im with metx, it just looks like there was excess flux not wiped off from soldering, a steel brush will take it off, from the pic I dont see evidence of pipes bursting and then being soldered closed, He simply soldered a support pipe there and then removed, the works not bad looking except for the excess flux left behind.
Yes, most of those joints were leaking and when you go to fix something, it will probably start leaking further down the line at on of the other joints. Someone mentioned Pex. That's the way to go. It's basically plastic tubing, you only need one joint at each end. It costs about $25 for 100 feet at Home Depot, then $5-$10 per fitting at the end. You cut your existing pipe, solder a connector to the pipe coming through your floor, then slip the Pex into the fitting. Done.
Now the reason that I wouldn't buy the house without some SERIOUS consideration is the electrical. A burst pipe won't kill your family, an electrical fire will. Those 3 yellow wires I see in every picture were not put in by an electrician and won't meet code ANYWHERE in the US. You might be okay if you're in Mexico or Canada, I have no idea of their codes. NO electrician would have drilled those 3 holes through the joists like that. The only reason to drill through the joists is if you're going to install sheetrock directly on the bottom of the joists. Since the plumbing makes that impossible, the wires should have been stapled to the bottom of the joists. Another thng that doesn't meet code is on a stud or joist where the wires go through, there needs to be a metal plate attached to the exposed edge so no one can drive a nail into the wire.
Issues this raises;
1) The rest of the electrical is in question. Are those yellow wires the proper guage to run whatever they're running, or can they overheat and burn? What other work was done by this same guy that's not code?
2) These wires will cause you re-sale problems later if the buyer gets a home inspection.
3) Insurance will not cover any electrical fire, or any other fire, if this electrical work is spotted.
4) I HIGHLY doubt if any town permits were pulled for that work. You will be responsible for paying for permits and fines, even though you didin't own the house at the time of the work, if you ever have an inspector in for future work. (Finishing the basement for example.) And you'd have to have the electrical brought up to code at that point.
As I stated above... AN ELECTRICAL FIRE WILL KILL YOUR FAMILY, A BURST PIPE WILL NOT.