Solution:
If $c<0$, there are no solutions. If $c>0$ then we have four possible systems of linear equations given by $x+y= \pm 99$, $x-y= \pm c$, giving four solutions $(x, y)$. So we must have $c=0$, and then we do get two solutions ($x=y$, so they must both equal $\pm 99 / 2$).