It's definitely possible, but for almost the same money, you could get a 600W PSU to power all of your system.
Finding a good PSU under $50 is almost impossible anyway. $50 is what a 620W Seasonic S12II costs, a very good PSU.
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Wiring isn't all that difficult, there is a wire in the ATX(green) connector that starts the PSU, that wire is then connected with a black wire(ground).
It's connected by taking a new wire, and putting one end in the green wire pin and the other end in the black wire pin.
From here, you can add a switch in the middle of the new wire, to connect or disconnect the two halves.
As for connections to the GPU, it's the same as always, connect the 6/8 pin to the GPU.
I don't recommend this over the first method, I would only do this, if I had a spare PSU laying around and really don't have any cash around for a new one.