Find an equation of the line that satisfies the given conditions. Through \( (-1,-2) \); perpendicular to the line \( 2 x+5 y+6=0 \)

To find the equation of the line that is perpendicular to \( 2x + 5y + 6 = 0 \) and passes through the point \( (-1, -2) \), follow these steps: 1. **Find the slope of the given line:** Start by rewriting the equation in slope-intercept form (\( y = mx + b \)): \[ 2x + 5y + 6 = 0 \implies 5y = -2x - 6 \implies y = -\frac{2}{5}x - \frac{6}{5} \] So, the slope (\( m_1 \)) of the given line is: \[ m_1 = -\frac{2}{5} \] 2. **Determine the slope of the perpendicular line:** The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore: \[ m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{2}{5}} = \frac{5}{2} \] 3. **Use the point-slope form to find the equation:** The point-slope form of a line is: \[ y - y_1 = m(x - x_1) \] Plugging in the point \( (-1, -2) \) and the slope \( \frac{5}{2} \): \[ y - (-2) = \frac{5}{2}(x - (-1)) \implies y + 2 = \frac{5}{2}(x + 1) \] 4. **Simplify to slope-intercept form:** Distribute and solve for \( y \): \[ y + 2 = \frac{5}{2}x + \frac{5}{2} \implies y = \frac{5}{2}x + \frac{5}{2} - 2 \implies y = \frac{5}{2}x + \frac{1}{2} \] **Final Equation:** \[ y = \frac{5}{2}x + \frac{1}{2} \]