Question

Determine whether the lines are parallel, perpendicular, or coinciding 6x-4y=24, 3y+2x=12

Answer 1

$Let\ k:y=m_1x+b_1\ l:y=m_2x+b_2,\ then\\\\l\ \parallel\ k\iff m_1=m_2\\\\l\ \perp\ k\iff m_1m_2=-1\\--------------------------------------\\6x-4y=24\qquad\text{subtract 6x from both sides}\\\\-4y=-6x+24\qquad\text{divide both sides by (-4)}\\\\y=\dfrac{-6}{-4}x+\dfrac{24}{-4}\\\\y=\dfrac{3}{2}x-6\to m_1=\dfrac{3}{2}\\----------------------$

$3y+2x=12\qquad\text{subtract 2x from both sides}\\\\3y=-2x+12\qquad\text{divide both sides by 3}\\\\y=\dfrac{-2}{3}x+\dfrac{12}{3}\\\\y=-\dfrac{2}{3}x+4\to m_2=-\dfra\dfrac{2}{3}\\-------------------------------\\\\m_1=\dfrac{3}{2},\ m_2=-\dfrac{2}{3}\to m_1\neq m_2-not\ parallel\\\\m_1m_2=\dfrac{3}{2}\left(-\dfrac{2}{3}\right)=-1\\\\Answer:\ The\ lines\ are\ perpendicular$