\(\frac{2x+1}{5}=\frac{3y-2}{7}=\frac{2x+3y-1}{6x}\)
\(\Rightarrow\frac{2x+1}{5}=k\rightarrow2x+1=5k\rightarrow2k=5k-1\)
\(\frac{3y-2}{7}=k\rightarrow3y-2=7k\rightarrow3y=2k+2\)
\(\frac{2x+3y-1}{6x}=k\rightarrow2x+3y-1=6x.k\)
\(\rightarrow5k-1+7k+2-1=k.3\left(5k-1\right)\)
\(\rightarrow12k=15k^2-3k\)
\(\rightarrow15k^2-15k=0\)
\(\rightarrow15k\left(k-1\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}k=0\rightarrow x=\frac{-1}{2};y=\frac{2}{3}\\k=1\rightarrow x=2;y=3\end{cases}}\)