Cruising headwind speed

When you have strong headwinds, can you go faster when cruising at lower altitudes? I notice that at higher altitudes the winds are always stronger and if you decrease altitude to around 32000 feet instead of 34000 the ground speed increases a bit

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Yes, you can then. The ground speed is the distance over the ground you covered, so, if it went up, you’ll get to the destination faster.

is this something that is commonly done, since most planes use the step climb method to save fuel? do pilots usually adjust altitude to increase speed

I believe so. @realnickchan

It honestly depends on what the winds are. For example, at FL410, the winds may be less than at FL360. If you want to find the optimal altitude to cruise at, use and check out what the winds are doing at different altitudes.
But to answer your question about ground speed, yes, if you fly with less of a headwind, you will have better ground speed. Groundspeed is a function of your TAS (true airspeed), altitude, wind speed, wind direction, and course.

If you want to figure out what pilots use to determine their course heading, this is a calculator I use to help plan flights IRL: E6BX | Aviation Calculators
As long as you know your info about the flight, you can figure out your groundspeed for yourself.