Use substitution
From second equation:-
x = 13 - 5y, Substitute for x in the first eqaution:
(13 - 5y)^2 + y^2 = 13
169 -130y + 25y^2 + y^2 - 13 = 0
26y^2 - 130y + 156 = 0
26(y^2 - 5y + 6) = 0
26(y - 3)(y - 2) = 0
y = 2 , y = 3
From the second equation when y = 2, x = 13 - 5(2) = 3
and when y = 3, x = 13 - 5(3) = -2
So the solution is [3, 2} and {-2, 3}