Find the equation of the line perpendicular to $LaTeX: \displaystyle y = \frac{4 x}{7} - 1$ that contains $LaTeX: \displaystyle \left( 2, \ 4\right)$ .

The slope of the line is $LaTeX: \displaystyle m = \frac{4}{7}$ so the perpendicular line has slope $LaTeX: \displaystyle m=- \frac{7}{4}$ . The equation has the form $LaTeX: \displaystyle y=- \frac{7}{4}x+b$ . Using the point $LaTeX: \displaystyle \left( 2, \ 4\right)$ gives the equation $LaTeX: \displaystyle 4=- \frac{7}{4}\left(2\right)+b$ Solving for $LaTeX: \displaystyle b$ gives $LaTeX: \displaystyle b = \frac{15}{2}$ . The equation of the perpendicular line is $LaTeX: \displaystyle y = \frac{15}{2} - \frac{7 x}{4}$ .