Are those actual examples then? If the weights are the same for all students, then that implies that x,y,z are the same for all of them. If so, then you can solve for x,y,z from a series of linear equations derived from any 3 columns. This is what I did in post 2, giving x=-1.4, y=4, and z=3.6. If you put those values into your equations from post 1, you'll see that all of them work out correctly. The problem now is that if you use those same values in subsequent columns, they do not generate the given total adjusted value. Probably a bigger issue is that -1.4 makes no sense as a weighting factor. It implies that the better the score on test 1, the worse you'll get on the overall rating.
So either my model is wrong, or else the total adjusted value for each column in your example has not been calculated consistently. I even considered that there might be some rounding going on, so I set up a Solver problem to get the "best fit". Using all 5 columns, it came up with x=2.5, y=1.25, z=.4. It's a good fit for columns C:E, but A is off by 2, and B is off by 4. So that's probably not the issue.