Giải thích các bước giải:
\(\begin{array}{l}
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{{x + y + z}}\\
\Leftrightarrow \frac{{xy + yz + xz}}{{xyz}} = \frac{1}{{x + y + z}}\\
\Leftrightarrow (xy + yz + xz)(x + y + z) = xyz\\
\Leftrightarrow {x^2}y + x{y^2} + xyz + xyz + {y^2}z + y{z^2} + {x^2}z + xyz + x{z^2} = xyz\\
\Leftrightarrow xy(x + y) + yz(x + y) + {z^2}(x + y) + xz(x + y) = 0\\
\Leftrightarrow (x + y)(y + z)(x + z) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
x = - y\\
y = - z\\
z = - x
\end{array} \right.
\end{array}\)
\(\Rightarrow \frac{1}{{{x^{2019}}}} + \frac{1}{{{y^{2019}}}} + \frac{1}{{{z^{2019}}}} = \frac{1}{{{x^{2019}} + {y^{2019}} + {z^{2019}}}}\)
