Answer
(a) $y=-\frac{1}{2}x-4$.
(b) $y=2x+1$.
Work Step by Step
The given line passes through the points $(-2,3)$ and $(0,2)$.
$\text{Slope of the given line}=\frac{2-3}{0-(-2)}=\frac{-1}{2}=-\frac{1}{2}$.
The point given is $(x_{1},y_{1})=(-2,-3)$.
(a) For parallel line, the slope is same.
Using point-slope form $y-y_{1}=m(x-x_{1})$, we have
$y-(-3)=-\frac{1}{2}(x-(-2))$
Using distributive property, we get
$y+3=-\frac{1}{2}x-1$
Subtracting $3$ from both sides, we obtain
$y=-\frac{1}{2}x-4$.
An equation of the line that passes through the given point and is parallel to the given line is $y=-\frac{1}{2}x-4$.
(b) For perpendicular lines, the slopes are negative reciprocals.
$\implies m=-(\frac{1}{-\frac{1}{2}})=2$.
Using point-slope form, we have
$y-(-3)=2(x-(-2))$
Using distributive property, we get
$y+3=2x+4$
$\implies y+3-3=2x+4-3$
$\implies y=2x+1$.
An equation of the line that passes through the given point and is perpendicular to the given line is $y=2x+1$.
