Question

Find the slope of a line perpendicular to the line whose equation is 2x+3y=242x+3y=24. Fully simplify your answer.

Answer 1

keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of the equation above

`2x+3y=24\implies 3y=-2x+24\implies y=\cfrac{-2x+24}{3} \\\\\\ y=\stackrel{\stackrel{m}{\downarrow }}{-\cfrac{2}{3}}x+8\qquad \impliedby \qquad \begin{array}ll \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array} \\\\[-0.35em] ~\dotfill`

`\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{ \cfrac{-2}{3}} ~\hfill \stackrel{reciprocal}{\cfrac{3}{-2}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{3}{-2} \implies \boxed{\cfrac{3}{ 2 }}}}`