Answer
Slopes of the lines $a$ and $b$ are same. So, they are parallel.
Work Step by Step
For line $a$: $3y-x=6$
$\implies 3y=6+x$
$\implies y=\frac{6+x}{3}=\frac{6}{3}+\frac{x}{3}$
$\implies y=\frac{1}{3}x+2$
Comparing the above equation with slope-intercept form $y=mx+b$, we find that
the slope of line $a$=$\frac{1}{3}$
For line $b$: $3y=x+18$
$\implies y=\frac{x}{3}+\frac{18}{3}$
$\implies y=\frac{1}{3}x+6$
Comparing the above equation with slope-intercept form $y=mx+b$, we find that
the slope of line $b$=$\frac{1}{3}$
For line $c$: $3y-2x=9$
$\implies 3y=2x+9$
$\implies y=\frac{2}{3}x+3$
Slope of line $c$= $\frac{2}{3}$
Slopes of the lines $a$ and $b$ are same. So, they are parallel.