Answer
For parallel
$a=-\frac{2}{3}$
For perpendicular
$a=6$
Work Step by Step
The given equations of the lines are
$6y=-2x+4$ ...... (1)
$2y=ax-5$ ...... (2)
Find the slope of the equation (1).
$\Rightarrow 6y=-2x+4$
Divide each side by $6$.
$\Rightarrow \frac{6y}{6}=\frac{-2x}{6}+\frac{4}{6}$
Simplify.
$\Rightarrow y=-\frac{1}{3}x+\frac{2}{3}$
This is the slope form $y=mx+b$.
The slope of the line is $m_1=-\frac{1}{3}$.
Find the slope of the equation (2).
$\Rightarrow 2y=ax-5$
Divide each side by $2$.
$\Rightarrow \frac{2y}{2}=\frac{ax}{2}-\frac{5}{2}$
Simplify.
$\Rightarrow y=\frac{a}{2}x-\frac{5}{2}$
This is the slope form $y=mx+b$.
The slope of the line is $m_2=\frac{a}{2}$.
Slopes are equal for parallel lines.
$\Rightarrow m_1=m_2$
$\Rightarrow -\frac{1}{3}=\frac{a}{2}$
Solve for $a$.
$\Rightarrow a=-\frac{2}{3}$
Slopes are negative reciprocal for perpendicular lines.
$\Rightarrow m_2=-\frac{1}{m_1}$
$\Rightarrow \frac{a}{2}=-\frac{1}{(-\frac{1}{3})}$
Solve for $a$.
$\Rightarrow a=6$
