Find an equation for the perpendicular bisector of the line segment whose endpoints are \( (-9,3) \) and \( (-3,-1) \).

To find the equation of the perpendicular bisector, first, we calculate the midpoint of the line segment joining the endpoints \((-9, 3)\) and \((-3, -1)\). The midpoint \(M\) is given by: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-9 + (-3)}{2}, \frac{3 + (-1)}{2} \right) = \left( \frac{-12}{2}, \frac{2}{2} \right) = (-6, 1). \] Next, we find the slope of the line segment. The slope \(m\) is calculated as follows: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 3}{-3 - (-9)} = \frac{-4}{6} = -\frac{2}{3}. \] The slope of the perpendicular bisector is the negative reciprocal of the slope of the original line, which will be \( \frac{3}{2} \). Now, we can use the point-slope form of a line equation, \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is the midpoint and \(m\) is the slope of the perpendicular bisector. Substituting in our values: \[ y - 1 = \frac{3}{2}(x + 6). \] Simplifying this, we have: \[ y - 1 = \frac{3}{2}x + 9 \implies y = \frac{3}{2}x + 10. \] Thus, the equation of the perpendicular bisector is: \[ \boxed{y = \frac{3}{2}x + 10}. \]