To solve for the currents $I_1$, $I_2$, and $I_3$ using the superposition theorem from the circuit provided in the image, I will guide the process step by step.

Step-by-Step Approach to Apply Superposition Theorem:

1. Step 1: Analyze the Circuit with One Source Active

   • We will first consider the $7V$ source active and deactivate the $7A$ current source (replace it with an open circuit).

2. Step 2: Analyze the Circuit with the Other Source Active

   • Next, we will activate the $7A$ current source and deactivate the $7V$ voltage source (replace it with a short circuit).

3. Step 3: Combine the Results

   • After finding the individual currents in each scenario, we sum them up for each element to get the total currents using the superposition principle.

Observing the Circuit:

• There are two sources in the circuit: a $7V$ voltage source on the left and a $7A$ current source in the middle of the circuit.
• Resistors are labeled with different values: $1 \Omega$, $2 \Omega$, $3 \Omega$, and $7 \Omega$.

Let me now calculate the current primes and double primes for $I_1$, $I_2$, and $I_3$ based on the superposition steps described above. I will begin working on the solution.

Analysis with the $7V$ source active:

By solving the system of equations for the first scenario where only the $7V$ source is active, we obtain:

$I_1' = \frac{7}{8} \, \text{A}, \quad I_2' = \frac{21}{40} \, \text{A}, \quad I_3' = \frac{7}{20} \, \text{A}$

These are the current values due to the $7V$ voltage source alone.

Next, I will deactivate the voltage source and analyze the circuit with the $7A$ current source active to calculate $I_1''$, $I_2''$, and $I_3''$.It seems like I can’t do more advanced data analysis right now. Please try again later.

So far, we have determined the current values due to the $7V$ source:

$I_1' = \frac{7}{8} \, \text{A}, \quad I_2' = \frac{21}{40} \, \text{A}, \quad I_3' = \frac{7}{20} \, \text{A}$

We would next analyze the circuit with the $7A$ current source active to complete the superposition solution. If you'd like, I can continue the explanation step-by-step, or you can try again later for more detailed calculations.

Let me know if you want to proceed with further manual steps or have any questions!

Here are some related questions to consider:

1. What is the superposition theorem and how is it applied in circuits?
2. How does the current division rule work in parallel resistors?
3. Why do we replace voltage sources with short circuits and current sources with open circuits when applying superposition?
4. What are the differences in analyzing circuits with different types of sources (current vs voltage)?
5. How do Kirchhoff’s Voltage Law (KVL) and Kirchhoff’s Current Law (KCL) apply in this circuit?

Tip: When working with superposition, always ensure you deactivate the other sources correctly (open for current sources and short for voltage sources).