The line joining (-5, 7) and (0, -2) is perpendicular to the line joining (1, -3) and (4, x). Find x.

Answer

To solve the problem, we need to find the value of \( x \) such that the line joining the points (-5, 7) and (0, -2) is perpendicular to the line joining the points (1, -3) and (4, x).

### Step-by-Step Solution:

1. **Identify the Points:**
- Let \( A(-5, 7) \) and \( B(0, -2) \) be the first line's endpoints.
- Let \( C(1, -3) \) and \( D(4, x) \) be the second line's endpoints.

2. **Calculate the Slope of Line AB:**
- The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
- For points \( A(-5, 7) \) and \( B(0, -2) \):
\[ m_1 = \frac{-2 - 7}{0 - (-5)} = \frac{-9}{5} \]

3. **Calculate the Slope of Line CD:**
- For points \( C(1, -3) \) and \( D(4, x) \):
\[ m_2 = \frac{x - (-3)}{4 - 1} = \frac{x + 3}{3} \]

4. **Use the Perpendicular Slope Condition:**
- Two lines are perpendicular if the product of their slopes is -1:
\[ m_1 \cdot m_2 = -1 \]
- Substitute the slopes:
\[ \left(-\frac{9}{5}\right) \cdot \left(\frac{x + 3}{3}\right) = -1 \]

5. **Solve for x:**
- Simplifying the equation:
\[ -\frac{9(x + 3)}{15} = -1 \]
- Multiply both sides by -15:
\[ 9(x + 3) = 15 \]
- Divide by 9:
\[ x + 3 = \frac{15}{9} = \frac{5}{3} \]
- Subtract 3 from both sides:
\[ x = \frac{5}{3} - 3 = \frac{5}{3} - \frac{9}{3} = -\frac{4}{3} \]

### Final Answer:
\[ x = -\frac{4}{3} \]