Answer
$x=\frac{b}{b-a},y=\frac{-a}{b-a}$
Work Step by Step
We solve the second equation for $x$:
$x+y=1$
$x=1-y$
Plugging this into the first equation, we get:
$ax+by=0\\a(1-y)+by=0\\a-ay+by=0\\y(b-a)=-a\\y=\frac{-a}{b-a}$
Thus, we have
$x=1-\frac{-a}{b-a}=\frac{b}{b-a}$