Problem Analysis:

The equation of the given line is $y = 4x + 2$, which has a slope $m = 4$. Additionally, you are provided with a point $(5, 8)$.

You are tasked to:

1. Write the equation of a line parallel to the given line passing through the point $(5, 8)$.
2. Write the equation of a line perpendicular to the given line passing through the point $(5, 8)$.

1. Equation of a Parallel Line:

A line parallel to the given line will have the same slope ($m = 4$).

The equation of the line in point-slope form is: $y - y_1 = m(x - x_1),$ where $(x_1, y_1) = (5, 8)$ and $m = 4$.

Substitute the values: $y - 8 = 4(x - 5).$

Simplify: $y - 8 = 4x - 20,$ $y = 4x - 12.$

So, the equation of the parallel line is: $y = 4x - 12.$

2. Equation of a Perpendicular Line:

A line perpendicular to the given line will have a negative reciprocal slope. If the slope of the given line is $4$, the perpendicular slope is: $m = -\frac{1}{4}.$

Using the point-slope form: $y - y_1 = m(x - x_1),$ where $(x_1, y_1) = (5, 8)$ and $m = -\frac{1}{4}$.

Substitute the values: $y - 8 = -\frac{1}{4}(x - 5).$

Simplify: $y - 8 = -\frac{1}{4}x + \frac{5}{4}.$

Add 8 to both sides: $y = -\frac{1}{4}x + \frac{5}{4} + 8.$

Convert $8$ to a fraction with denominator 4: $8 = \frac{32}{4}.$

$y = -\frac{1}{4}x + \frac{5}{4} + \frac{32}{4}.$

$y = -\frac{1}{4}x + \frac{37}{4}.$

So, the equation of the perpendicular line is: $y = -\frac{1}{4}x + \frac{37}{4}.$

Final Answer:

1. Equation of the parallel line: $y = 4x - 12$.
2. Equation of the perpendicular line: $y = -\frac{1}{4}x + \frac{37}{4}$.

Would you like further explanation or help with graphing these lines? Here are 5 related questions to consider:

1. What is the significance of slopes for parallel and perpendicular lines?
2. How can you verify that two lines are perpendicular using their slopes?
3. Can we rewrite the perpendicular line equation in decimal form instead of fractions?
4. What changes if the point given was $(0, 0)$?
5. How do we determine whether a point lies on a given line?

Tip: Always use the slope-intercept form for easier comparison of lines and slopes!