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To solve this problem, let's go through the steps required to find the equation of line \( q \), which is perpendicular to line \( p \) and passes through the point \( (-10, -3) \).

### Given Information
- Equation of line \( p \): \( y = \frac{5}{3}x - 4 \)
- Line \( q \) passes through the point \( (-10, -3) \) and is perpendicular to line \( p \).

### Steps to Solution

1. **Identify the Slope of Line \( p \):**
   - Line \( p \) is in slope-intercept form \( y = mx + b \), where \( m \) is the slope.
   - For line \( p \), the slope \( m_p = \frac{5}{3} \).

2. **Determine the Slope of Line \( q \):**
   - Since line \( q \) is perpendicular to line \( p \), its slope \( m_q \) will be the negative reciprocal of \( m_p \).
   - Therefore, \( m_q = -\frac{3}{5} \).

3. **Use the Point-Slope Form to Find the Equation of Line \( q \):**
   - We have the point \( (-10, -3) \) and the slope \( -\frac{3}{5} \).
   - The point-slope form of a line is:
     \[
     y - y_1 = m(x - x_1)
     \]
     Substituting \( x_1 = -10 \), \( y_1 = -3 \), and \( m = -\frac{3}{5} \):
     \[
     y - (-3) = -\frac{3}{5}(x - (-10))
     \]
     Simplify the equation:
     \[
     y + 3 = -\frac{3}{5}(x + 10)
     \]

4. **Simplify to Slope-Intercept Form \( y = mx + b \):**
   - Distribute the slope on the right side:
     \[
     y + 3 = -\frac{3}{5}x - \frac{3}{5} \cdot 10
     \]
   - Calculate \( -\frac{3}{5} \cdot 10 = -6 \):
     \[
     y + 3 = -\frac{3}{5}x - 6
     \]
   - Subtract 3 from both sides to isolate \( y \):
     \[
     y = -\frac{3}{5}x - 9
     \]

### Final Answer
The equation of line \( q \) in slope-intercept form is:
\[
y = -\frac{3}{5}x - 9
\]

Would you like further details on any step, or do you have any questions?

Here are some related questions to expand your understanding:
1. How do you find the equation of a line parallel to a given line?
2. What is the significance of the negative reciprocal in perpendicular slopes?
3. How would you rewrite the equation if the point given was different?
4. Can you convert the equation into standard form?
5. What happens to the line if you change the slope while keeping the same point?

**Tip:** When working with perpendicular lines, remember that their slopes are always negative reciprocals. This concept is useful in geometry and analytic geometry problems.

Line p has an equation of y = 5/3x - 4. Line q includes the point (-10, -3) and is perpendicular to line p. What is the equation of line q?

Learn how to find the equation of a line perpendicular to y = 5/3x - 4 that passes through the point (-10, -3). This problem covers key algebraic concepts including slope-intercept form, point-slope form, and perpendicular slopes. Suitable for Grade 9 students.

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