Solving systems of equations is solving where the equations are equal. Dependency of variables is not the same as dependence of equations themselves. Equality is an equivalence relation meaning that a = b implies a + c = b + c, ra = rb for real r, and if a = b and c = d then a + c = b + d. These properties allow us to manipulate them in such a way that we can eliminate variables while still being true. Testing variables doesn’t always work as they can imply failure of the test.