This thread has some excellent top-level answers on the benefits of linear tactics and also on their origins, which appears to have been in the 15th century at the earliest though it was a gradual development that the users in this thread greatly expand upon.

EDIT: Here's a good, concise answer from /u/tyn_peddler that touches upon major aspects of linear tactics and how and why they evolved from the 17th century onwards.

I think if you want a Guiness Book of Recrods style 'first time' then it would precede most known written records. Its unlikely to have occurred before some degree of organised army existed and only when some degree of armour was introduce.

Most early armies made extensive use of hunting and skirmish weapons. Of these the bow was popular. Until extensive bronze armour and shields could be made close order warriors would have been quite vulnerable to arrows, effective leather/hide not offering the same degree of protection.

I've had a quick look and I dont see much evidence that Egypt would be a good source. Their extensive art shows little sign of such warfare. So perhaps the greeks/persians would be the best bet.

So overall i'm thinking late bronze/early iron age to be the start of such tactics because armour and shields work very well when in close formation.