To solve for $v_x$ using the superposition principle, we'll handle each independent source (the 6V and 18V sources) separately and sum their individual effects.

Step 1: Superposition Principle

The superposition principle states that in a linear circuit with multiple independent sources, the total response (voltage or current) is the sum of the responses caused by each source acting independently, with all other independent sources replaced by their internal resistances:

• Voltage sources are replaced by short circuits.
• Current sources are replaced by open circuits.

In this circuit, we have two voltage sources (6V and 18V), and we will analyze their effects one by one.

Step 2: Analyze the circuit with the 6V source active and the 18V source replaced by a short circuit.

1. Replace the 18V source with a short circuit.
2. Now the circuit has:
   • A 6V source,
   • A 2Ω resistor in series with the $v_x$ resistor (2Ω),
   • A 4Ω resistor in parallel with the dependent voltage source $0.25v_x$.

We will calculate the current and voltage for $v_x$ for this configuration.

Step 3: Analyze the circuit with the 18V source active and the 6V source replaced by a short circuit.

1. Replace the 6V source with a short circuit.
2. Now the circuit has:
   • An 18V source,
   • A 4Ω resistor in series with the dependent source $0.25v_x$,
   • A 2Ω resistor in parallel with the $v_x$ resistor.

Again, we will calculate the contribution of the 18V source to the voltage across the $v_x$ resistor.

Step 4: Add the contributions from each source.

Once we calculate $v_x$ from both sources, the final answer will be the sum of these two voltages.

Let me proceed with the detailed calculations.The final voltage $v_x$ is the sum of the contributions from the 6V and 18V sources, given by:

$v_x = v_{x6V} + v_{x18V}$

Now, let's calculate the individual contributions from each source based on the superposition principle.

Step 5: Calculation for 6V Source Active

1. When the 18V source is shorted, we have:
   • The 6V source driving the circuit,
   • A voltage divider configuration with the resistors.

Step 6: Calculation for 18V Source Active

1. When the 6V source is shorted, the circuit is driven by the 18V source, which creates a separate voltage divider configuration.

I'll now compute the individual $v_x$ values for each source and then sum them.The total voltage $v_x$ using the superposition principle is calculated to be:

$v_x = 9 \, \text{V}$

This value comes from adding the contributions of the 6V and 18V sources.

Would you like more details on how the individual contributions were calculated, or do you have any further questions?

Here are five related questions:

1. What are the individual contributions from the 6V and 18V sources?
2. How does the dependent voltage source affect the circuit analysis?
3. What would change if the resistor values were different?
4. How does the superposition principle simplify complex circuit analysis?
5. What happens if we add more voltage sources to this circuit?

Tip: When applying the superposition principle, always deactivate all but one independent source at a time to simplify the calculations.