Giải thích các bước giải:
$\begin{array}{l}
\frac{{3a - 2b}}{5} = \frac{{2c - 5a}}{3} = \frac{{5b - 3c}}{2}\\
\Rightarrow \left\{ \begin{array}{l}
\frac{{34a}}{{15}} = \frac{{2c}}{3} + \frac{{2b}}{5}\\
\frac{{13c}}{6} = \frac{{5b}}{2} + \frac{{5a}}{3}
\end{array} \right.\\
\Rightarrow \left\{ \begin{array}{l}
a = \frac{{5c}}{{17}} + \frac{{3b}}{{17}}\\
\frac{{13c}}{6} = \frac{{5b}}{2} + \frac{{5a}}{3}
\end{array} \right.\\
\Rightarrow \left\{ \begin{array}{l}
a = \frac{{5c}}{{17}} + \frac{{3b}}{{17}}\\
\frac{{13c}}{6} = \frac{{5b}}{2} + \frac{5}{3}(\frac{{5c}}{{17}} + \frac{{3b}}{{17}})
\end{array} \right.\\
\Rightarrow \left\{ \begin{array}{l}
a = \frac{{5c}}{{17}} + \frac{{3b}}{{17}}\\
\frac{{57c}}{{34}} = \frac{{95b}}{{34}}
\end{array} \right.\\
\Rightarrow \left\{ \begin{array}{l}
a = \frac{{5c}}{{17}} + \frac{{3b}}{{17}}\\
c = \frac{{5b}}{3}
\end{array} \right.\\
\Rightarrow \left\{ \begin{array}{l}
a = \frac{5}{{17}}.\frac{{5b}}{3} + \frac{{3b}}{{17}}\\
c = \frac{{5b}}{3}
\end{array} \right.\\
\Rightarrow \left\{ \begin{array}{l}
a = \frac{{2b}}{3}\\
c = \frac{{5b}}{3}
\end{array} \right.
\end{array}$
Vì ${a^2} + 275 = bc$ nên:
$\begin{array}{l}
{\frac{{2b}}{3}^2} + 275 = b.\frac{{5b}}{3}\\
\Leftrightarrow \frac{{11b}}{9} = 275\\
\Leftrightarrow b = 225\\
\Rightarrow a = 150,c = 375
\end{array}$
