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To solve for \( V_a \) and \( V_b \), we will solve the two simultaneous equations derived from the node voltages in what looks like a resistive circuit. These equations are:

For Node \( a \):
\[
\frac{V_a - 3}{4700} + \frac{V_a}{1200} + \frac{V_a - V_b}{5600} + \frac{V_a - V_b}{1707.22} = 0
\]

For Node \( b \):
\[
\frac{V_b - V_a}{1707.22} + \frac{V_b - V_a}{5600} + \frac{V_b}{1500} = 0
\]

### Step 1: Simplify the Equations

#### For Node \( a \):

We will combine the terms over a common denominator:

\[
\frac{V_a - 3}{4700} + \frac{V_a}{1200} + \frac{V_a - V_b}{5600} + \frac{V_a - V_b}{1707.22} = 0
\]

#### For Node \( b \):

Similarly, combine terms for \( V_b \):

\[
\frac{V_b - V_a}{1707.22} + \frac{V_b - V_a}{5600} + \frac{V_b}{1500} = 0
\]

### Step 2: Substitute numerical values and solve

I will now solve the system of equations numerically.The solution for the node voltages is:

\[
V_a = 0.455 \, \text{V}
\]
\[
V_b = 0.243 \, \text{V}
\]

Would you like further details on how these were calculated, or do you have any specific questions?

Here are 5 related questions for further exploration:
1. How does the choice of resistances affect the node voltages?
2. Can this circuit be simplified using Thevenin's theorem?
3. How would adding another resistor between the nodes affect \(V_a\) and \(V_b\)?
4. How do you calculate current in each branch once the node voltages are known?
5. What role does the 3V voltage source play in this circuit?

**Tip:** When solving systems of equations in circuit analysis, keep track of the reference (ground) node to simplify voltage assignments.

Solve for Va and Vb in the given electrical circuit using the node voltage method.

Learn how to solve for node voltages Va and Vb using the node voltage method in a resistor network circuit. This problem demonstrates the application of Kirchhoff's Current Law (KCL) and Ohm's Law, involving resistors and voltage sources. Suitable for undergraduate electrical engineering students.

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