To find the equation of a line that is parallel to the given line $y = -4x + 1$ and passes through the point $C(4, 8)$, we follow these steps:

1. Understand the slope of the given line: The slope of the line $y = -4x + 1$ is $-4$, since it is in slope-intercept form $y = mx + b$, where $m$ is the slope.

2. Parallel lines have the same slope: A line parallel to this given line must have the same slope, $m = -4$.

3. Use the point-slope form of a line equation: The point-slope form is given by: $y - y_1 = m(x - x_1)$ where $(x_1, y_1)$ is the point through which the line passes and $m$ is the slope. In this case, the point $C(4, 8)$ gives us $x_1 = 4$ and $y_1 = 8$, and the slope is $m = -4$.

4. Substitute the values: $y - 8 = -4(x - 4)$ Simplify the equation: $y - 8 = -4x + 16$ $y = -4x + 16 + 8$ $y = -4x + 24$

Thus, the equation of the line parallel to $y = -4x + 1$ and passing through $C(4, 8)$ is: $y = -4x + 24$

Would you like further clarification or details on any part of this solution?

Here are five related questions to expand your understanding:

1. How do you find the equation of a line perpendicular to a given line?
2. What is the significance of the y-intercept in a linear equation?
3. How do you convert between point-slope form and slope-intercept form?
4. Can you derive the equation of a line given two points?
5. How does the slope of a line affect its steepness?

Tip: When working with parallel lines, remember that the slopes are always identical, while the intercepts will differ unless the lines are the same!