Find the equation of the line that is perpendicular to the line given by \(y = -\frac{1}{2}x + 3\) and passes through the point (4, 2).

To find the equation of the line that is perpendicular to the given line \( y = -\frac{1}{2}x + 3 \) and passes through the point \((4, 2)\), 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. For \( y = -\frac{1}{2}x + 3 \), the slope \( m_1 \) is: \[ m_1 = -\frac{1}{2} \] 2. **Find the slope of the perpendicular line:** The slope of a line perpendicular to another is the **negative reciprocal** of the original slope. Therefore, the slope \( m_2 \) of the perpendicular line is: \[ m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{1}{2}} = 2 \] 3. **Use the point-slope form to find the equation:** The point-slope form is given by: \[ y - y_1 = m(x - x_1) \] Plugging in the slope \( m_2 = 2 \) and the point \( (4, 2) \): \[ y - 2 = 2(x - 4) \] Simplifying this: \[ y - 2 = 2x - 8 \] \[ y = 2x - 6 \] **Final Answer:** \[ y = 2x - 6 \]