Question

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

4x+10y=-140.
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

`4x+10y=-140\implies 10y=-4x-140\implies y=\cfrac{-4x-140}{10} \\\\\\ y=\cfrac{-4x}{10}-\cfrac{140}{10}\implies y=\stackrel{\stackrel{m}{\downarrow }}{-\cfrac{2}{5}}x-14\qquad \impliedby \begin{array} \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}{5}} ~\hfill \stackrel{reciprocal}{\cfrac{5}{-2}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{5}{-2}\implies {\Large \begin{array}{llll} \cfrac{5}{2} \end{array}}}}`