The idea of using linear equations is an old and obvious one. You show a particular example for which it works.
However, it has never worked in general for a very simple reason:
In standard (i.e. 9x9) Sudoku, you have:
- 9x3 = 27 linear equations (9 rows, 9 blocks, 9 columns)
- 81 - #givens unknowns, which is in general much more than 27
so that the obvious linear system is largely under-constrained.
You must add:
- constraints 1 ≤ x ≤ 9 for all the unknowns
- constraints about uniqueness in each unit
which together take you far from mere linear algebra