Final answer:
Using Kirchhoff's rules to solve the given circuit equations, we find that I1 equals 3.00 A, I2 equals -3.50 A, and I3 equals 8.25 A, and these values are verified by checking their consistency with the power in the circuit.
Step-by-step explanation:
To solve the given set of equations for the unknown currents I1, I2, and I3, we can use Kirchhoff's rules which are essential in circuit analysis. The question requires employing both the junction rule and the loop rule to obtain a set of equations that can determine the current in each branch of the circuit.
Starting at point a, Kirchhoff's first rule, also known as the junction rule, gives us an equation that relates I1, I2, and I3. Furthermore, by manipulating the given equations and applying Kirchhoff's second rule, also known as the loop rule, we can eliminate variables and solve for the currents sequentially.
After manipulation, we find that I1 equals 3.00 A, I2 equals -3.50 A, indicating it flows in the opposite direction to the assumed one, and I3 equals 8.25 A. To verify the correctness of our solutions, we substitute these current values into the equations and check for consistency, particularly looking at the power supplied and dissipated in the circuit.