Prompt:
Consider an electrical circuit with 5 nodes, labeled 0 to 4.  The circuit has the following edges:\n  Edge 1: R=2.00 Ohm, E=-9.00 V, in branch 0-4 (E is the Electromotive Force in branch 0-4; positive if the source's positive terminal is at node 4 and negative terminal at node 0.)
  Edge 2: R=7.00 Ohm, E=-1.00 V, in branch 4-2 (E is the Electromotive Force in branch 4-2; positive if the source's positive terminal is at node 2 and negative terminal at node 4.)
  Edge 3: R=7.00 Ohm, E=3.00 V, in branch 2-1 (E is the Electromotive Force in branch 2-1; positive if the source's positive terminal is at node 1 and negative terminal at node 2.)
  Edge 4: R=10.00 Ohm, E=10.00 V, in branch 1-3 (E is the Electromotive Force in branch 1-3; positive if the source's positive terminal is at node 3 and negative terminal at node 1.)
  Edge 5: R=7.00 Ohm, E=4.00 V, in branch 0-3 (E is the Electromotive Force in branch 0-3; positive if the source's positive terminal is at node 3 and negative terminal at node 0.)
  Edge 6: R=6.00 Ohm, E=-1.00 V, in branch 3-2 (E is the Electromotive Force in branch 3-2; positive if the source's positive terminal is at node 2 and negative terminal at node 3.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 5 nodes, labeled 0 to 4.  The circuit has the following edges:\n  Edge 1: R=7.00 Ohm, E=3.00 V, in branch 3-0 (E is the Electromotive Force in branch 3-0; positive if the source's positive terminal is at node 0 and negative terminal at node 3.)
  Edge 2: R=7.00 Ohm, E=2.00 V, in branch 0-2 (E is the Electromotive Force in branch 0-2; positive if the source's positive terminal is at node 2 and negative terminal at node 0.)
  Edge 3: R=3.00 Ohm, E=8.00 V, in branch 2-4 (E is the Electromotive Force in branch 2-4; positive if the source's positive terminal is at node 4 and negative terminal at node 2.)
  Edge 4: R=8.00 Ohm, E=0.00 V, in branch 4-1 (E is the Electromotive Force in branch 4-1; positive if the source's positive terminal is at node 1 and negative terminal at node 4.)
  Edge 5: R=3.00 Ohm, E=3.00 V, in branch 2-3 (E is the Electromotive Force in branch 2-3; positive if the source's positive terminal is at node 3 and negative terminal at node 2.)
  Edge 6: R=5.00 Ohm, E=-6.00 V, in branch 1-0 (E is the Electromotive Force in branch 1-0; positive if the source's positive terminal is at node 0 and negative terminal at node 1.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 6 nodes, labeled 0 to 5.  The circuit has the following edges:\n  Edge 1: R=7.00 Ohm, E=-7.00 V, in branch 4-1 (E is the Electromotive Force in branch 4-1; positive if the source's positive terminal is at node 1 and negative terminal at node 4.)
  Edge 2: R=4.00 Ohm, E=-1.00 V, in branch 1-2 (E is the Electromotive Force in branch 1-2; positive if the source's positive terminal is at node 2 and negative terminal at node 1.)
  Edge 3: R=10.00 Ohm, E=9.00 V, in branch 2-5 (E is the Electromotive Force in branch 2-5; positive if the source's positive terminal is at node 5 and negative terminal at node 2.)
  Edge 4: R=2.00 Ohm, E=9.00 V, in branch 5-3 (E is the Electromotive Force in branch 5-3; positive if the source's positive terminal is at node 3 and negative terminal at node 5.)
  Edge 5: R=2.00 Ohm, E=8.00 V, in branch 3-0 (E is the Electromotive Force in branch 3-0; positive if the source's positive terminal is at node 0 and negative terminal at node 3.)
  Edge 6: R=6.00 Ohm, E=-4.00 V, in branch 1-5 (E is the Electromotive Force in branch 1-5; positive if the source's positive terminal is at node 5 and negative terminal at node 1.)
  Edge 7: R=1.00 Ohm, E=-1.00 V, in branch 5-0 (E is the Electromotive Force in branch 5-0; positive if the source's positive terminal is at node 0 and negative terminal at node 5.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 5 nodes, labeled 0 to 4.  The circuit has the following edges:\n  Edge 1: R=7.00 Ohm, E=-4.00 V, in branch 4-2 (E is the Electromotive Force in branch 4-2; positive if the source's positive terminal is at node 2 and negative terminal at node 4.)
  Edge 2: R=2.00 Ohm, E=3.00 V, in branch 2-0 (E is the Electromotive Force in branch 2-0; positive if the source's positive terminal is at node 0 and negative terminal at node 2.)
  Edge 3: R=5.00 Ohm, E=-7.00 V, in branch 0-3 (E is the Electromotive Force in branch 0-3; positive if the source's positive terminal is at node 3 and negative terminal at node 0.)
  Edge 4: R=5.00 Ohm, E=-5.00 V, in branch 3-1 (E is the Electromotive Force in branch 3-1; positive if the source's positive terminal is at node 1 and negative terminal at node 3.)
  Edge 5: R=2.00 Ohm, E=9.00 V, in branch 2-1 (E is the Electromotive Force in branch 2-1; positive if the source's positive terminal is at node 1 and negative terminal at node 2.)
