Question

tìm giá trị của biểu thức

\(\dfrac{x+y}{z}+\dfrac{x+z}{y}+\dfrac{y+z}{x}\)

biết \(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=0\)

Answer 1

Ta có: \(A=\dfrac{x+y}{z}+\dfrac{x+z}{y}+\dfrac{y+z}{x}\)

\(\Rightarrow A+3=\dfrac{x+y}{z}+1+\dfrac{x+z}{y}+1+\dfrac{y+z}{x}+1\)

\(=\dfrac{x+y+z}{z}+\dfrac{x+y+z}{y}+\dfrac{x+y+z}{x}\)

\(=\left(x+y+z\right)\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\)

Mà \(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=0\Rightarrow A+3=0\) \(\Rightarrow A=-3\)