OK so,

lets first rearrange the second equation: x^2-xy+y^2=64.

and we know that x+y=10.

we can rewrite the second equation like this

(x+y)^2-3xy=64

and since x+y=10,

100-3xy=64

which means that xy=12.

NOW

we solve x+y=10 and xy=12.

we can substitute x as 10-y

so we have

y(10-y)=12

10y-y^2=12

and we get the quadratic

y^2-10y+12=0.

this isnt easily factorable so i will use the quadratic formula.

we get that y=5+sqrt 13, 5-sqrt 13.

so that means

(x,y)=(5+sqrt 13, 5-sqrt 13) and (5-sqrt 13, 5+sqrt 13).

:)