Lời giải:
\(\text{HPT}\Leftrightarrow \left\{\begin{matrix} \frac{1}{a}=\frac{1}{x}-\frac{1}{y}-\frac{1}{z}(1)\\ \frac{1}{b}=\frac{1}{y}-\frac{1}{z}-\frac{1}{x}(2)\\ \frac{1}{c}=\frac{1}{z}-\frac{1}{x}-\frac{1}{y}(3)\end{matrix}\right.\Rightarrow -\left [\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right]=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}(4)\)
Lấy \((1),(2),(3)+(4)\Rightarrow \left\{\begin{matrix} -\left(\frac{1}{b}+\frac{1}{c}\right)=\frac{2}{x}\\ -\left(\frac{1}{a}+\frac{1}{c}\right)=\frac{2}{y}\\ -\left(\frac{1}{a}+\frac{1}{b}\right)=\frac{2}{z}\end{matrix}\right.\rightarrow \left\{\begin{matrix} x=\frac{-2bc}{b+c}\\ y=\frac{-2ac}{a+c}\\ z=\frac{-2ab}{a+b}\end{matrix}\right.\)
Vậy nghiệm của HPT là \((x,y,z)=\left(\frac{-2bc}{b+c},\frac{-2ac}{a+c},\frac{-2ab}{a+b}\right)\)
