Question

What is the slope of a line perpendicular to the line whose equation is

6x+5y=−30
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

`6x+5y=-30\implies 5y=-6x-30\implies y=\cfrac{-6x-30}{5} \\\\\\ y=\stackrel{\stackrel{m}{\downarrow }}{-\cfrac{6}{5}}x-6\qquad \impliedby \qquad \begin{array}c \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{-6}{5}} ~\hfill \stackrel{reciprocal}{\cfrac{5}{-6}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{5}{-6}\implies {\Large \begin{array}{llll} \cfrac{5}{6} \end{array}}}}`