x+y/x的平方-xy+y的平方=3/7,求它的整数解

x^2表示x²
(x+y)/(x^2-xy+y^2)=3/7
设x+y=3t
x^2-xy+y^2=7t
t为整数
于是
x^2+2xy+y^2=9t^2
3xy=9t^2-7t
(x-y)^2=(28t-9t^2)/3
则t是3的倍数,于是,设t=3k
则(x-y)^2=28k-27k^2=k(28-27k)≥0
又k是整数,于是k=0或1
当k=0时,分母为0,舍弃,于是k=1
则x-y=±1,x+y=3t=9k=9
x=4,y=5或x=5,y=4