# What factors/equation(s) are used to relate aircraft weights and cruise speeds?

With the recent coronavirus issues, there have been lots of empty seats flying around, especially on long haul flights. I checked the seat map for a recent SFO-SYD flight (777-300) and there were 224 empty seats out of 340 available. This obviously means the aircraft was much lighter than usual for this run - a quick estimate gives me 25 tonnes less weight just in people/baggage. To get to the actual question, when doing flight planning what equations or other mathematical factors go into computing how this impacts the climb rate and cruise speed for the flight? I'm familiar with the concepts of induced drag but I don't have an intuitive or empirical understanding of the actual impacts the reduction in weight would have.

Does taking 25 tonnes of weight from the airplane move the climb rate by 100 ft/min, or 1000 ft/min?

If we have 25 tonnes less "cargo", this means we also need less fuel to lift and carry the aircraft... but how much less? How do we compute the fuel savings for the lower take off weights? I know there have to be formulas (or at the very least charts/curves) but I didn't have much luck finding them.

Is the cruise speed meaningfully increased by the lighter weight? Or does it just require less thrust (i.e. lower fuel burn) to fly at the same cruise speed?

I realize these are framed as discrete questions, but I think they're all inter-related in a way that's close enough to put them into a single post here. Moderators feel free to disagree ;-)