  Edge 6: R=5.00 Ohm, E=6.00 V, in branch 4-1 (E is the Electromotive Force in branch 4-1; positive if the source's positive terminal is at node 1 and negative terminal at node 4.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 6 nodes, labeled 0 to 5.  The circuit has the following edges:\n  Edge 1: R=5.00 Ohm, E=8.00 V, in branch 5-0 (E is the Electromotive Force in branch 5-0; positive if the source's positive terminal is at node 0 and negative terminal at node 5.)
  Edge 2: R=9.00 Ohm, E=-3.00 V, in branch 0-3 (E is the Electromotive Force in branch 0-3; positive if the source's positive terminal is at node 3 and negative terminal at node 0.)
  Edge 3: R=1.00 Ohm, E=-9.00 V, in branch 3-2 (E is the Electromotive Force in branch 3-2; positive if the source's positive terminal is at node 2 and negative terminal at node 3.)
  Edge 4: R=1.00 Ohm, E=-7.00 V, in branch 2-1 (E is the Electromotive Force in branch 2-1; positive if the source's positive terminal is at node 1 and negative terminal at node 2.)
  Edge 5: R=3.00 Ohm, E=0.00 V, in branch 1-4 (E is the Electromotive Force in branch 1-4; positive if the source's positive terminal is at node 4 and negative terminal at node 1.)
  Edge 6: R=9.00 Ohm, E=-10.00 V, in branch 2-5 (E is the Electromotive Force in branch 2-5; positive if the source's positive terminal is at node 5 and negative terminal at node 2.)
  Edge 7: R=5.00 Ohm, E=-8.00 V, in branch 2-4 (E is the Electromotive Force in branch 2-4; positive if the source's positive terminal is at node 4 and negative terminal at node 2.)
  Edge 8: R=7.00 Ohm, E=-6.00 V, in branch 5-4 (E is the Electromotive Force in branch 5-4; positive if the source's positive terminal is at node 4 and negative terminal at node 5.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 6 nodes, labeled 0 to 5.  The circuit has the following edges:\n  Edge 1: R=10.00 Ohm, E=7.00 V, in branch 0-1 (E is the Electromotive Force in branch 0-1; positive if the source's positive terminal is at node 1 and negative terminal at node 0.)
  Edge 2: R=1.00 Ohm, E=8.00 V, in branch 1-2 (E is the Electromotive Force in branch 1-2; positive if the source's positive terminal is at node 2 and negative terminal at node 1.)
  Edge 3: R=9.00 Ohm, E=3.00 V, in branch 2-3 (E is the Electromotive Force in branch 2-3; positive if the source's positive terminal is at node 3 and negative terminal at node 2.)
  Edge 4: R=5.00 Ohm, E=2.00 V, in branch 3-5 (E is the Electromotive Force in branch 3-5; positive if the source's positive terminal is at node 5 and negative terminal at node 3.)
  Edge 5: R=10.00 Ohm, E=4.00 V, in branch 5-4 (E is the Electromotive Force in branch 5-4; positive if the source's positive terminal is at node 4 and negative terminal at node 5.)
  Edge 6: R=9.00 Ohm, E=6.00 V, in branch 0-3 (E is the Electromotive Force in branch 0-3; positive if the source's positive terminal is at node 3 and negative terminal at node 0.)
  Edge 7: R=10.00 Ohm, E=4.00 V, in branch 3-4 (E is the Electromotive Force in branch 3-4; positive if the source's positive terminal is at node 4 and negative terminal at node 3.)
  Edge 8: R=7.00 Ohm, E=-2.00 V, in branch 0-5 (E is the Electromotive Force in branch 0-5; positive if the source's positive terminal is at node 5 and negative terminal at node 0.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 6 nodes, labeled 0 to 5.  The circuit has the following edges:\n  Edge 1: R=7.00 Ohm, E=-10.00 V, in branch 4-2 (E is the Electromotive Force in branch 4-2; positive if the source's positive terminal is at node 2 and negative terminal at node 4.)
  Edge 2: R=3.00 Ohm, E=6.00 V, in branch 2-5 (E is the Electromotive Force in branch 2-5; positive if the source's positive terminal is at node 5 and negative terminal at node 2.)
  Edge 3: R=7.00 Ohm, E=-3.00 V, in branch 5-1 (E is the Electromotive Force in branch 5-1; positive if the source's positive terminal is at node 1 and negative terminal at node 5.)
  Edge 4: R=6.00 Ohm, E=-3.00 V, in branch 1-0 (E is the Electromotive Force in branch 1-0; positive if the source's positive terminal is at node 0 and negative terminal at node 1.)
  Edge 5: R=1.00 Ohm, E=10.00 V, in branch 0-3 (E is the Electromotive Force in branch 0-3; positive if the source's positive terminal is at node 3 and negative terminal at node 0.)
  Edge 6: R=5.00 Ohm, E=5.00 V, in branch 0-2 (E is the Electromotive Force in branch 0-2; positive if the source's positive terminal is at node 2 and negative terminal at node 0.)
  Edge 7: R=1.00 Ohm, E=9.00 V, in branch 5-4 (E is the Electromotive Force in branch 5-4; positive if the source's positive terminal is at node 4 and negative terminal at node 5.)
  Edge 8: R=9.00 Ohm, E=2.00 V, in branch 4-1 (E is the Electromotive Force in branch 4-1; positive if the source's positive terminal is at node 1 and negative terminal at node 4.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 6 nodes, labeled 0 to 5.  The circuit has the following edges:\n  Edge 1: R=8.00 Ohm, E=-9.00 V, in branch 2-1 (E is the Electromotive Force in branch 2-1; positive if the source's positive terminal is at node 1 and negative terminal at node 2.)
