Question

The equation of line d is y =-4/7x+5/2.

The equation of line e is y = 7/4x+ 5/7.
= Are line d and
line e parallel or perpendicular?

Answer 1

`y=\stackrel{\stackrel{m}{\downarrow }}{-\cfrac{4}{7}}x+\cfrac{5}{2}\hspace{5em}y=\stackrel{\stackrel{m}{\downarrow }}{\cfrac{7}{4}}x+\cfrac{5}{7}\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}`

now, parallel lines have exactly the same slope, well, clearly this isn't the case, well, and perpendicular ones have negative reciprocal slopes, let's check hmmm what's the negative reciprocal of -4/7?

`\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{ \cfrac{-4}{7}} ~\hfill \stackrel{reciprocal}{\cfrac{7}{-4}} ~\hfill \stackrel{negative~reciprocal}{\underset{ \textit{\large perpendicular \checkmark} }{-\cfrac{7}{-4} \implies \cfrac{7}{ 4 }}}}`