Answer :

-3.25

KCL: The algebraic sum of currents flowing into a node is equal to the algebraic sum of currents flowing out of that same node.

Apply KCL at node x, assuming all currents leaving the node

$$\begin{gathered} \frac{V_x-6}{1}+\frac{V_x}{4}+\frac{V_x-V_y}{2}=0 \\ 4V_x-24+V_x+2V_x-2Vy=0 \\ 7V_x-2V_y=24....(1) \end{gathered}$$

Given V_y = 0 V

Then V_x = 24/7 V

Apply KCL at node y, assuming all currents leaving the node

$$\begin{gathered} \frac{V_y-V_x}{2}-2+\frac{V_y-\beta V_x}{3}=0 \\ 3V_y-3V_x-12+2V_y-2\beta V_x=0 \\ 5V_y-(3+2\beta )V_x=12.....(2) \end{gathered}$$

Substitute V_x and V_y in equation(2), we get β

β = - 3.25