  Edge 2: R=5.00 Ohm, E=-9.00 V, in branch 1-0 (E is the Electromotive Force in branch 1-0; positive if the source's positive terminal is at node 0 and negative terminal at node 1.)
  Edge 3: R=2.00 Ohm, E=9.00 V, in branch 0-5 (E is the Electromotive Force in branch 0-5; positive if the source's positive terminal is at node 5 and negative terminal at node 0.)
  Edge 4: R=1.00 Ohm, E=-4.00 V, in branch 5-3 (E is the Electromotive Force in branch 5-3; positive if the source's positive terminal is at node 3 and negative terminal at node 5.)
  Edge 5: R=5.00 Ohm, E=-6.00 V, in branch 3-4 (E is the Electromotive Force in branch 3-4; positive if the source's positive terminal is at node 4 and negative terminal at node 3.)
  Edge 6: R=2.00 Ohm, E=-3.00 V, in branch 3-2 (E is the Electromotive Force in branch 3-2; positive if the source's positive terminal is at node 2 and negative terminal at node 3.)
  Edge 7: R=9.00 Ohm, E=-2.00 V, in branch 0-4 (E is the Electromotive Force in branch 0-4; positive if the source's positive terminal is at node 4 and negative terminal at node 0.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 5 nodes, labeled 0 to 4.  The circuit has the following edges:\n  Edge 1: R=3.00 Ohm, E=5.00 V, in branch 4-2 (E is the Electromotive Force in branch 4-2; positive if the source's positive terminal is at node 2 and negative terminal at node 4.)
  Edge 2: R=8.00 Ohm, E=-5.00 V, in branch 2-1 (E is the Electromotive Force in branch 2-1; positive if the source's positive terminal is at node 1 and negative terminal at node 2.)
  Edge 3: R=10.00 Ohm, E=10.00 V, in branch 1-3 (E is the Electromotive Force in branch 1-3; positive if the source's positive terminal is at node 3 and negative terminal at node 1.)
  Edge 4: R=8.00 Ohm, E=-1.00 V, in branch 3-0 (E is the Electromotive Force in branch 3-0; positive if the source's positive terminal is at node 0 and negative terminal at node 3.)
  Edge 5: R=2.00 Ohm, E=-2.00 V, in branch 0-4 (E is the Electromotive Force in branch 0-4; positive if the source's positive terminal is at node 4 and negative terminal at node 0.)
  Edge 6: R=10.00 Ohm, E=7.00 V, in branch 4-3 (E is the Electromotive Force in branch 4-3; positive if the source's positive terminal is at node 3 and negative terminal at node 4.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 3 nodes, labeled 0 to 2.  The circuit has the following edges:\n  Edge 1: R=4.00 Ohm, E=-6.00 V, in branch 1-0 (E is the Electromotive Force in branch 1-0; positive if the source's positive terminal is at node 0 and negative terminal at node 1.)
  Edge 2: R=2.00 Ohm, E=2.00 V, in branch 0-2 (E is the Electromotive Force in branch 0-2; positive if the source's positive terminal is at node 2 and negative terminal at node 0.)
  Edge 3: R=1.00 Ohm, E=9.00 V, in branch 1-2 (E is the Electromotive Force in branch 1-2; positive if the source's positive terminal is at node 2 and negative terminal at node 1.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 4 nodes, labeled 0 to 3.  The circuit has the following edges:\n  Edge 1: R=7.00 Ohm, E=-2.00 V, in branch 1-2 (E is the Electromotive Force in branch 1-2; positive if the source's positive terminal is at node 2 and negative terminal at node 1.)
  Edge 2: R=10.00 Ohm, E=-1.00 V, in branch 2-3 (E is the Electromotive Force in branch 2-3; positive if the source's positive terminal is at node 3 and negative terminal at node 2.)
  Edge 3: R=5.00 Ohm, E=5.00 V, in branch 3-0 (E is the Electromotive Force in branch 3-0; positive if the source's positive terminal is at node 0 and negative terminal at node 3.)
  Edge 4: R=2.00 Ohm, E=-5.00 V, in branch 0-1 (E is the Electromotive Force in branch 0-1; positive if the source's positive terminal is at node 1 and negative terminal at node 0.)
  Edge 5: R=4.00 Ohm, E=7.00 V, in branch 0-2 (E is the Electromotive Force in branch 0-2; positive if the source's positive terminal is at node 2 and negative terminal at node 0.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 6 nodes, labeled 0 to 5.  The circuit has the following edges:\n  Edge 1: R=8.00 Ohm, E=8.00 V, in branch 1-2 (E is the Electromotive Force in branch 1-2; positive if the source's positive terminal is at node 2 and negative terminal at node 1.)
  Edge 2: R=1.00 Ohm, E=-10.00 V, in branch 2-3 (E is the Electromotive Force in branch 2-3; positive if the source's positive terminal is at node 3 and negative terminal at node 2.)
  Edge 3: R=2.00 Ohm, E=8.00 V, in branch 3-5 (E is the Electromotive Force in branch 3-5; positive if the source's positive terminal is at node 5 and negative terminal at node 3.)
  Edge 4: R=5.00 Ohm, E=5.00 V, in branch 5-4 (E is the Electromotive Force in branch 5-4; positive if the source's positive terminal is at node 4 and negative terminal at node 5.)
  Edge 5: R=4.00 Ohm, E=-7.00 V, in branch 4-0 (E is the Electromotive Force in branch 4-0; positive if the source's positive terminal is at node 0 and negative terminal at node 4.)
