What is the equation of the line perpendicular to the line with equ ation \( 7 x+2 y=14 \) and passing through \( (7,-6) 7 \)

To find the equation of the line that is perpendicular to the line given by the equation \( 7x + 2y = 14 \) and passes through the point \( (7, -6) \), we can follow these steps: 1. **Find the slope of the given line**: We start by rewriting the equation \( 7x + 2y = 14 \) in slope-intercept form \( y = mx + b \), where \( m \) is the slope. \[ 2y = -7x + 14 \] \[ y = -\frac{7}{2}x + 7 \] From this, we see that the slope \( m \) of the given line is \( -\frac{7}{2} \). 2. **Determine 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: \[ m_{\perp} = -\frac{1}{-\frac{7}{2}} = \frac{2}{7} \] 3. **Use the point-slope form to find the equation of the perpendicular line**: The point-slope form of a line 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 \( (7, -6) \) and the slope \( \frac{2}{7} \): \[ y - (-6) = \frac{2}{7}(x - 7) \] \[ y + 6 = \frac{2}{7}(x - 7) \] 4. **Simplify the equation**: Distributing the slope on the right side: \[ y + 6 = \frac{2}{7}x - 2 \] Now, subtract 6 from both sides: \[ y = \frac{2}{7}x - 2 - 6 \] \[ y = \frac{2}{7}x - 8 \] 5. **Final equation**: The equation of the line that is perpendicular to \( 7x + 2y = 14 \) and passes through the point \( (7, -6) \) is: \[ \boxed{y = \frac{2}{7}x - 8} \]