Let's solve for \( I_1 \) using source transformation in **2 steps**, with explanations: --- ### **Step 1: Source Transformation of Current Sources and Parallel Resistors** **Explanation:** We convert the given current sources with parallel resistors into their equivalent voltage sources in series with resistors. - **Left Side:** - Current Source: 1 mA - Parallel Resistor: 47 kΩ - **Equivalent Voltage Source:** \( V = I \times R = 1\,\text{mA} \times 47\,\text{k}\Omega = 47\,\text{V} \) - **Series Resistance:** 47 kΩ - **Right Side:** - Current Source: 2 mA - Parallel Resistor: 10 kΩ - **Equivalent Voltage Source:** \( V = I \times R = 2\,\text{mA} \times 10\,\text{k}\Omega = 20\,\text{V} \) - **Series Resistance:** 10 kΩ Now, the circuit becomes: - 47V (left) in series with 47kΩ - 3.3kΩ resistor in the middle (across which \( I_1 \) flows and the dependent source \( 4I_1 \) is defined) - 20V (right) in series with 10kΩ --- ### **Step 2: Apply KVL and Solve for \( I_1 \)** **Explanation:** Apply Kirchhoff's Voltage Law (KVL) around the loop and include the dependent source. Let the current through the loop be \( I_1 \) (from left to right across 3.3kΩ). Write the KVL equation: \[ 47V - 47k\Omega \cdot I_1 - (4I_1) - 3.3k\Omega \cdot I_1 - 10k\Omega \cdot I_1 - 20V = 0 \] But \( 4I_1 \) is a voltage source (since it’s in volts), and it should be added or subtracted based on polarity (assuming it's in the same direction as the loop): So, the equation becomes: \[ 47 - 47,000I_1 - 4I_1 - 3,300I_1 - 10,000I_1 - 20 = 0 \] Combine terms: \[ (47 - 20) - (47,000 + 3,300 + 10,000)I_1 - 4I_1 = 0 \] \[ 27 - 60,300I_1 - 4I_1 = 0 \] \[ 27 - 60,304I_1 = 0 \] \[ I_1 = \frac{27}{60,304} \approx 0.000448 \text{ A} = \boxed{0.448\,\text{mA}} \] --- ## **Final Answer:** ### **\( \boxed{I_1 = 0.448\,\text{mA}} \)** --- **Summary of Steps:** 1. **Source Transformation:** Convert current sources and parallel resistors into voltage sources in series with resistors. 2. **KVL Equation:** Write and solve the KVL equation including the dependent source to find \( I_1 \).