Line \( v \) passes through points \( (9,9) \) and \( (2,11) \). Line \( w \) is perpendicular to \( v \). What is the slope of line \( w \) ? Simplify your answer and write it as a proper fraction, improper fraction, or integer.

To find the slope of line \( w \) which is perpendicular to line \( v \), we first need to determine the slope of line \( v \). The slope \( m \) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] For line \( v \), we have the points \((9,9)\) and \((2,11)\): \[ m_v = \frac{11 - 9}{2 - 9} = \frac{2}{-7} = -\frac{2}{7} \] Since line \( w \) is perpendicular to line \( v \), the slope of line \( w \) is the negative reciprocal of the slope of line \( v \). Thus, \[ m_w = -\frac{1}{m_v} = -\frac{1}{-\frac{2}{7}} = \frac{7}{2} \] Therefore, the slope of line \( w \) is: \[ \frac{7}{2} \]