x²+6xy-7y²
=(x+7y)(x-y)
再将2009因素分解
2009=1×2009
2009=287×7
2009=41×7×7
1、
x+7y=2009
x-y=1
x=252
y=251
2、x+7y=287
x-y=7
x=42
y=35
3、x+7y=49
x-y=41
x=42
y=1
x^2+6xy-7y^2=2009
x^2+6xy+9y^2-16y^2=2009
(x+3y)^2-16y^2=2009
(x+3y+4y)(x+3y-4y)=2009
(x+7y)(x-y)=2009
令x-y=1 则x=y+1
即(y+1+7y)=2009
8y+1=2009
y=251
x=252
x^2+6xy-7y^2=(x-y)(x+7y)=2009
可知道令a=x-y,b=x+7y,ab=2009=7*7*41*1
b-a=8y
有2009=7*287=41*49=2009*1三种情况,代入均符合
得到三组解:x=42,y=1;x=42,y=35;x=252,y=251.