Answer
$(6,8),(8,6)$
Work Step by Step
$x+y-14=0\\y=14-x$
Thus plugging this into the first equation we get: $x^2+(14-x)^2-100=0\\2x^2+96-28x=0\\x^2-14x+48=0\\(x-6)(x-8)=0$
Thus $x=6$ or $x=8$. Thus if $x=6$, then $y=8$, and if $x=8$, then $y=6$.
Thus the solutions are: $(6,8),(8,6)$
