Like with some pre built systems, if it just relied on the on board graphics solution of the motherboard for graphics display.
Or used a very low end Pci-e graphics card Stapler, that required no extra power from the Psu direct to the graphics card via the provided 12v Pci-e power connectors.
You must check that they if the Rx 470 card requires any extra power are connected.
Often with pre built systems that previously only used the on board video solution of the motherboard.
The original power supply fitted, and the E-atx, Eps eight, or four pin 12v power connector direct from the power supply is not connected to the motherboard.
Because it is not initially connected, even when you replace the power supply in the system with a higer rated power supply unit of a better wattage out put.
It is often missed.
You need too look at the motherboard inside the system case Stapler.
Click on the link bellow, where the main picture shows the motherboard of the Gateway DX 4870 system you have.
To the top left of the picture of the board you can see a four pin power plug on the motherboard.
https://www.google.co.uk/search?q=DX+4870&source=lnms&tbm=isch&sa=X&ved=0ahUKEwixsJylkMbOAhXGOBoKHf_WA9EQ_AUICigD&biw=1773&bih=1101&dpr=0.9#imgrc=muvmdMzXt5IG2M%3A
Get the four pin 12v power feed from the PSU, and connect it to that power point on the motherboard.
And connect the six pin 12v Pci-e of the power supply to tthe outer edge of the RX 470 card that will be facing you when put in the Pci-e graphics slot of the motherboard.
If you do this the system should power up for you.
If the four pin was missing or power to the graphics card via the six pin 12v Pci-e power connector it is why your system did not power up right.
Make sure both are connected.
Once done.
1. Your next step is to connect the video lead from the monitor to the motherboard video out put ports on the back main I/O ports.
You should then get a video signal to the monitor.
2. You need to them press the key on your keyboard when the system first powers up that the board states when posting to enter your bios of the motherboard.
Going through the bios menus, find the section that deals with graphics, and the first graphics display device to initialise first of all.
3. Set the current mode you see if IGPFX to Pci-e or Peg mode.
4. After the change, save the new changes in the bios before you exit it Stapler.
5. When done, swap the video monitor lead from the back of the motherboard to one of the video display output ports of the RX 470 Pci-e video card.
6. Reset the system and your done, The system should power up right.
And you should have a display image on your monitor when the video lead from it is connected to the video output ports of your new RX 470 card.