Find the equation of the line perpendicular to $LaTeX: \displaystyle y = \frac{4 x}{3} + 6$ that contains $LaTeX: \displaystyle \left( -7, \ -8\right)$ .

The slope of the line is $LaTeX: \displaystyle m = \frac{4}{3}$ so the perpendicular line has slope $LaTeX: \displaystyle m=- \frac{3}{4}$ . The equation has the form $LaTeX: \displaystyle y=- \frac{3}{4}x+b$ . Using the point $LaTeX: \displaystyle \left( -7, \ -8\right)$ gives the equation $LaTeX: \displaystyle -8=- \frac{3}{4}\left(-7\right)+b$ Solving for $LaTeX: \displaystyle b$ gives $LaTeX: \displaystyle b = - \frac{53}{4}$ . The equation of the perpendicular line is $LaTeX: \displaystyle y = - \frac{3 x}{4} - \frac{53}{4}$ .