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

To find the equation of the line that is perpendicular to the line given by the equation \(y = 2x - 4\) and passes through the point (3, 1), we can follow these steps: 1. **Determine the slope of the given line**: The equation \(y = 2x - 4\) is in slope-intercept form \(y = mx + b\), where \(m\) is the slope. Here, the slope \(m\) is 2. 2. **Find the slope of the perpendicular line**: The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope \(m_{\perp}\) of the perpendicular line is: \[ m_{\perp} = -\frac{1}{m} = -\frac{1}{2} \] 3. **Use the point-slope form of the equation of a line**: The point-slope form of a line's equation is given by: \[ y - y_1 = m(x - x_1) \] where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Here, we have the point (3, 1) and the slope \(-\frac{1}{2}\). 4. **Substituting the values into the point-slope form**: \[ y - 1 = -\frac{1}{2}(x - 3) \] 5. **Simplifying the equation**: \[ y - 1 = -\frac{1}{2}x + \frac{3}{2} \] \[ y = -\frac{1}{2}x + \frac{3}{2} + 1 \] \[ y = -\frac{1}{2}x + \frac{3}{2} + \frac{2}{2} \] \[ y = -\frac{1}{2}x + \frac{5}{2} \] Thus, the equation of the line that is perpendicular to \(y = 2x - 4\) and passes through the point (3, 1) is: \[ \boxed{y = -\frac{1}{2}x + \frac{5}{2}} \]