6 Equations, 6 Unknowns, Guess and Check?

We did this problem in my Advanced Algebra class today. Yesterday we took the test on systems of equations including up to 3 equations with 3 unknowns. Today I put them in groups of four and didn’t give them any clue about how to solve this.

I figured that it wasn’t fair to just do it on my calculator, so I started to work it out knowing full well that I was likely to mess something up along the way… A negative sign, a miss calculation, or a typo. Sure enough I did…

Then a kid came up and said he got it…. What???
Check it out…

What are the chances? I didn’t give any clue about the answers.