Answer
a) $b=15$
b) $b=3$ (As long as $b \ne 15$, the system would have no solution.)
Work Step by Step
a)
$x+y=5$
$3x+3y=b$
$3x+3y=b$
$3*(x+y=b/3)$
For there to be an infinite number of solutions, we need $b/3 =5$.
$b/3=5$
$b=15$
b)
$x+y=5$
$3x+3y=b$
For there to be no solutions to the system, we need $b/3\ne5$. (We want two parallel lines, since parallel lines do not intersect.)
$b/3 \ne5$
$b \ne 15$
$x+y=5$
$3x+3y=b$
If we let $b=3$,
$3x+3y=3$
$3(x+y=1)$
$x+y=5$
$x+y=1$
$x+y=1$
$-1*(x+y=1)$
$-x-y=-1$
$x+y=5$
$-x-y=-1$
$x-x+y-y=5-1$
$0=4$ (false statement, so we know the system has no solution)
