Ta có: $2a=3b\Rightarrow \frac{a}{3}=\frac{b}{2}\Rightarrow \frac{a}{21}=\frac{b}{14}$
$5b=7c\Rightarrow \frac{b}{7}=\frac{c}{5}\Rightarrow \frac{b}{14}=\frac{c}{10}$
Khi đó: $\frac{a}{21}=\frac{b}{14}=\frac{c}{10}\Rightarrow \frac{3a}{63}=\frac{7b}{98}=\frac{5c}{50}$
Áp dụng tính chất dãy tỉ số bằng nhau:
$\frac{3a}{63}=\frac{7b}{98}=\frac{5c}{50}=\frac{3a+7b+5c}{63+98+50}=\frac{30}{211}$
$\Rightarrow \left\{ \begin{align}
& a=\frac{30}{211}.21=\frac{630}{211} \\
& b=\frac{30}{211}.14=\frac{420}{211} \\
& c=\frac{30}{211}.10=\frac{300}{211} \\
\end{align} \right.$
Vậy $\left( a;b;c \right)=\left( \frac{630}{211};\frac{420}{211};\frac{300}{211} \right)$