Find the equation of the line perpendicular to $LaTeX: \displaystyle y = \frac{3 x}{2} + 6$ that contains $LaTeX: \displaystyle \left( 8, \ 9\right)$ .

The slope of the line is $LaTeX: \displaystyle m = \frac{3}{2}$ so the perpendicular line has slope $LaTeX: \displaystyle m=- \frac{2}{3}$ . The equation has the form $LaTeX: \displaystyle y=- \frac{2}{3}x+b$ . Using the point $LaTeX: \displaystyle \left( 8, \ 9\right)$ gives the equation $LaTeX: \displaystyle 9=- \frac{2}{3}\left(8\right)+b$ Solving for $LaTeX: \displaystyle b$ gives $LaTeX: \displaystyle b = \frac{43}{3}$ . The equation of the perpendicular line is $LaTeX: \displaystyle y = \frac{43}{3} - \frac{2 x}{3}$ .