We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions of these results. Finally, we obtain the first nontrivial upper bounds for the fundamental problem of the maximal size of independent systems. These bounds depend quadratically on the size of the shortest equation. No methods of having such bounds have been known before.