Nodal analysis

Nodal analysis is a recipe for solving any circuit by picking the node voltages as your unknowns and writing one Kirchhoff-current-law equation per node. Instead of guessing branch currents one by one, you label every junction's voltage relative to a chosen ground, then demand that current balances at each node — turning the whole circuit into a tidy set of simultaneous equations.

You choose a reference (ground) node, give the other nodes voltages V1, V2, …, express each branch current as (V_a − V_b)/R using Ohm's law, and set the sum at each node to zero. With N non-reference nodes you get N equations in N unknowns, solvable by hand or by matrix methods (modern simulators like SPICE do exactly this with a conductance matrix). Nodal analysis is usually the fastest route when a circuit has many parallel branches and few independent loops.