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=(-6)/(-4)x+(24)/(-4)\\\\y=(3)/(2)x-6\to m_1=(3)/(2)\\----------------------`

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