Question

Determine the given pair of lines of parallel, perpendicular, or intersecting.

Answer 1

$k:\ y=m_1\cdot x+b_1\ \ \ \ and\ \ \ \ l:\ y=m_2\cdot x+b_2 \\\\k\ ||\ l\ \ \ \Leftrightarrow\ \ \ m_1=m_2\ \ \ \ \ \ and \ \ \ \ \ \ k\ \bot\ l\ \ \ \Leftrightarrow\ \ \ \ m_1\cdot m_2=-1\\----------------------------\\1.\\ k:\ y= \frac{7}{3} x-8\ \ \ and\ \ \ l:\ y=\frac{7}{3} x+3\\\\m_1=m_2=\frac{7}{3}\ \ \ \Rightarrow\ \ \ k\ ||\ l$

$2.\\k:\ y=2x-1\ \ and\ \ l:\ -x+2y=6\ \Rightarrow\ \ 2y=x+6\ \ \Rightarrow\ \ y= \frac{1}{2}x+3 \\\\m_1 \neq m_2\ \ \ and\ \ \ m_1\cdot m_2 \neq -1\ \ \ \Rightarrow\ \ \ the\ lines\ intersect\\\\3.\\k:\ y= \frac{5}{2} x-17\ \ \ and\ \ \ l:\ y=- \frac{5}{2} x-1\ \ \ \Rightarrow\ \ \ the\ lines\ intersect\\\\4.\\k:\ y= \frac{5}{2} x-4\ \ and\ \ l:\ -5x+2y=6\ \Rightarrow \ 2y=5x+6\ \Rightarrow \ y= \frac{5}{2}x +3\\\\m_1=m_2= \frac{5}{2} \ \ \ \Rightarrow\ \ \ k\ ||\ l$

$5.\\k:\ y=- \frac{2}{3} x+12\ \ \ and\ \ l:\ y= \frac{3}{2} x+3\\\\m_1\cdot m_2=- \frac{2}{3}\cdot \frac{3}{2} =-1\ \ \ \Rightarrow\ \ \ k\ ||\ l\\\\6.\\k:\ y= \frac{1}{3}x-13\ \ \ and\ \ \ l:\ y=3x+2\ \ \ \Rightarrow\ \ \ the\ lines\ intersect \\\\7.\\k:\ y=-4x+5\ \ ,\ \ l: -x+4y=-16\ \Rightarrow\ 4y=x-16\ \Rightarrow\ y= \frac{1}{4}x-4 \\\\m_1\cdot m_2=-4\cdot \frac{1}{4} =-1\ \ \ \Rightarrow\ \ \ k\ \bot\ l$

$8.\\k:\ y= \frac{2}{3} x+3 \ ,\ \ l:\ -2x+3y=21 \ \Rightarrow\ \ 3y=2x+21\ \Rightarrow\ y= \frac{2}{3} x+7\\\\m_1=m_2= \frac{2}{3} \ \ \ \Rightarrow\ \ \ k\ ||\ l$