To determine the equation of a line that is perpendicular to the given line $y = -3x + 2$ and passes through the point $(6, 8)$, we proceed as follows:

Step 1: Find the slope of the perpendicular line.

The slope of the given line $y = -3x + 2$ is $m = -3$.
The slope of a line perpendicular to another is the negative reciprocal of the original slope. Thus, the slope of the perpendicular line is:

$m_{\text{perpendicular}} = \frac{-1}{-3} = \frac{1}{3}.$

Step 2: Write the point-slope form of the equation.

The point-slope form of a line is:

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

where $m$ is the slope, and $(x_1, y_1)$ is a point on the line. Substituting $m = \frac{1}{3}$ and $(x_1, y_1) = (6, 8)$, we have:

$y - 8 = \frac{1}{3}(x - 6).$

Step 3: Simplify the equation.

Expand and simplify:

$y - 8 = \frac{1}{3}x - 2,$

$y = \frac{1}{3}x + 6.$

Thus, the equation of the line is:

$\boxed{y = \frac{1}{3}x + 6}.$

Step 4: Match the options.

The correct answer is:

$\textbf{D. } y = \frac{1}{3}x + 6.$

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Do you want a step-by-step graphical explanation or details about slope properties?
Here are 5 related questions for deeper exploration:

1. How do you find the slope of a line given two points?
2. What is the general relationship between slopes of perpendicular lines?
3. How do you determine if two lines are parallel?
4. How can the point-slope form be converted into the slope-intercept form?
5. What are real-life applications of perpendicular lines in geometry?

Tip: Always verify perpendicular slopes by multiplying them; their product should equal $-1$.