Consider the circuit that was solved using the superposition theorem in the previous section. Using this alternative method the directions of the currents and the voltage drops are guessed to begin with. If these guesses prove to be incorrect, the values obtained from the calculations will turn out to be negative and so they can then be corrected.

Kirchhoff's voltage law can be applied to any loop in the circuit.

(Kirchhoff's voltage law states that around any loop in the circuit, the sum of the e.m.f. s is equal to the sum of the voltage
drops.)

- Applying Kirchoff's voltage law to loop 1, gives E
_{1}= V_{1}+ V_{2}. - Applying Kirchoff's voltage law to loop 2, gives E
_{2}= V_{3}+ V_{2}.

using Ohm's law, V = IR, to replace V_{1}, V_{2} and V_{3} in the above expressions gives.

- E
_{1}= I_{1}R_{1}+ I_{2}R_{2}. - E
_{2}= I_{3}R_{3}+ I_{2}R_{2}.

Putting in the values for the resistors and the e.m.f.s gives;

- 10 = 4I
_{1}+ 6I_{2}(1) . - 15 = 10I
_{3}+ 6I_{2}(2) .

Applying Kirchhoff's current law to the junction:

- I
_{2}= I_{1}+ I_{3}. - Therefore I
_{3}= I_{2}- I_{1}(3) .

substituting for I_{3} in equation (2) gives;

- 15 = 10(I
_{2}- I_{1}) + 6I_{2}. - 15 = 10I
_{2}- 10I_{1}+ 6I_{2}. - 15 = - 10I
_{1}+ 16I_{2}(4) .

multiplying equation (1) by 10

& multiplying equation (4) by 4 gives:

- 100 = 40I
_{1}+ 60I_{2}(5) . - 60 = -40I
_{1}+ 64I_{2}(6) .

adding equation (5) to equation (6) .

- 160 = 124I
_{2}. - I
_{2}= 160/124 =**1.29 A**.

substituting this value of I_{2} into equation (1) .

- 10 = 4I
_{1}+ 6 x 1.29 . - 10 = 4I
_{1}+ 7.74 . - 10 - 7.74 = 4I
_{1}. - 2.26 = 4I
_{1}. - I
_{1}=2.26/4 =**0.565 A**.

putting the values for I_{1} and I_{2} into eq (3).

- I
_{3}= I_{2}- I_{1}. - Therefore I
_{3}= 1.29 - 0.565 =**0.725 A**.

This gives the same answers (to within 0.001A due to rounding errors) as the superposition theorem.

In this case all the current values were positive, indicating that the initial assumed directions of current flow
were correct.

The voltage drops could now be found using ohms law as shown in the superposition theorem example.