through: \( (3,3) \), perp. to \( y=\frac{3}{2} x+4 \)

To find the equation of a line that passes through the point \( (3,3) \) and is perpendicular to the line given by \( y = \frac{3}{2} x + 4 \), we first need to determine the slope of the perpendicular line. The slope of the given line is \( \frac{3}{2} \), so the slope of the perpendicular line will be the negative reciprocal, which is \( -\frac{2}{3} \). Using the point-slope form of a linear equation, \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is our point \( (3, 3) \) and \( m \) is the slope \( -\frac{2}{3} \), we have: \[ y - 3 = -\frac{2}{3}(x - 3) \] Distributing and rearranging gives us: \[ y - 3 = -\frac{2}{3}x + 2 \] Adding 3 to both sides leads to: \[ y = -\frac{2}{3}x + 5 \] So the equation of the line that goes through the point \( (3,3) \) and is perpendicular to \( y=\frac{3}{2} x+4 \) is: \[ y = -\frac{2}{3}x + 5 \]