The equation of the line perpendicular to \( 3 x-2 y-6=0 \) and passing through \( P(0 ; ;-3) \) is... (A) \( 3 y=-x+1 \) (B) \( 3 y=2 x-9 \) (C) \( 3 y=-2 x-9 \) (D) \( 3 y=2 x \) (E) \( 3 y=-2 x \)

To find the equation of the line perpendicular to \( 3x - 2y - 6 = 0 \), we first need to determine the slope of the given line. Rearranging it into slope-intercept form gives: \[ 2y = 3x - 6 \quad \Rightarrow \quad y = \frac{3}{2}x - 3 \] The slope of the line is \( \frac{3}{2} \). The slope of a line that is perpendicular to this line will be the negative reciprocal, which is \( -\frac{2}{3} \). 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 point through which the line passes, which in this case is \( (0, -3) \): \[ y - (-3) = -\frac{2}{3}(x - 0) \quad \Rightarrow \quad y + 3 = -\frac{2}{3}x \] Rearranging this equation, we get: \[ y = -\frac{2}{3}x - 3 \] To convert this equation to a more familiar form, we can multiply everything by 3 to eliminate the fraction: \[ 3y = -2x - 9 \] which can be rearranged to: \[ 2x + 3y + 9 = 0 \] Now let’s compare this with the answer choices given. The final equation \( 3y = -2x - 9 \) matches option (C): \( 3y = -2x - 9 \) So the correct answer is \( \text{(C) } 3y = -2x - 9 \).