From the first equation: 1/y2=7-4/x2=(7x2-4)/x2.
Substitute for 1/y2 in the other equation:
2/x2-2(7x2-4)/x2=-4,
2-14x2+8=-4x2,
10x2=10, x2=1, x=±1.
Substitute x2=1 in either of the given equations:
1/y2=7-4/x2=7-4=3, y2=⅓, y=±1/√3=±√3/3.
So there are 4 solutions: (x,y)=(-1,-√3/3),(-1,√3/3),(1,-√3/3),(1,√3/3).