To find the equation of a line that is parallel to the given line \( y = -4x + 1 \) and passes through the point \( (4, -2) \), we first need to note that parallel lines have the same slope.
The slope of the line given by the equation \( y = -4x + 1 \) is \(-4\).
Using the point-slope form of a line's equation, which is:
\[ y - y_1 = m(x - x_1) \]
where \( m \) is the slope and \( (x_1, y_1) \) is the point the line passes through, we can substitute in our values:
- \( m = -4 \)
- \( (x_1, y_1) = (4, -2) \)
Substituting these values in, we get:
\[ y - (-2) = -4(x - 4) \]
This simplifies to:
\[ y + 2 = -4(x - 4) \]
Distributing the \(-4\):
\[ y + 2 = -4x + 16 \]
Now, subtracting \( 2 \) from both sides gives:
\[ y = -4x + 14 \]
Thus, the equation of the line that passes through the point \( (4, -2) \) and is parallel to the line \( y = -4x + 1 \) is:
\[ \boxed{y = -4x + 14} \]
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