Đặt $z=x+yi,\,\left( x,y\in \mathbb{R} \right)\Rightarrow \bar{z}=x-yi\Rightarrow -2\bar{z}=-2x+2yi$

$x+yi-2x+2yi=3+4i\Leftrightarrow -x+3yi=3+4i\Leftrightarrow \left\{ \begin{align} & -x=3 \\ & 3y=4 \\ \end{align} \right.$$\Leftrightarrow \left\{ \begin{align} & x=-3 \\ & y=\frac{4}{3} \\ \end{align} \right.$

Vậy $z=-3+\frac{4}{3}i\Rightarrow \left| z \right|=\sqrt{{{\left( -3 \right)}^{2}}+{{\left( \frac{4}{3} \right)}^{2}}}=\sqrt{\frac{97}{9}}=\frac{\sqrt{97}}{3}$