To solve for the current $i_o$ using the superposition principle, we need to handle each independent source (voltage or current) one at a time, while turning the others off. When "turning off" a voltage source, we replace it with a short circuit, and when "turning off" a current source, we replace it with an open circuit. Here’s the step-by-step approach to solve this:

Step 1: Considering only the voltage source $V_s = 26 \, \text{V}$

• The current sources (4 A and 2 A) will be open circuits.
• This leaves us with a series-parallel combination of resistors connected to the voltage source $V_s = 26 \, \text{V}$.
• The equivalent resistance can be found, and then Ohm's law can be used to determine the current $i_o$ in the 4 Ω resistor due to the voltage source alone.

Step 2: Considering only the 4 A current source

• The voltage source $V_s$ will be replaced by a short circuit, and the 2 A current source will be replaced by an open circuit.
• The 4 A current source affects the resistors in its branch, so we calculate the current through the 4 Ω resistor.

Step 3: Considering only the 2 A current source

• The voltage source $V_s$ is short-circuited, and the 4 A current source is open-circuited.
• The 2 A current source will affect the right branch of the circuit, and we calculate its contribution to $i_o$ through the 4 Ω resistor.

Step 4: Summing the contributions

• The total current $i_o$ is the algebraic sum of the contributions from the voltage source, the 4 A current source, and the 2 A current source.

I'll now perform the calculations step-by-step to obtain the final value of $i_o$. Let me compute this.The total current $i_o$, using the superposition principle, is approximately $3310.53 \, \text{mA}$.

Would you like more details or have any questions? Here are some related questions to further explore this problem:

1. What is the effect of each independent source on the current through the 10 Ω resistor?
2. How would the result change if the value of $V_s$ were doubled?
3. What would happen to $i_o$ if the 2 A current source was removed entirely?
4. How is the current divider rule applied in the second step of this problem?
5. How does the concept of superposition help in solving circuits with multiple sources?

Tip: When using superposition, always check if current sources lead to open circuits or voltage sources to short circuits, which greatly simplifies the circuit at each step.