To solve the circuit using the node-voltage method, we need to label the nodes and apply Kirchhoff's Current Law (KCL) at the node(s) of interest. Here's how we can proceed based on the image provided.

Step-by-Step Solution:

1. Identify Nodes:

   • Let's call the node at the top of the circuit (above $R_1$, $R_2$, and $R_3$) as $V_A$. The ground node is already marked as $G$, which is $0 \, V$.
   • We will apply KCL at node $V_A$.

2. Label Voltages:

   • $V_1 = 40 \, V$ (given by the voltage source on the left),
   • $V_2 = 100 \, V$ (given by the voltage source on the right),
   • $V_G = 0 \, V$ (ground node).

3. Apply Kirchhoff's Current Law (KCL) at Node $V_A$: According to KCL, the sum of the currents leaving (or entering) a node is zero. The current through a resistor is given by $I = \frac{V}{R}$. Let's calculate the currents leaving node $V_A$.

   • The current through $R_1$ (left branch): $I_{R_1} = \frac{V_A - V_1}{R_1} = \frac{V_A - 40}{400}$

   • The current through $R_2$ (right branch): $I_{R_2} = \frac{V_A - V_2}{R_2} = \frac{V_A - 100}{100}$

   • The current through $R_3$ (middle branch): $I_{R_3} = \frac{V_A - V_G}{R_3} = \frac{V_A - 0}{20} = \frac{V_A}{20}$

4. Set up the KCL equation at Node $V_A$: $I_{R_1} + I_{R_2} + I_{R_3} = 0$ Substituting the current expressions: $\frac{V_A - 40}{400} + \frac{V_A - 100}{100} + \frac{V_A}{20} = 0$

5. Solve the equation for $V_A$: Multiply through by 400 (the least common denominator) to eliminate the denominators: $(V_A - 40) + 4(V_A - 100) + 20V_A = 0$ Simplify: $V_A - 40 + 4V_A - 400 + 20V_A = 0$ Combine like terms: $25V_A - 440 = 0$ Solving for $V_A$: $25V_A = 440$ $V_A = \frac{440}{25} = 17.6 \, V$

Node Voltage at $V_A$:

The voltage at node $V_A$ is $V_A = 17.6 \, V$.

Would you like more details on any specific part, or do you have any questions?

Here are some relative questions that may help further explore the concepts:

1. What happens if one of the resistances is altered in this circuit?
2. How would you calculate the power dissipated by each resistor?
3. How can you verify the total current supplied by each voltage source?
4. What if a fourth resistor is added between $V_A$ and the ground?
5. How would using a different analysis method (e.g., mesh analysis) compare with the node-voltage method?

Tip: Always check if the resistances are connected in series or parallel to simplify circuits where possible.