Đáp án:
\[\left\{ \begin{array}{l}
x = \frac{1}{2}\\
y = \frac{5}{6}\\
z = \frac{{ - 5}}{6}
\end{array} \right.\]
Giải thích các bước giải:
Áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\begin{array}{l}
\frac{x}{{y + z + 1}} = \frac{y}{{x + z + 2}} = \frac{z}{{x + y - 3}} = x + y + z = \frac{{x + y + z}}{{\left( {y + z + 1} \right) + \left( {x + z + 2} \right) + \left( {x + y - 3} \right)}} = \frac{{x + y + z}}{{2\left( {x + y + z} \right)}} = \frac{1}{2}\\
\Rightarrow \left\{ \begin{array}{l}
\frac{x}{{y + z + 1}} = \frac{1}{2}\\
\frac{y}{{x + z + 2}} = \frac{1}{2}\\
\frac{z}{{x + y - 3}} = \frac{1}{2}\\
x + y + z = \frac{1}{2}
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
\frac{x}{{\frac{1}{2} - x + 1}} = \frac{1}{2}\\
\frac{y}{{\frac{1}{2} - y + 2}} = \frac{1}{2}\\
\frac{z}{{\frac{1}{2} - z - 3}} = \frac{1}{2}
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x = \frac{1}{2}\\
y = \frac{5}{6}\\
z = \frac{{ - 5}}{6}
\end{array} \right.
\end{array}\)
