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The problem asks to find the currents \( i_1 \), \( i_2 \), and \( i_3 \) in the circuit, which consists of resistors and three voltage sources. Let's approach this step-by-step:

### Step 1: Label the Nodes
- Node \( A \) connects the 5Ω, 6Ω, and 4Ω resistors.
- Node \( B \) connects the 8Ω resistor and 5V voltage source.
- Node \( C \) connects the 6Ω, 8Ω, and 16Ω resistors.
- Node \( D \) connects the 16Ω resistor and 2V voltage source.
- Node \( E \) connects the 21V source, 5Ω, and 11Ω resistors.

### Step 2: Apply Kirchhoff's Current Law (KCL)
We will define the currents as follows:
- \( i_1 \) flows through the 21V source and 5Ω resistor.
- \( i_2 \) flows through the 5V source and 4Ω resistor.
- \( i_3 \) flows through the 2V source and 16Ω resistor.

### Step 3: Apply Kirchhoff's Voltage Law (KVL)
We need to write KVL equations for loops:

#### Loop 1: Left loop (containing the 21V source)
\[ -21V + 5i_1 + 6(i_1 - i_2) + 11i_1 = 0 \]

Simplifying:
\[ 22i_1 - 6i_2 = 21 \] \(\quad \text{(Equation 1)}\)

#### Loop 2: Right loop (containing the 5V source)
\[ 5V + 4i_2 + 6(i_2 - i_1) + 8i_2 = 0 \]

Simplifying:
\[ -6i_1 + 18i_2 = -5 \] \(\quad \text{(Equation 2)}\)

#### Loop 3: Bottom loop (containing the 2V source)
\[ -2V + 16i_3 + 8(i_3 - i_2) = 0 \]

Simplifying:
\[ 24i_3 - 8i_2 = 2 \] \(\quad \text{(Equation 3)}\)

### Step 4: Solve the system of equations
We now have three equations:

1. \( 22i_1 - 6i_2 = 21 \)
2. \( -6i_1 + 18i_2 = -5 \)
3. \( 24i_3 - 8i_2 = 2 \)

We can solve these using substitution or matrix methods.

Let me compute the solution for \( i_1 \), \( i_2 \), and \( i_3 \).The solutions for the currents are:

- \( i_1 = \frac{29}{30} \, \text{A} \approx 0.967 \, \text{A} \)
- \( i_2 = \frac{2}{45} \, \text{A} \approx 0.0444 \, \text{A} \)
- \( i_3 = \frac{53}{540} \, \text{A} \approx 0.0981 \, \text{A} \)

Would you like further details or clarification on any step?

Here are 5 related questions to consider:
1. How can Kirchhoff's Current Law (KCL) be applied at different nodes?
2. What happens if the polarity of one of the voltage sources is reversed?
3. How do you determine the direction of current flow in a circuit like this?
4. Can these methods be applied to circuits with capacitors or inductors?
5. What are the advantages of using matrix methods to solve systems of equations?

**Tip:** Always double-check the direction of currents when applying Kirchhoff's Laws to avoid sign errors in equations.

Find the values of i1, i2, and i3 in the given circuit with resistors and voltage sources.

Learn how to solve a circuit problem involving Kirchhoff's Voltage and Current Laws to find the values of i1, i2, and i3. This step-by-step guide explains how to analyze the circuit using fundamental electrical principles, including resistances, voltage sources, and Ohm's law.

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