  Edge 6: R=5.00 Ohm, E=9.00 V, in branch 2-4 (E is the Electromotive Force in branch 2-4; positive if the source's positive terminal is at node 4 and negative terminal at node 2.)
  Edge 7: R=3.00 Ohm, E=-1.00 V, in branch 3-0 (E is the Electromotive Force in branch 3-0; positive if the source's positive terminal is at node 0 and negative terminal at node 3.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 3 nodes, labeled 0 to 2.  The circuit has the following edges:\n  Edge 1: R=9.00 Ohm, E=4.00 V, in branch 0-2 (E is the Electromotive Force in branch 0-2; positive if the source's positive terminal is at node 2 and negative terminal at node 0.)
  Edge 2: R=1.00 Ohm, E=5.00 V, in branch 2-1 (E is the Electromotive Force in branch 2-1; positive if the source's positive terminal is at node 1 and negative terminal at node 2.)
  Edge 3: R=3.00 Ohm, E=-8.00 V, in branch 1-0 (E is the Electromotive Force in branch 1-0; positive if the source's positive terminal is at node 0 and negative terminal at node 1.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 5 nodes, labeled 0 to 4.  The circuit has the following edges:\n  Edge 1: R=9.00 Ohm, E=4.00 V, in branch 2-3 (E is the Electromotive Force in branch 2-3; positive if the source's positive terminal is at node 3 and negative terminal at node 2.)
  Edge 2: R=5.00 Ohm, E=3.00 V, in branch 3-0 (E is the Electromotive Force in branch 3-0; positive if the source's positive terminal is at node 0 and negative terminal at node 3.)
  Edge 3: R=10.00 Ohm, E=-3.00 V, in branch 0-1 (E is the Electromotive Force in branch 0-1; positive if the source's positive terminal is at node 1 and negative terminal at node 0.)
  Edge 4: R=5.00 Ohm, E=5.00 V, in branch 1-4 (E is the Electromotive Force in branch 1-4; positive if the source's positive terminal is at node 4 and negative terminal at node 1.)
  Edge 5: R=10.00 Ohm, E=-8.00 V, in branch 0-4 (E is the Electromotive Force in branch 0-4; positive if the source's positive terminal is at node 4 and negative terminal at node 0.)
  Edge 6: R=1.00 Ohm, E=-1.00 V, in branch 4-2 (E is the Electromotive Force in branch 4-2; positive if the source's positive terminal is at node 2 and negative terminal at node 4.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 5 nodes, labeled 0 to 4.  The circuit has the following edges:\n  Edge 1: R=6.00 Ohm, E=0.00 V, in branch 2-3 (E is the Electromotive Force in branch 2-3; positive if the source's positive terminal is at node 3 and negative terminal at node 2.)
  Edge 2: R=9.00 Ohm, E=5.00 V, in branch 3-0 (E is the Electromotive Force in branch 3-0; positive if the source's positive terminal is at node 0 and negative terminal at node 3.)
  Edge 3: R=8.00 Ohm, E=10.00 V, in branch 0-1 (E is the Electromotive Force in branch 0-1; positive if the source's positive terminal is at node 1 and negative terminal at node 0.)
  Edge 4: R=1.00 Ohm, E=5.00 V, in branch 1-4 (E is the Electromotive Force in branch 1-4; positive if the source's positive terminal is at node 4 and negative terminal at node 1.)
  Edge 5: R=3.00 Ohm, E=-10.00 V, in branch 1-3 (E is the Electromotive Force in branch 1-3; positive if the source's positive terminal is at node 3 and negative terminal at node 1.)
  Edge 6: R=5.00 Ohm, E=8.00 V, in branch 4-3 (E is the Electromotive Force in branch 4-3; positive if the source's positive terminal is at node 3 and negative terminal at node 4.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 4 nodes, labeled 0 to 3.  The circuit has the following edges:\n  Edge 1: R=1.00 Ohm, E=-2.00 V, in branch 1-3 (E is the Electromotive Force in branch 1-3; positive if the source's positive terminal is at node 3 and negative terminal at node 1.)
  Edge 2: R=8.00 Ohm, E=3.00 V, in branch 3-2 (E is the Electromotive Force in branch 3-2; positive if the source's positive terminal is at node 2 and negative terminal at node 3.)
  Edge 3: R=4.00 Ohm, E=-5.00 V, in branch 2-0 (E is the Electromotive Force in branch 2-0; positive if the source's positive terminal is at node 0 and negative terminal at node 2.)
  Edge 4: R=3.00 Ohm, E=8.00 V, in branch 1-0 (E is the Electromotive Force in branch 1-0; positive if the source's positive terminal is at node 0 and negative terminal at node 1.)
  Edge 5: R=8.00 Ohm, E=1.00 V, in branch 1-2 (E is the Electromotive Force in branch 1-2; positive if the source's positive terminal is at node 2 and negative terminal at node 1.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 6 nodes, labeled 0 to 5.  The circuit has the following edges:\n  Edge 1: R=1.00 Ohm, E=-3.00 V, in branch 1-5 (E is the Electromotive Force in branch 1-5; positive if the source's positive terminal is at node 5 and negative terminal at node 1.)
  Edge 2: R=5.00 Ohm, E=-6.00 V, in branch 5-4 (E is the Electromotive Force in branch 5-4; positive if the source's positive terminal is at node 4 and negative terminal at node 5.)
