To determine the equation of a line that is perpendicular to the given line and passes through a specific point, we follow these steps:
1. Identify the slope of the given line:
The given line is \( y + 1 = -3(x - 5) \). This can be written in slope-intercept form \( y = mx + b \).
Start by expanding the equation:
[tex]\[
y + 1 = -3(x - 5)
\][/tex]
[tex]\[
y + 1 = -3x + 15
\][/tex]
[tex]\[
y = -3x + 14
\][/tex]
Hence, the slope of the given line is \( -3 \).
2. Find the slope of the perpendicular line:
The slopes of perpendicular lines are negative reciprocals of each other. Therefore, the slope \( m \) of the line perpendicular to the given line is:
[tex]\[
m = -\frac{1}{-3} = \frac{1}{3}
\][/tex]
3. Use the point-slope form to find the equation of the new line:
The point-slope form of a line is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope.
We have \( m = \frac{1}{3} \) and the point \( (4, -6) \). Substitute these values into the point-slope form:
[tex]\[
y - (-6) = \frac{1}{3}(x - 4)
\][/tex]
Simplify to:
[tex]\[
y + 6 = \frac{1}{3}(x - 4)
\][/tex]
4. Identify the correct choice:
Among the given options, the equation \( y + 6 = \frac{1}{3}(x - 4) \) corresponds to option D.
Therefore, the equation of the line that is perpendicular to \( y + 1 = -3(x - 5) \) and passes through the point \( (4, -6) \) is:
[tex]\[
\boxed{y + 6 = \frac{1}{3}(x - 4)}
\][/tex]
Thus, the correct answer is [tex]\( D \)[/tex].
