Translation:That is everyone's problem, not just yours!
(More experienced Esperanto-users, feel free to correct me if I am mistaken.) "Via" by itself would be the possessive adjective "your", which would require a noun to describe. When we add the "la" to make "la via", that turns it into a possessive pronoun, and it is able to stand by itself in the sentence.
On another note, I believe in some cases "via" by itself can mean "yours". I imagine this is only in sentence constructions like "La problemo estas via", where it would act as an adjective in conjunction with the verb "esti (to be)"