To solve for $v_x$ and $i_x$ in the circuit using nodal analysis, here is the summary of the approach:

Node Voltage Assignment:

1. Assign a reference node (ground) at the bottom of the circuit.
2. Assign the node at the top-left corner (between the $50 \, V$ source and $10 \, \Omega$ resistor) as the first node.
3. The node in the middle (where $v_x$ is) will be the second node for nodal analysis.

Nodal Equations:

• The currents into each node must sum to zero, according to Kirchhoff's Current Law (KCL).

We can now define the two nodal equations to solve for $v_x$ and $i_x$.

1. At the first node (top-left): $\frac{50 - v_x}{10} = i_x + 3A$ where $i_x = \frac{v_x}{5}$.

2. At the second node (middle node where $v_x$): $\frac{v_x - 50}{10} + \frac{v_x}{5} + \frac{v_x}{2} = \frac{v_x}{4}$

Solving the system of equations will give us $v_x$ and $i_x$.

Now let's calculate.It appears that no valid solution was found due to the nature of the equation setup. Let me review the system and reattempt solving it with the correct formulation. Specifically, I will handle the dependent sources and simplify the expressions further before reattempting.It seems the solution did not return a dictionary, but a list instead. I'll handle this result properly and recalculate both $v_x$ and $i_x$. Let's extract the correct values and present them.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you have any specific questions or want a detailed explanation, feel free to ask.

Here are 5 related questions for deeper understanding:

1. What are the key principles of nodal analysis in electrical circuits?
2. How do dependent sources affect the nodal analysis process?
3. Why is it important to assign a reference node when using nodal analysis?
4. How can Kirchhoff’s Current Law (KCL) be applied in more complex circuits?
5. What role does the supernode concept play in solving circuits with voltage sources?

Tip: Always double-check boundary conditions and dependent source relationships in circuit analysis to avoid errors in the final solution.