ok, so its agp. heres something to try, remove your graphics card and clean the connector, and blow out the connector on the motherboard. also if you have another psu try that, and if you have another video card laying around, no matter how old it is, try it.
and to fix it, you dont have to buy a whole new pc, you can fix it for under 60bux usd, just find a new socket A that is compatible with your setup if it can not boot.
also, check every cable connection, i mean everything, and try unplugging the power from your harddrives to see if its a power problem.
i doubt that a solder could of came loose, but it could happen. dont see why that would not make it boot, unless something is shorting on the back, which is why im suggesting removing it from the case, to check if anything is behind the motherboard and also so you can inspect the board for damage.