Find the equation of the line perpendicular to $LaTeX: \displaystyle y = - \frac{x}{3} - 6$ that contains $LaTeX: \displaystyle \left( 5, \ 7\right)$ .

The slope of the line is $LaTeX: \displaystyle m = - \frac{1}{3}$ so the perpendicular line has slope $LaTeX: \displaystyle m=3$ . The equation has the form $LaTeX: \displaystyle y=3x+b$ . Using the point $LaTeX: \displaystyle \left( 5, \ 7\right)$ gives the equation $LaTeX: \displaystyle 7=3\left(5\right)+b$ Solving for $LaTeX: \displaystyle b$ gives $LaTeX: \displaystyle b = -8$ . The equation of the perpendicular line is $LaTeX: \displaystyle y = 3 x - 8$ .