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

The slope of the line is $LaTeX: \displaystyle m = 1$ so the perpendicular line has slope $LaTeX: \displaystyle m=-1$ . The equation has the form $LaTeX: \displaystyle y=-1x+b$ . Using the point $LaTeX: \displaystyle \left( 3, \ 2\right)$ gives the equation $LaTeX: \displaystyle 2=-1\left(3\right)+b$ Solving for $LaTeX: \displaystyle b$ gives $LaTeX: \displaystyle b = 5$ . The equation of the perpendicular line is $LaTeX: \displaystyle y = 5 - x$ .