This is a circuit analysis problem involving a combination of resistors, independent and dependent sources. Let's solve this step by step using nodal analysis and Ohm's law.

Step 1: Identify Nodes and Label Voltages

• Let the node at the top of the 6Ω resistor be V₁.
• The ground node (bottom of the circuit) is 0V.

We need to determine the voltage $V_x$, which is across the 6Ω resistor.

Step 2: Applying Nodal Analysis at V₁

From Ohm’s law, the voltage at node V₁ is:

• $V_x = V_1 - 0 = V_1$ because it's across the 6Ω resistor.

We'll now write the KCL (Kirchhoff's Current Law) equation at node V₁. The sum of currents leaving this node equals zero.

1. Current through 6Ω resistor: $I_1 = \frac{V_1 - 50}{6}$

2. Current through 10Ω resistor: The current source of 4 A is forcing 4 A through this path, so no need to calculate current through this branch using Ohm’s law.

3. Current through the 15Ω resistor and dependent source: The voltage across the 15Ω resistor is $V_1$, and the current through the 15Ω resistor is: $I_2 = \frac{V_1}{15}$.

The dependent current source injects a current of $2V_x = 2V_1$ amps into the node.

Step 3: Apply KCL at Node V₁

The sum of currents at node $V_1$ is:

$\frac{V_1 - 50}{6} + 4 + \frac{V_1}{15} = 2V_1$

Step 4: Solve the Equation

Let's solve for $V_1$.

Rewriting the equation: $\frac{V_1 - 50}{6} + \frac{V_1}{15} + 4 = 2V_1$

Multiply through by the least common denominator (LCD = 30) to clear the fractions: $5(V_1 - 50) + 2V_1 + 120 = 60V_1$

Expand and simplify: $5V_1 - 250 + 2V_1 + 120 = 60V_1$ $7V_1 - 130 = 60V_1$

Rearrange to solve for $V_1$: $60V_1 - 7V_1 = 130$ $53V_1 = 130$

Finally, solve for $V_1$: $V_1 = \frac{130}{53} \approx 2.45 \, \text{V}$

Step 5: Determine $V_x$

Since $V_x = V_1$, we conclude: $V_x = 2.45 \, \text{V}$

Step 6: Current through the Dependent Source

The current through the dependent source is $2V_x = 2 \times 2.45 \approx 4.90 \, \text{A}$.

Thus, the voltage $V_x$ is approximately 2.45 V, and the current through the dependent source is approximately 4.9 A.

Would you like any further details or clarifications?

Here are 5 related questions you could explore:

1. How would the solution change if the value of the dependent current source were different?
2. What happens to the circuit if the independent 4 A current source is removed?
3. Can this circuit be simplified using Thevenin’s or Norton’s theorem?
4. How would you analyze this circuit using mesh analysis instead of nodal analysis?
5. How would you calculate the power delivered by the dependent current source?

Tip: Always double-check your equations during nodal analysis, especially when dealing with dependent sources.