What is the equation of the line perpendicular to \( y=\frac{-1}{5} x-25 \) that passes through the point \( (-2,-4) \) ?

To find the equation of the line perpendicular to \( y = \frac{-1}{5}x - 25 \) that passes through the point \( (-2, -4) \), follow these steps: 1. **Determine the slope of the given line:** The equation \( y = \frac{-1}{5}x - 25 \) is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. \[ m_1 = -\frac{1}{5} \] 2. **Find the slope of the perpendicular line:** The slope of a line perpendicular to another is the negative reciprocal of the original slope. \[ m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{1}{5}} = 5 \] 3. **Use the point-slope form to write the equation of the perpendicular line:** The point-slope form is: \[ y - y_1 = m(x - x_1) \] Plugging in the slope \( m = 5 \) and the point \( (-2, -4) \): \[ y - (-4) = 5(x - (-2)) \\ y + 4 = 5(x + 2) \] 4. **Simplify to slope-intercept form:** \[ y + 4 = 5x + 10 \\ y = 5x + 10 - 4 \\ y = 5x + 6 \] **Final Equation:** \[ \boxed{y = 5x + 6} \]