\(\dfrac{3}{7}=\dfrac{7x}{7a}=\dfrac{6y}{6b}=\dfrac{z}{c}=\dfrac{7x+6y+z}{7a+6b+c}=\dfrac{3a}{7a+6b+c}\)
\(\Rightarrow3\left(7a+6b+c\right)=3a.7\)
\(\Rightarrow21a+18b+3c=21a\)
\(\Rightarrow18b+3c=0\)
\(\Rightarrow6b=-c\)
\(\Rightarrow\dfrac{b}{c}=-\dfrac{1}{6}\)