已知a,b,c都是有理数,满足：a(1/b+1/c)+b(1/a+1/c)+c(1/a+1/b)=-3且a+b+c不为0求：1/a+1/b+1/c的值

Question

已知a,b,c都是有理数,满足：
a(1/b + 1/c)+ b(1/a + 1/c)+ c(1/a + 1/b)=-3
且a+b+c不为0
求：1/a + 1/b + 1/c的值

Answer 1

a(1/b + 1/c)+ b(1/a + 1/c)+ c(1/a + 1/b)=-3
由乘法分配律
a/b+a/c+b/a+b/c+c/a+c/b+3=0
a/b+a/c+b/a+b/c+c/a+c/b+1+1+1=0
(a/b+c/b+1)+(b/a+c/a+1)+(a/c+b/c+1)=0
(a+b+c)/b+(a+b+c)/a+(a+b+c)/c=0
(a+b+c)(1/a+1/b+1/c)=0
a+b+c≠0
所以1/a+1/b+1/c=0