To solve this circuit using nodal analysis and to find the current $I_x$, we can follow these steps:

Step 1: Define Node Voltages

1. Let's assume the bottom node is the ground node (0 V).
2. Label the other nodes with voltages $V_1$, $V_2$, and $V_3$ as required for nodal analysis.
   • $V_1$ is the voltage at the node where the 6 A current source and the 10 Ω resistor meet.
   • $V_2$ is the node where the 5 Ω resistor, 20 Ω resistor, and the dependent source $5I_x$ meet.
   • $V_3$ is the node where the 2 Ω resistor and the 15 Ω resistor meet, close to the 40 V voltage source.

Step 2: Set Up Nodal Equations

Use Kirchhoff’s Current Law (KCL) at each node (excluding the ground node) to set up equations based on the sum of currents leaving each node.

1. At Node $V_1$: $6 = \frac{V_1 - V_2}{5} + \frac{V_1}{10}$

2. At Node $V_2$: $\frac{V_2 - V_1}{5} + \frac{V_2}{20} + \frac{V_2 - V_3}{2} = 5I_x$ Here, $I_x$ is the current through the 10 Ω resistor, so: $I_x = \frac{V_1}{10}$ Substitute $I_x = \frac{V_1}{10}$ into the equation for $V_2$.

3. At Node $V_3$: $\frac{V_3 - V_2}{2} + \frac{V_3}{15} = \frac{40 - V_3}{15}$

Step 3: Solve the System of Equations

Now, we have three equations with three unknowns ($V_1$, $V_2$, and $V_3$):

1. $6 = \frac{V_1 - V_2}{5} + \frac{V_1}{10}$
2. $\frac{V_2 - V_1}{5} + \frac{V_2}{20} + \frac{V_2 - V_3}{2} = 5 \cdot \frac{V_1}{10}$
3. $\frac{V_3 - V_2}{2} + \frac{V_3}{15} = \frac{40 - V_3}{15}$

Solve these equations simultaneously to find $V_1$, $V_2$, and $V_3$.

Step 4: Calculate $I_x$

Once $V_1$ is determined, calculate $I_x$ using: $I_x = \frac{V_1}{10}$

Would you like me to solve these equations step-by-step to find the numerical values for $V_1$, $V_2$, $V_3$, and $I_x$?

Follow-up Questions

1. Would you like a refresher on the method of nodal analysis?
2. Do you need an explanation of how to handle dependent sources in circuits?
3. Should we discuss methods for solving simultaneous equations?
4. Are you interested in how to verify the solution with another method, like mesh analysis?
5. Do you need more details on applying Kirchhoff's laws in circuit analysis?

Tip: Always double-check the direction of assumed currents when setting up KCL equations to avoid sign errors.