x^2-y^2=5(x+y)
===> (x+y)(x-y)-5(x+y)=0
===> (x+y)(x-y-5)=0
===> x=-y,或者x=y+5
①当x=-y时,带入第二个有：x^2-x^2+x^2=43
===> x^2=43
===> x=±√43
则,y=±√43
②当x=y+5时,带入第二个有：(y+5)^2+(y+5)*y+y^2=43
===> y^2+10y+25+y^2+5y+y^2=43
===> 3y^2+15y-18=0
===> y^2+5y-6=0
===> (y-1)(y+6)=0
===> y=1,或者y=-6
所以,x=6,或者x=-1
综上：
①x=√43,y=-√43
②x=-√43,y=√43
③x=6,y=-1
④x=-6,y=1