To find a line that is perpendicular to the line \( y = \frac{3}{5}x + 6 \) and passes through the point \((-3, 6)\), we need to determine the slope of the perpendicular line.
1. **Determine the slope of the given line:** The given line has a slope of \( \frac{3}{5} \).
2. **Find the slope of the perpendicular line:** The slope of a line that is perpendicular to another line is the negative reciprocal of the other line's slope. Therefore, if the slope of the given line is \( \frac{3}{5} \), the slope of the perpendicular line \( m \) is:
\[
m = -\frac{1}{\left(\frac{3}{5}\right)} = -\frac{5}{3}
\]
3. **Use the point-slope form of a line to find the equation of the perpendicular line:** The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope.
Here, we use the point \((-3, 6)\), so \(x_1 = -3\) and \(y_1 = 6\):
\[
y - 6 = -\frac{5}{3}(x + 3)
\]
4. **Now, simplify the equation:**
\[
y - 6 = -\frac{5}{3}x - 5
\]
\[
y = -\frac{5}{3}x + 1
\]
Thus, the equation of the line that is perpendicular to \( y = \frac{3}{5}x + 6 \) and passes through the point \((-3, 6)\) is:
\[
y = -\frac{5}{3}x + 1
\]
**So the correct option is \( y = -\frac{5}{3}x + 1 \).**