  Edge 3: R=2.00 Ohm, E=-6.00 V, in branch 4-3 (E is the Electromotive Force in branch 4-3; positive if the source's positive terminal is at node 3 and negative terminal at node 4.)
  Edge 4: R=1.00 Ohm, E=-9.00 V, in branch 3-2 (E is the Electromotive Force in branch 3-2; positive if the source's positive terminal is at node 2 and negative terminal at node 3.)
  Edge 5: R=5.00 Ohm, E=10.00 V, in branch 2-0 (E is the Electromotive Force in branch 2-0; positive if the source's positive terminal is at node 0 and negative terminal at node 2.)
  Edge 6: R=7.00 Ohm, E=4.00 V, in branch 0-4 (E is the Electromotive Force in branch 0-4; positive if the source's positive terminal is at node 4 and negative terminal at node 0.)
  Edge 7: R=2.00 Ohm, E=1.00 V, in branch 4-1 (E is the Electromotive Force in branch 4-1; positive if the source's positive terminal is at node 1 and negative terminal at node 4.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 6 nodes, labeled 0 to 5.  The circuit has the following edges:\n  Edge 1: R=10.00 Ohm, E=-9.00 V, in branch 1-0 (E is the Electromotive Force in branch 1-0; positive if the source's positive terminal is at node 0 and negative terminal at node 1.)
  Edge 2: R=3.00 Ohm, E=-9.00 V, in branch 0-4 (E is the Electromotive Force in branch 0-4; positive if the source's positive terminal is at node 4 and negative terminal at node 0.)
  Edge 3: R=2.00 Ohm, E=-3.00 V, in branch 4-3 (E is the Electromotive Force in branch 4-3; positive if the source's positive terminal is at node 3 and negative terminal at node 4.)
  Edge 4: R=1.00 Ohm, E=-1.00 V, in branch 3-2 (E is the Electromotive Force in branch 3-2; positive if the source's positive terminal is at node 2 and negative terminal at node 3.)
  Edge 5: R=5.00 Ohm, E=-9.00 V, in branch 2-5 (E is the Electromotive Force in branch 2-5; positive if the source's positive terminal is at node 5 and negative terminal at node 2.)
  Edge 6: R=10.00 Ohm, E=-8.00 V, in branch 1-5 (E is the Electromotive Force in branch 1-5; positive if the source's positive terminal is at node 5 and negative terminal at node 1.)
  Edge 7: R=6.00 Ohm, E=-9.00 V, in branch 1-4 (E is the Electromotive Force in branch 1-4; positive if the source's positive terminal is at node 4 and negative terminal at node 1.)
  Edge 8: R=1.00 Ohm, E=-3.00 V, in branch 2-4 (E is the Electromotive Force in branch 2-4; positive if the source's positive terminal is at node 4 and negative terminal at node 2.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 5 nodes, labeled 0 to 4.  The circuit has the following edges:\n  Edge 1: R=5.00 Ohm, E=-2.00 V, in branch 4-2 (E is the Electromotive Force in branch 4-2; positive if the source's positive terminal is at node 2 and negative terminal at node 4.)
  Edge 2: R=10.00 Ohm, E=5.00 V, in branch 2-3 (E is the Electromotive Force in branch 2-3; positive if the source's positive terminal is at node 3 and negative terminal at node 2.)
  Edge 3: R=1.00 Ohm, E=-4.00 V, in branch 3-0 (E is the Electromotive Force in branch 3-0; positive if the source's positive terminal is at node 0 and negative terminal at node 3.)
  Edge 4: R=4.00 Ohm, E=-7.00 V, in branch 0-1 (E is the Electromotive Force in branch 0-1; positive if the source's positive terminal is at node 1 and negative terminal at node 0.)
  Edge 5: R=3.00 Ohm, E=-3.00 V, in branch 1-2 (E is the Electromotive Force in branch 1-2; positive if the source's positive terminal is at node 2 and negative terminal at node 1.)
  Edge 6: R=5.00 Ohm, E=-3.00 V, in branch 4-1 (E is the Electromotive Force in branch 4-1; positive if the source's positive terminal is at node 1 and negative terminal at node 4.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 5 nodes, labeled 0 to 4.  The circuit has the following edges:\n  Edge 1: R=4.00 Ohm, E=2.00 V, in branch 4-0 (E is the Electromotive Force in branch 4-0; positive if the source's positive terminal is at node 0 and negative terminal at node 4.)
  Edge 2: R=10.00 Ohm, E=10.00 V, in branch 0-1 (E is the Electromotive Force in branch 0-1; positive if the source's positive terminal is at node 1 and negative terminal at node 0.)
  Edge 3: R=2.00 Ohm, E=10.00 V, in branch 1-2 (E is the Electromotive Force in branch 1-2; positive if the source's positive terminal is at node 2 and negative terminal at node 1.)
  Edge 4: R=9.00 Ohm, E=3.00 V, in branch 2-3 (E is the Electromotive Force in branch 2-3; positive if the source's positive terminal is at node 3 and negative terminal at node 2.)
  Edge 5: R=3.00 Ohm, E=9.00 V, in branch 3-1 (E is the Electromotive Force in branch 3-1; positive if the source's positive terminal is at node 1 and negative terminal at node 3.)
  Edge 6: R=8.00 Ohm, E=0.00 V, in branch 3-4 (E is the Electromotive Force in branch 3-4; positive if the source's positive terminal is at node 4 and negative terminal at node 3.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 4 nodes, labeled 0 to 3.  The circuit has the following edges:\n  Edge 1: R=8.00 Ohm, E=8.00 V, in branch 0-3 (E is the Electromotive Force in branch 0-3; positive if the source's positive terminal is at node 3 and negative terminal at node 0.)
