To solve this circuit problem, we need to use nodal analysis and other circuit analysis techniques to find the node voltages and the value of $V_x$.

Let's go through the solution step-by-step.

1. Define Node Voltages

Let's label the top node as $V_1$, the node where $V_x$ is located as $V_2$, and the ground node at the bottom.

2. Apply KCL (Kirchhoff's Current Law) at Each Node

Node $V_1$:

At node $V_1$, we have a 10V voltage source, a 3Ω resistor to $V_2$, and a dependent current source (5 times the current $i_x$) connected to ground. Applying KCL:

$\frac{V_1 - 10}{6} + \frac{V_1 - V_2}{3} + 5i_x = 0$

Since $i_x$ is defined as the current through the 1Ω resistor (from $V_2$ to ground), we can express $i_x$ as:

$i_x = \frac{V_2}{1} = V_2$

Substituting $i_x = V_2$ into the equation:

$\frac{V_1 - 10}{6} + \frac{V_1 - V_2}{3} + 5V_2 = 0$

Node $V_2$:

At node $V_2$, we have a 2Ω resistor to ground, a 1Ω resistor to ground, and a dependent voltage source (2 times $i_x$), with voltage $V_x$ across it. Applying KCL:

$\frac{V_2 - V_1}{3} + \frac{V_2}{2} + i_x = 0$

Substitute $i_x = V_2$ into the equation:

$\frac{V_2 - V_1}{3} + \frac{V_2}{2} + V_2 = 0$

3. Solve the System of Equations

We now have two equations:

1. $\frac{V_1 - 10}{6} + \frac{V_1 - V_2}{3} + 5V_2 = 0$
2. $\frac{V_2 - V_1}{3} + \frac{V_2}{2} + V_2 = 0$

Solving these equations simultaneously will give us the values of $V_1$ and $V_2$.

Final Answer

Node Voltages:

• $V_1 = \text{calculated value}$
• $V_2 = \text{calculated value}$

Finding $V_x$:

Since $V_x = 2 \cdot i_x$ and $i_x = V_2$, then:

$V_x = 2 \cdot V_2$

Let me calculate these values.The solutions for the node voltages are:

• $V_1 = \frac{110}{89} \approx 1.236 \, \text{V}$
• $V_2 = \frac{20}{89} \approx 0.225 \, \text{V}$

Finding $V_x$:

Since $V_x = 2 \cdot V_2$:

$V_x = 2 \cdot \frac{20}{89} = \frac{40}{89} \approx 0.449 \, \text{V}$

Final Answers

1. Node Voltages:

   • $V_1 \approx 1.236 \, \text{V}$
   • $V_2 \approx 0.225 \, \text{V}$

2. $V_x \approx 0.449 \, \text{V}$

Would you like further details on any step, or have any questions?

Here are some related questions you might find helpful:

1. How do you apply KCL and KVL in complex circuits?
2. What are dependent sources and how do they affect circuit analysis?
3. How is nodal analysis different from mesh analysis?
4. How do you handle multiple sources in circuit analysis?
5. How can circuit simulators help in verifying these results?

Tip: When working with dependent sources, always express the dependency in terms of known voltages or currents to simplify your equations.