I am relocating my dryer from the first floor to the second floor of my house can I splice the new wire into the old wire?

U can use the old outlet box as a junction box. The old box will have to be exposed. It can not be cover or hidden in wall. Use a blank cover plate.

Be sure to turn off breaker when working on this.

Chuck

You can only do this if your circuit contains four wires, two hots a ground and a neutral.
If it is an older 3-wire circuit you cannot splice and lengthen it.