Hi suitcase,
It's possible to re-use some of the parts from the Gateway PC, but you might not be able to re-use the case or power supply. The case might be a standard ATX case or it might be made to only fit a proprietary Gateway motherboard. The motherboard power connector from the Gateway power supply might be wired differently from a standard ATX power connector too...it could fry your new motherboard if it's doesn't follow the standard ATX wiring layout.
If the Gateway is an older one it might follow the standard ATX format, but the newer ones might be more proprietary.
It might be easier to just go with a standard ATX case and power supply, install the new parts and then transfer the parts you want from the Gateway.
Cricket