Question

Does the given equation of line form parallel lines or not

Answer 1

$k:y=m_1x+b_1;\ l:y=m_2x+b_2\\\\k\ \perp\ l\iff m_1\cdot m_2=-1\\\\k\ ||\ l\iff m_1=m_2$

$k:x+4y=10\ \ \ |-x\\\\4y=-x+10\ \ \ \ |:4\\\\y=-\dfrac{1}{4}x+\dfrac{10}{4}\to m_1=-\dfrac{1}{4}\\\\l:3x-2y=5\ \ \ \ |-3x\\\\-2y=-3x+5\ \ \ \ |:(-2)\\\\y=\dfrac{3}{2}x-\dfrac{5}{2}\to m_2=\dfrac{3}{2}$

$m_1\neq m_2\to\text{ no parallel}\\\\m_1\cdot m_2=-\dfrac{1}{4}\cdot\dfrac{3}{2}\neq-1\to\text{ no perpendicular}$

Answer: NOT PARALLEL