To find the equation of a line that is perpendicular to the given line \( y + 1 = -3(x - 5) \) and passes through the point \((4, -6)\), let's follow these steps:
1. Determine the slope of the original line:
- The given equation \( y + 1 = -3(x - 5) \) can be rewritten in slope-intercept form \( y = mx + b \).
- Starting with \( y + 1 = -3(x - 5) \), distribute the \(-3\):
[tex]\[
y + 1 = -3x + 15
\][/tex]
- Isolate \( y \) to get:
[tex]\[
y = -3x + 14
\][/tex]
- From this, we see that the slope \( m \) of the original line is \(-3\).
2. Find the slope of the perpendicular line:
- The slope of a line perpendicular to another is the negative reciprocal of the original line's slope.
- The negative reciprocal of \(-3\) is \(\frac{1}{3}\). Hence, the slope \( m \) of the perpendicular line is \(\frac{1}{3}\).
3. Use the point-slope formula to find the equation:
- The point-slope form of a line's equation is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1) \) is a point on the line.
- Here, \( (x_1, y_1) = (4, -6) \) and \( m = \frac{1}{3} \).
- Substitute these values into the point-slope form:
[tex]\[
y - (-6) = \frac{1}{3}(x - 4)
\][/tex]
- Simplify this equation:
[tex]\[
y + 6 = \frac{1}{3}(x - 4)
\][/tex]
4. Select the correct answer choice:
- Comparing this final form with the given options, we see that it matches option D:
[tex]\[
y + 6 = \frac{1}{3}(x - 4)
\][/tex]
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]
