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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!

Write an equation for the line parallel to the given line that contains C. C(4, 8); y = -4x + 1

This problem teaches how to find the equation of a line parallel to y = -4x + 1 and passing through point C(4, 8). Learn how to use the point-slope form and identify the slope for parallel lines. Suitable for students in Grades 8-10.

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