x^2-y^2=55
this means that
(x+y)(x-y)=55
we know that x-y=11 so we can write
(x+y)(11)=55
x+y=55/11 we divide by 11 and we obtain
x+y=5
x-y=11 this means
x=y+11 we replace in the other equation (x+y=5)
y+11+y=5
2y=5-11
2y=-6
y=-3
and you find out that also
x=8
