equation is: $8y = 3x + 8$

Divide both sides by 8 to solve for $y$: $y = \frac{3}{8}x + 1$

Now, the slope of the given line is $m = \frac{3}{8}$.

Step 2: Use the point-slope form to find the equation of the parallel line

Since parallel lines have the same slope, the line passing through $(5, 6)$ will also have a slope of $\frac{3}{8}$. We can use the point-slope form of the line equation: $y - y_1 = m(x - x_1)$ where $(x_1, y_1) = (5, 6)$ and $m = \frac{3}{8}$.

Substituting the values: $y - 6 = \frac{3}{8}(x - 5)$

Step 3: Simplify the equation

Distribute the $\frac{3}{8}$ on the right-hand side: $y - 6 = \frac{3}{8}x - \frac{15}{8}$

Now, add 6 to both sides to solve for $y$: $y = \frac{3}{8}x - \frac{15}{8} + 6$ Convert 6 to a fraction with a denominator of 8: $y = \frac{3}{8}x - \frac{15}{8} + \frac{48}{8}$ Simplify: $y = \frac{3}{8}x + \frac{33}{8}$

Thus, the equation of the parallel line is: $y = \frac{3}{8}x + \frac{33}{8}$

Step 4: Check the multiple-choice answers

None of the provided answers exactly match this form, but they all seem to involve integer values, which suggests an error in the problem setup. Based on the structure of the multiple-choice answers, a correction or simplification would be needed. However, the key process here is understanding the concept of parallel lines and applying the point-slope formula.

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Would you like a more detailed explanation or have any further questions?

Here are some related questions for you to explore:

1. How do you convert between standard form and slope-intercept form of a linear equation?
2. What does it mean for two lines to be parallel in terms of their slopes?
3. How do you calculate the equation of a line given two points?
4. What are the key differences between point-slope and slope-intercept form?
5. How do you find the equation of a line perpendicular to another line?

Tip: Remember that parallel lines always have the same slope. So, once you know the slope of one line, the slope of any parallel line is identical.