Hi I bought my mother a new electric dryer and I simply want to know if I can add new wire on to the existing wire. The old dryer was directly hooked up to the 220 volt line in the basement and I need to put the dryer upstairs for her now and I need to run about 20ft of new wire and I did not know if I could splice it together like 110volt lines

Older drying wiring used only 3 wires. 2 hots and a neutral. If you have that kind of wiring it is grandfathered in to allow replacing the dryer.

However, that kind of wiring can not be extended or moved. It needs to be replaced with a modern 4 wire circuit. 2 Hots, neutral, and GROUND.

The NEC changed this in 1996. But in some areas it was required much earlier.

The dryers, themself, can be setup for either type and come with the instructions.

If you have a 4 wire circuit then it can be extended. Just make sure that you use #10 wire and the that wirenuts or other connects are correctly sized.