  Edge 2: R=10.00 Ohm, E=-2.00 V, in branch 3-1 (E is the Electromotive Force in branch 3-1; positive if the source's positive terminal is at node 1 and negative terminal at node 3.)
  Edge 3: R=3.00 Ohm, E=0.00 V, in branch 1-2 (E is the Electromotive Force in branch 1-2; positive if the source's positive terminal is at node 2 and negative terminal at node 1.)
  Edge 4: R=5.00 Ohm, E=-8.00 V, in branch 0-1 (E is the Electromotive Force in branch 0-1; positive if the source's positive terminal is at node 1 and negative terminal at node 0.)
  Edge 5: R=4.00 Ohm, E=-10.00 V, in branch 0-2 (E is the Electromotive Force in branch 0-2; positive if the source's positive terminal is at node 2 and negative terminal at node 0.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 3 nodes, labeled 0 to 2.  The circuit has the following edges:\n  Edge 1: R=10.00 Ohm, E=3.00 V, in branch 2-1 (E is the Electromotive Force in branch 2-1; positive if the source's positive terminal is at node 1 and negative terminal at node 2.)
  Edge 2: R=6.00 Ohm, E=-9.00 V, in branch 1-0 (E is the Electromotive Force in branch 1-0; positive if the source's positive terminal is at node 0 and negative terminal at node 1.)
  Edge 3: R=1.00 Ohm, E=0.00 V, in branch 2-0 (E is the Electromotive Force in branch 2-0; positive if the source's positive terminal is at node 0 and negative terminal at node 2.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 5 nodes, labeled 0 to 4.  The circuit has the following edges:\n  Edge 1: R=5.00 Ohm, E=7.00 V, in branch 1-3 (E is the Electromotive Force in branch 1-3; positive if the source's positive terminal is at node 3 and negative terminal at node 1.)
  Edge 2: R=2.00 Ohm, E=3.00 V, in branch 3-2 (E is the Electromotive Force in branch 3-2; positive if the source's positive terminal is at node 2 and negative terminal at node 3.)
  Edge 3: R=8.00 Ohm, E=-7.00 V, in branch 2-4 (E is the Electromotive Force in branch 2-4; positive if the source's positive terminal is at node 4 and negative terminal at node 2.)
  Edge 4: R=3.00 Ohm, E=-3.00 V, in branch 4-0 (E is the Electromotive Force in branch 4-0; positive if the source's positive terminal is at node 0 and negative terminal at node 4.)
  Edge 5: R=2.00 Ohm, E=-4.00 V, in branch 3-0 (E is the Electromotive Force in branch 3-0; positive if the source's positive terminal is at node 0 and negative terminal at node 3.)
  Edge 6: R=9.00 Ohm, E=-1.00 V, in branch 1-2 (E is the Electromotive Force in branch 1-2; positive if the source's positive terminal is at node 2 and negative terminal at node 1.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 4 nodes, labeled 0 to 3.  The circuit has the following edges:\n  Edge 1: R=6.00 Ohm, E=-3.00 V, in branch 3-1 (E is the Electromotive Force in branch 3-1; positive if the source's positive terminal is at node 1 and negative terminal at node 3.)
  Edge 2: R=5.00 Ohm, E=-4.00 V, in branch 1-2 (E is the Electromotive Force in branch 1-2; positive if the source's positive terminal is at node 2 and negative terminal at node 1.)
  Edge 3: R=8.00 Ohm, E=2.00 V, in branch 2-0 (E is the Electromotive Force in branch 2-0; positive if the source's positive terminal is at node 0 and negative terminal at node 2.)
  Edge 4: R=9.00 Ohm, E=-1.00 V, in branch 2-3 (E is the Electromotive Force in branch 2-3; positive if the source's positive terminal is at node 3 and negative terminal at node 2.)
  Edge 5: R=3.00 Ohm, E=4.00 V, in branch 1-0 (E is the Electromotive Force in branch 1-0; positive if the source's positive terminal is at node 0 and negative terminal at node 1.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 3 nodes, labeled 0 to 2.  The circuit has the following edges:\n  Edge 1: R=7.00 Ohm, E=-6.00 V, in branch 1-0 (E is the Electromotive Force in branch 1-0; positive if the source's positive terminal is at node 0 and negative terminal at node 1.)
  Edge 2: R=9.00 Ohm, E=-9.00 V, in branch 0-2 (E is the Electromotive Force in branch 0-2; positive if the source's positive terminal is at node 2 and negative terminal at node 0.)
  Edge 3: R=8.00 Ohm, E=8.00 V, in branch 2-1 (E is the Electromotive Force in branch 2-1; positive if the source's positive terminal is at node 1 and negative terminal at node 2.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 3 nodes, labeled 0 to 2.  The circuit has the following edges:\n  Edge 1: R=8.00 Ohm, E=3.00 V, in branch 2-1 (E is the Electromotive Force in branch 2-1; positive if the source's positive terminal is at node 1 and negative terminal at node 2.)
  Edge 2: R=7.00 Ohm, E=9.00 V, in branch 1-0 (E is the Electromotive Force in branch 1-0; positive if the source's positive terminal is at node 0 and negative terminal at node 1.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 3 nodes, labeled 0 to 2.  The circuit has the following edges:\n  Edge 1: R=9.00 Ohm, E=1.00 V, in branch 2-1 (E is the Electromotive Force in branch 2-1; positive if the source's positive terminal is at node 1 and negative terminal at node 2.)
