State Kirchhoff's rules. Using these rules, find the current flowing through branch FC in the given circuit.

Kirchhoff's rules: Junction rule: At any junction, the sum of currents entering equals the sum of currents leaving (conservation of charge). Loop rule: The algebraic sum of changes in potential around any closed loop is zero (conservation of energy).Finding the current through branch FC: For loop AFCBA: 3i₁ + i₁ − i₂ = −$\displaystyle 3$ + $\displaystyle 5$ − $\displaystyle 2$ ⟹ 4i₁ − i₂ = $\displaystyle 0$ ⟹ 4i₁ = i₂ … For loop FEDCF: i₂ + $\displaystyle 4$(i₁ + i₂) = $\displaystyle 2$ ⟹ 4i₁ + 5i₂ = $\displaystyle 2$ … Solving ($\displaystyle 1$) and ($\displaystyle 2$): i₁ = $\displaystyle 1$/$\displaystyle 12$ A and i₂ = $\displaystyle 1$/$\displaystyle 3$ A. Current through branch FC = i₁ + i₂ = $\displaystyle 1$/$\displaystyle 12$ + $\displaystyle 1$/$\displaystyle 3$ = $\displaystyle 5$/$\displaystyle 12$ A.