Question

Cho \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\)   Tính A = \(\frac{\left(x+y\right)}{z}+\frac{\left(x+z\right)}{y}+\frac{\left(y+z\right)}{x}\)

Answer 1

\(A+3=\left(1+\frac{x+y}{z}\right)+\left(1+\frac{x+z}{y}\right)+\left(1+\frac{y+z}{x}\right)\)

\(A+3=\left(x+y+z\right).\left(\frac{1}{z}+\frac{1}{y}+\frac{1}{x}\right)\)

\(A+3=\left(x+y+z\right).0=0\Rightarrow A=-3\)

Answer 2

\(A=\frac{x+y}{z}+\frac{x+z}{y}+\frac{y+z}{x}=\left(\frac{x+y}{z}+1\right)+\left(\frac{x+z}{y}+1\right)+\left(\frac{y+z}{x}+1\right)-3\)

\(=\frac{x+y+z}{z}\cdot\frac{x+y+z}{y}\cdot\frac{x+y+z}{x}-3=\left(x+y+z\right)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)-3=-3\)