Find the equation of the line, in the form \( y=m x+b \), or \( x=c \), having the given properties. Perpendicular to the line \( y=2 x+4 \) and passing through \( (-2,2) \). \( y= \)

To find the equation of the line that is perpendicular to \( y = 2x + 4 \) and passes through the point \( (-2, 2) \), follow these steps: 1. **Determine the slope of the given line:** The given line is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. \[ m = 2 \] 2. **Find the slope of the perpendicular line:** Perpendicular lines have slopes that are negative reciprocals of each other. Therefore, the slope \( m' \) of the perpendicular line is: \[ m' = -\frac{1}{2} \] 3. **Use the point-slope form to find the equation of the perpendicular line:** The point-slope form is: \[ y - y_1 = m'(x - x_1) \] Plugging in the slope \( m' = -\frac{1}{2} \) and the point \( (-2, 2) \): \[ y - 2 = -\frac{1}{2}(x + 2) \] 4. **Simplify to the slope-intercept form \( y = mx + b \):** \[ y - 2 = -\frac{1}{2}x - 1 \] \[ y = -\frac{1}{2}x - 1 + 2 \] \[ y = -\frac{1}{2}x + 1 \] **Final Equation:** \[ y = -\frac{1}{2}x + 1 \]