$$\textbf{Step 1 : Write the given regression equations .}$$
$$\begin{align} 2Y - X - 50 = 0\end{align}...............(i) \ and$$
$$\begin{align} 3Y - 2X - 10 = 0\end{align}...............(ii)$$
$$\textbf{Step 2 : Determine means of X and Y }$$$$\textbf{ by solving two regression equation .}$$
$$\begin{align} 2Y - X = 50 \end{align}...............(multiply \ by \ 3 ) \ and$$
$$\begin{align} 3Y - 2X = 10 \end{align}...............(multiply \ by \ (-2))$$
$$\text{we get ,}$$ $$\begin{align} 6Y - 3X = 150 \end{align} $$
$$\begin{align} -6Y + 4X = -20 \end{align}$$
$$\Rightarrow$$ $$\begin{align} X = 130 \end{align}$$
$$\text{put X = 130 in eq (i) we get ,}$$
$$\begin{align} 2Y - 130 = 50 \end{align}$$
$$\Rightarrow$$ $$\begin{align} Y = 90 \end{align}$$
$$\textbf{Step 3 : Determining }$$ $$\mathbf{b_{xy}}$$ $$\textbf{and}$$ $$\mathbf{b_{yx} .}$$
$$\text{Consider equation (i)}$$
$$\begin{align} 2Y - X - 50 = 0\end{align}$$
$$\begin{align} - X = 50 - 2Y \end{align}$$
$$\begin{align} X = -50 + 2Y \end{align}$$
$$\therefore$$ $$\mathbb{b_{xy} = 2}$$
$$\text{Consider equation (ii)}$$
$$\begin{align} 3Y - 2X - 10 = 0\end{align}$$
$$\begin{align} 3Y = 2X + 10\end{align}$$
$$\begin{align} Y = \dfrac{2}{3}X + \dfrac{10}{3}\end{align}$$
$$\therefore$$ $$\mathbb{b_{yx} = \dfrac{2}{3}}$$
$$\textbf{Step 4 : Determining the correlation coefficient .}$$
$$\text{Correlation coefficient =}$$ $$\mathbb{\sqrt{b_{yx}\times b_{xy}}}$$
$$= \mathbb{\sqrt{\dfrac{2}{3}\times 2}}$$
$$= \mathbb{\sqrt{\dfrac{4}{3}}}$$
$$= \mathbb{{\dfrac{2}{\sqrt{3}}}}$$
$$\therefore$$$$\textbf{Correlation coefficient is}$$ $$ \mathbf{{\dfrac{2}{\sqrt{3}}}}$$
$$\textbf{Hence , Means of X = 130 and Y = 90 , and correlation coefficient is}$$ $$\mathbf{\dfrac{2}{\sqrt{3}}}$$