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To find the equation of a line that is perpendicular to a given line and passes through a specific point, we need to go through these steps:

1. **Identify the Slope of the Given Line**: 
   The given line equation is:
   \[
   y = \frac{3}{5}x + 10
   \]
   This line has a slope of \(\frac{3}{5}\).

2. **Find the Slope of the Perpendicular Line**:
   For a line to be perpendicular, its slope must be the negative reciprocal of the slope of the given line. Therefore, the perpendicular slope is:
   \[
   -\frac{5}{3}
   \]

3. **Use the Point-Slope Form**:
   Since the perpendicular line passes through the point \((15, 5)\), we can use the point-slope formula:
   \[
   y - y_1 = m(x - x_1)
   \]
   Substituting \(m = -\frac{5}{3}\), \(x_1 = 15\), and \(y_1 = 5\):
   \[
   y - 5 = -\frac{5}{3}(x - 15)
   \]

4. **Simplify to Slope-Intercept Form**:
   Expanding this equation:
   \[
   y - 5 = -\frac{5}{3}x + 25
   \]
   Adding 5 to both sides:
   \[
   y = -\frac{5}{3}x + 30
   \]

The equation of the line perpendicular to \(y = \frac{3}{5}x + 10\) and passing through \((15, 5)\) is:
\[
y = -\frac{5}{3}x + 30
\]

### Answer:
The correct choice is:
\[
\boxed{y = -\frac{5}{3}x + 30}
\]

Would you like further details on the steps? Here are some additional questions for deeper understanding:

1. How do you find the slope of a line given two points on that line?
2. Why do perpendicular lines have slopes that are negative reciprocals of each other?
3. What is the purpose of using the point-slope form in linear equations?
4. How can you verify if two lines are indeed perpendicular by their equations?
5. In what scenarios might you need to find the equation of a perpendicular line?

**Tip:** When finding the slope of a perpendicular line, remember that you flip the fraction of the original slope and change its sign.

What is the equation of the line that is perpendicular to the line y = (3/5)x + 10 and passes through the point (15, 5)?

Determine the equation of a line perpendicular to y = (3/5)x + 10 that passes through (15, 5). This problem covers concepts of perpendicular slopes, point-slope form, and slope-intercept form. Ideal for students in Grades 9-10.

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