  Edge 2: R=1.00 Ohm, E=-4.00 V, in branch 1-0 (E is the Electromotive Force in branch 1-0; positive if the source's positive terminal is at node 0 and negative terminal at node 1.)
  Edge 3: R=2.00 Ohm, E=1.00 V, in branch 0-2 (E is the Electromotive Force in branch 0-2; positive if the source's positive terminal is at node 2 and negative terminal at node 0.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 5 nodes, labeled 0 to 4.  The circuit has the following edges:\n  Edge 1: R=9.00 Ohm, E=-1.00 V, in branch 2-1 (E is the Electromotive Force in branch 2-1; positive if the source's positive terminal is at node 1 and negative terminal at node 2.)
  Edge 2: R=7.00 Ohm, E=2.00 V, in branch 1-4 (E is the Electromotive Force in branch 1-4; positive if the source's positive terminal is at node 4 and negative terminal at node 1.)
  Edge 3: R=2.00 Ohm, E=-10.00 V, in branch 4-0 (E is the Electromotive Force in branch 4-0; positive if the source's positive terminal is at node 0 and negative terminal at node 4.)
  Edge 4: R=10.00 Ohm, E=9.00 V, in branch 0-3 (E is the Electromotive Force in branch 0-3; positive if the source's positive terminal is at node 3 and negative terminal at node 0.)
  Edge 5: R=4.00 Ohm, E=-8.00 V, in branch 2-3 (E is the Electromotive Force in branch 2-3; positive if the source's positive terminal is at node 3 and negative terminal at node 2.)
  Edge 6: R=6.00 Ohm, E=10.00 V, in branch 3-1 (E is the Electromotive Force in branch 3-1; positive if the source's positive terminal is at node 1 and negative terminal at node 3.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 6 nodes, labeled 0 to 5.  The circuit has the following edges:\n  Edge 1: R=2.00 Ohm, E=7.00 V, in branch 5-2 (E is the Electromotive Force in branch 5-2; positive if the source's positive terminal is at node 2 and negative terminal at node 5.)
  Edge 2: R=8.00 Ohm, E=-2.00 V, in branch 2-0 (E is the Electromotive Force in branch 2-0; positive if the source's positive terminal is at node 0 and negative terminal at node 2.)
  Edge 3: R=5.00 Ohm, E=-10.00 V, in branch 0-1 (E is the Electromotive Force in branch 0-1; positive if the source's positive terminal is at node 1 and negative terminal at node 0.)
  Edge 4: R=4.00 Ohm, E=-4.00 V, in branch 1-3 (E is the Electromotive Force in branch 1-3; positive if the source's positive terminal is at node 3 and negative terminal at node 1.)
  Edge 5: R=9.00 Ohm, E=-4.00 V, in branch 3-4 (E is the Electromotive Force in branch 3-4; positive if the source's positive terminal is at node 4 and negative terminal at node 3.)
  Edge 6: R=10.00 Ohm, E=-3.00 V, in branch 3-2 (E is the Electromotive Force in branch 3-2; positive if the source's positive terminal is at node 2 and negative terminal at node 3.)
  Edge 7: R=7.00 Ohm, E=-7.00 V, in branch 1-4 (E is the Electromotive Force in branch 1-4; positive if the source's positive terminal is at node 4 and negative terminal at node 1.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 4 nodes, labeled 0 to 3.  The circuit has the following edges:\n  Edge 1: R=2.00 Ohm, E=-9.00 V, in branch 2-3 (E is the Electromotive Force in branch 2-3; positive if the source's positive terminal is at node 3 and negative terminal at node 2.)
  Edge 2: R=6.00 Ohm, E=2.00 V, in branch 3-0 (E is the Electromotive Force in branch 3-0; positive if the source's positive terminal is at node 0 and negative terminal at node 3.)
  Edge 3: R=1.00 Ohm, E=5.00 V, in branch 0-1 (E is the Electromotive Force in branch 0-1; positive if the source's positive terminal is at node 1 and negative terminal at node 0.)
  Edge 4: R=7.00 Ohm, E=7.00 V, in branch 2-0 (E is the Electromotive Force in branch 2-0; positive if the source's positive terminal is at node 0 and negative terminal at node 2.)
  Edge 5: R=9.00 Ohm, E=-7.00 V, in branch 1-2 (E is the Electromotive Force in branch 1-2; positive if the source's positive terminal is at node 2 and negative terminal at node 1.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 4 nodes, labeled 0 to 3.  The circuit has the following edges:\n  Edge 1: R=10.00 Ohm, E=-10.00 V, in branch 0-1 (E is the Electromotive Force in branch 0-1; positive if the source's positive terminal is at node 1 and negative terminal at node 0.)
  Edge 2: R=10.00 Ohm, E=0.00 V, in branch 1-2 (E is the Electromotive Force in branch 1-2; positive if the source's positive terminal is at node 2 and negative terminal at node 1.)
  Edge 3: R=10.00 Ohm, E=-5.00 V, in branch 2-3 (E is the Electromotive Force in branch 2-3; positive if the source's positive terminal is at node 3 and negative terminal at node 2.)
  Edge 4: R=8.00 Ohm, E=4.00 V, in branch 2-0 (E is the Electromotive Force in branch 2-0; positive if the source's positive terminal is at node 0 and negative terminal at node 2.)
