It sort of works if you mean that, say, her co-workers have lunch at different places, but she eats right here. Which is why we accept that answer. These two word orders are not interchangeable, though, meaning that in a particular situation you choose the one that makes sense.
A bonus fact: in Ukrainian these words are different (Russian and Ukrainian are closely related). The is-word sounds like "yeah" while the inifitive of eat is like "yisti".
If you ignore the different endings, you can see the vowels are different. They were somewhat close; the two merged in Russian but drifted further apart in Ukrainian (this is why you have ru лес = uk ліс , ru небо = uk небо)