Trying to find the voltage of a circut

You're not keeping proper track of the signs. It makes a difference when you make a loop whether you're going through from + to - or from - to +. (You also have a couple of small algebra errors. By your equations, V3 should be a -10V, and I'm not sure about the others because your variables seem to change meaning partway through. This is a common problem, especially when you're using a lot of things that look very much alike.)

Let's look at this logically. Considering your diagram, the 10V source is shorted across v3. What does that tell you about the voltage across v3?

Now - if we removed the v2 and v3 tracks, you have the 24V source opposite to the 10 and 12v sources, which are acting in series. This means that the 10 and 12v sources are acting like an effective 22V source, opposing a 24V source. What would this tell you about the remaining voltage to be dissipated across v1?

The equations I used:

10 - v3 = 0 (loop inside the right-hand circuit, going counter-clockwise)

12 + 10 + v1 - 24 = 0 (loop around the outside, going counter-clockwise)

24 - v1 + v2 = 0 (loop inside the upper left-hand circuit, going counter-clockwise)

It's more a coincidence than anything else that I went counter-clockwise on all three. All that would change would be the signs on the numbers. Notice also that v1 changes sign between equation 2 and equation 3, reflecting the fact that I'm going a different direction through it in each case.

One other point: if you end up with a negative voltage somewhere, it simply means you chose the wrong signs for either end of the circuit element. This is clearly the case for V2 - can you see why V2 is obviously mislabelled?

Does this help?