Battle of Zama

Cornelius Scipio
placed his troops the usual Roman way; with some exceptions. He placed his manipels of Hastati, Principes and Triarii, not staggered but in straight lines, creating lanes from the front till the rear side of his army. He covered these lanes by placing his Velites in those lanes at the front between the Hastati. He ordered his Velites to start the fighting and lure the elephants into those lanes and attack them there from all sides.
The 1600 cavalry he had brought from Sicily he placed at his left flank commanded by his trusted Laelius. Masinissa and his horse, probably even more than 1600 at the right flank.

Hannibal
placed his elephants in front of his army. If he, as suggested in the ancient texts, covered his whole front of approximately 1600 m there was one elephant every 20m. This would have made them vulnerable for side attacks. This indicates the elephants were not there in the first place for the assault but for the morale of the troops behind them. Those could feel themselves relatively safe behind those animals. A wise decision and fits in Hannibal's usual defensive battle tactics.
The fact he placed his best troops, the core that had served him over the Alps and in Italy, in the rear stresses even more he intended to block and withstand an attack and not prepared a serious attack himself. Ca 200 m before these troops he placed his African and Carthaginian troops and before them as first line behind the elephants, mercenaries from everywhere: Ligurians, Celts, Balearians and Mauretanians.
This time his cavalry didn't outnumber the Roman cavalry and that would make the difference once more.

Phase II
A non moving elephant is of little use in a battle so Hannibal ordered his elephants to attack. The Romans were ready for them. They were familiar with attacking elephants and knew: a charging elephant was a frightening sight, but elephants were much more vulnerable than they seemed to be. The Velites attacked the animals with their javelins and lured them into the lanes between the the manipels, where those were attacked with javelins and spears from all sides, as was foreseen by Cornelius Scipio. Some of the elephants panicked, turned and charged their own troops.
At the same time the Roman horse attacked Hannibal's and met little trouble there. Hannibal's cavalry fled and was persued by Masinissa and Laelius.

Phase III
The Romans hacked their way through the first line of the mercenaries. The Carthaginians in the second line didn't give much help. This made the terrain full of dead bodies and Scipio recalled his Hastati and reorganized his army.
Hannibal prevented fleeing soldiers to retire into his main body and forced them to move to the flanks.

Phase IV
Cornelius Scipio now moved his Principii and Triarii to the flanks where they next to the Hastati moved forward in close ranks. Doing this he avoided his second and third line having to climb over the corpses and gave an answer to Hannibal's troops moving to the flanks. For quite some time Hannibal's troops stand firm and the outcome seemed indecisive. But then Masinissa and Laelius returned from their pursuit and attacked Hannibal's troops in the back. This was too much and Hannibal's army was routed. Hannibal himself fled to Hadrumetum