To solve for $v_1$ in the given circuit, we can apply Kirchhoff's Voltage Law (KVL) and Ohm's Law. Here's how to proceed:

Step-by-Step Approach:

1. Label the circuit elements and unknowns:

   • The two current sources are given: $1 \text{ A}$ on the left and $2 \text{ A}$ on the right.
   • Resistor values: $20 \ \Omega$, $10 \ \Omega$, and $5 \ \Omega$.
   • Unknown node voltages: $v_1$ at the node between the $20 \ \Omega$ resistor and the $10 \ \Omega$ resistor, and $v_2$ at the node between the $5 \ \Omega$ resistor and the same $10 \ \Omega$ resistor.

2. Apply Kirchhoff's Current Law (KCL):

   • At node $v_1$, the sum of the currents leaving the node should equal the total current entering.
   • Similarly, apply KCL at node $v_2$.

3. Write equations:

   For node $v_1$: $\frac{v_1 - v_2}{10} + \frac{v_1}{20} = 1$ This comes from the current through the $10 \ \Omega$ resistor and the $20 \ \Omega$ resistor.

   For node $v_2$: $\frac{v_2 - v_1}{10} + \frac{v_2}{5} = 2$ This comes from the current through the $10 \ \Omega$ and $5 \ \Omega$ resistors.

4. Solve the system of equations:

Let's solve these two equations to find $v_1$ and $v_2$. I'll handle the calculation now.The solution for $v_1$ is:

$v_1 = \frac{100}{7} \text{ V} \approx 14.29 \text{ V}$

And for reference, $v_2 = \frac{80}{7} \text{ V} \approx 11.43 \text{ V}$.

Would you like further details on the method used, or do you have any questions?

Related Questions:

1. How does Kirchhoff's Current Law work in circuit analysis?
2. Can Kirchhoff's Voltage Law be applied in this problem instead?
3. What would change if the current sources were voltage sources?
4. How do you calculate total power dissipated in this circuit?
5. How would you solve this if resistances were in parallel?

Tip: Always double-check units when working with current and voltage to avoid confusion in complex circuits.