  Edge 5: R=9.00 Ohm, E=5.00 V, in branch 3-1 (E is the Electromotive Force in branch 3-1; positive if the source's positive terminal is at node 1 and negative terminal at node 3.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 3 nodes, labeled 0 to 2.  The circuit has the following edges:\n  Edge 1: R=3.00 Ohm, E=-1.00 V, in branch 0-2 (E is the Electromotive Force in branch 0-2; positive if the source's positive terminal is at node 2 and negative terminal at node 0.)
  Edge 2: R=3.00 Ohm, E=-5.00 V, in branch 2-1 (E is the Electromotive Force in branch 2-1; positive if the source's positive terminal is at node 1 and negative terminal at node 2.)
  Edge 3: R=7.00 Ohm, E=-10.00 V, in branch 0-1 (E is the Electromotive Force in branch 0-1; positive if the source's positive terminal is at node 1 and negative terminal at node 0.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 4 nodes, labeled 0 to 3.  The circuit has the following edges:\n  Edge 1: R=5.00 Ohm, E=2.00 V, in branch 1-2 (E is the Electromotive Force in branch 1-2; positive if the source's positive terminal is at node 2 and negative terminal at node 1.)
  Edge 2: R=8.00 Ohm, E=6.00 V, in branch 2-3 (E is the Electromotive Force in branch 2-3; positive if the source's positive terminal is at node 3 and negative terminal at node 2.)
  Edge 3: R=9.00 Ohm, E=-5.00 V, in branch 3-0 (E is the Electromotive Force in branch 3-0; positive if the source's positive terminal is at node 0 and negative terminal at node 3.)
  Edge 4: R=6.00 Ohm, E=-7.00 V, in branch 0-2 (E is the Electromotive Force in branch 0-2; positive if the source's positive terminal is at node 2 and negative terminal at node 0.)
  Edge 5: R=6.00 Ohm, E=2.00 V, in branch 3-1 (E is the Electromotive Force in branch 3-1; positive if the source's positive terminal is at node 1 and negative terminal at node 3.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 5 nodes, labeled 0 to 4.  The circuit has the following edges:\n  Edge 1: R=7.00 Ohm, E=-4.00 V, in branch 4-0 (E is the Electromotive Force in branch 4-0; positive if the source's positive terminal is at node 0 and negative terminal at node 4.)
  Edge 2: R=8.00 Ohm, E=-7.00 V, in branch 0-1 (E is the Electromotive Force in branch 0-1; positive if the source's positive terminal is at node 1 and negative terminal at node 0.)
  Edge 3: R=5.00 Ohm, E=0.00 V, in branch 1-3 (E is the Electromotive Force in branch 1-3; positive if the source's positive terminal is at node 3 and negative terminal at node 1.)
  Edge 4: R=6.00 Ohm, E=-1.00 V, in branch 3-2 (E is the Electromotive Force in branch 3-2; positive if the source's positive terminal is at node 2 and negative terminal at node 3.)
  Edge 5: R=4.00 Ohm, E=5.00 V, in branch 3-4 (E is the Electromotive Force in branch 3-4; positive if the source's positive terminal is at node 4 and negative terminal at node 3.)
  Edge 6: R=5.00 Ohm, E=-7.00 V, in branch 0-2 (E is the Electromotive Force in branch 0-2; positive if the source's positive terminal is at node 2 and negative terminal at node 0.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.


Prompt:
Consider an electrical circuit with 3 nodes, labeled 0 to 2.  The circuit has the following edges:\n  Edge 1: R=4.00 Ohm, E=9.00 V, in branch 2-0 (E is the Electromotive Force in branch 2-0; positive if the source's positive terminal is at node 0 and negative terminal at node 2.)
  Edge 2: R=4.00 Ohm, E=-4.00 V, in branch 0-1 (E is the Electromotive Force in branch 0-1; positive if the source's positive terminal is at node 1 and negative terminal at node 0.)
  Edge 3: R=1.00 Ohm, E=-1.00 V, in branch 1-2 (E is the Electromotive Force in branch 1-2; positive if the source's positive terminal is at node 2 and negative terminal at node 1.)\nFor this circuit, your task is to formulate the set of equations based on Kirchhoff's Laws that can be used to solve for all branch currents.\nYou are NOT required to solve these equations or provide the numerical values for currents.\n
Let's think step by step. Follow these instructions to formulate the equations:

1. **Analyze the Circuit Structure:** Identify all nodes and branches in the circuit. Determine how many independent loops exist.
2. **Formulate Equations using Kirchhoff's Laws with Branch Currents as Unknowns:**
   - Assign a branch current variable to each edge. I_1 represents the current through Edge 1, I_2 represents the current through Edge 2, and so on. The assumed direction of each current aligns with the u -> v direction of the edge definition as provided: 'positive current is defined to flow from the first node towards the second node listed in the edge description'.
   - Apply Kirchhoff's Current Law (KCL) at n-1 independent nodes (where n is the total number of nodes) to get a set of equations.
   - Apply Kirchhoff's Voltage Law (KVL) around each independent loop to get another set of equations. Ensure you correctly account for the voltage drops across resistors (V=IR) and the EMFs of voltage sources (E), paying attention to their polarities relative to the loop traversal direction.
3. **Output the Equations:** Use the following format for your answer, listing all formulated KCL and KVL equations clearly:

```
Equations:
KCL at Node 1: <equation_node_1>
KCL at Node 2: <equation_node_2>
...
KVL for Loop 1: <equation_loop_1>
KVL for Loop 2: <equation_loop_2>
...
```

Focus solely on providing the correct set of equations based on the circuit description.