Answer
a) $b=2$
b) $b=4$ is one possible answer. (As long as $b\ne2$, there is one solution to the system.)
Work Step by Step
a)
$x+y=4$
$2x+by=8$
$2x+by=8$
$2*(x+b/2*y=4)$
Since we want there to be an infinite number of solutions, we want $1 = b/2$ (have the coefficients of $y$ in both equations match).
$b/2=1$
$b=2$
b)
We want a single solution of the system, so we don't want $b=2$.
If we let $b=4$, then we have
$x+y=4$
$2x+4y=8$
$x+y=4$
$-2*(x+y=4)$
$-2x-2y=-8$
$2x+4y=8$
$-2x-2y=-8$
$2x-2x+4y-2y=8-8$
$2y=0$
$2y/2=0/2$
$y=0$
$x+y=4$
$x+0=4$
$x=4$
