Real heating jobs are journeys with legs, and the heating curve is the map: a slope (warming, priced by $Q = mc\,\Delta T$), then a flat plateau (a phase change, priced by $Q = mL$), then another slope. The rule is simply to price every leg and add them. Ice from the deep freeze climbs to 0 °C, crosses the melting plateau, and climbs again as water — three legs, three lines, one total. Once a journey has more than one leg the subscripts number them in order: $\Delta T_1$ is the first slope's climb in C°, $\Delta T_2$ the second slope's, and $m_2$ means only the mass that crosses the plateau — the steam that actually leaves, not the whole kettle.

The classic slip — worth naming out loud — is skipping the plateau, because the thermometer stops advertising it: the reading sits still, so the leg looks free. It is the opposite of free. Melting charges 334 kJ for every kilogram, more than the whole climb from freezer-cold usually costs. Watch for it below; this lesson sets that trap on purpose, with love.