Answer
The equation of the parallel line is $y=-\frac{2}{3}x-\frac{1}{3}$
The equation of the perpendicular line is $y=\frac{3}{2}x+3$
Work Step by Step
First, let's rearrange the line into the form y=mx+b:
$2x+3y=6$
$3y=-2x+6$
$y=-\frac{2}{3}x+2$
Parallel lines have the same slope. Since we have the slope and a point, we can find the equation of the parallel line by first finding the y-intercept (b):
$-1=-\frac{2}{3}(1)+b$
$-1=-\frac{2}{3}+b$
$-1+\frac{2}{3}=b$
$-\frac{1}{3}=b$
So, the equation of the parallel line is $y=-\frac{2}{3}x-\frac{1}{3}$
The slope of a perpendicular line is the negative inverse: $-\frac{2}{3} \rightarrow \frac{3}{2}$
Now that we have the slope and a point, we can find the equation of the perpendicular line by first finding the y-intercept (b):
$3=\frac{3}{2}(0)+b$
$b=3$
So, the equation of the perpendicular line is $y=\frac{3}{2}x+3$
