Question

Is 2x+3y=4 and y=2/3x+3 perpendicular parallel or neither

Answer 1

keeping in mind that perpendicular lines have negative reciprocal slopes, and that parallel lines have exactly the same slope, let's check for the slope of the equations above

`y = \stackrel{\stackrel{m}{\downarrow }}{\cfrac{2}{3}}x+3\qquad \impliedby \qquad \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`

`2x+3y=4\implies 3y=-2x+4\implies y=\cfrac{-2x+4}{3}\implies y=\stackrel{ \stackrel{m}{\downarrow } }{-\cfrac{2}{3}} x+\cfrac{4}{3} \\\\\\ ~\hfill~\textit{slopes are the same, \underline{parallel}}~\hfill~`