Abuelazam, who was expressionless as he responded to questions from Judge Richard Hicks, first told Hicks he wanted to stay in Georgia and face the charges. But Hicks told him he would have to return to Michigan if he wanted to fight them.
After Hicks explained the process further, Abuelazam agreed to waive an extradition fight, a process that could take months, and go back to Michigan.
The stabbings began in May, with the suspect approaching men on empty roads at night and ask for directions or help with a broken-down car. Then he would pull out a knife, stab the victim and flee.
The stabbings were in Michigan, Ohio and Virginia.
Abuelazam was arrested on Wednesday night in Atlanta before a flight to Israel. He had apparently slipped out of police hands twice before he was finally arrested.
Just last week, he was arrested during a traffic stop on an outstanding warrant outside D.C. Police even found a knife and hammer inside his blazer. Still, he posted bail and was released.
Abuelazam's mother, Iyam al-Azzam, told Israel Radio that she talked to him by phone before he was supposed to board a plane in Atlanta "and he sounded the same as usual, quiet and calm."
"I do not believe these charges are true," she said. "Elias, my son, is a religious, God-fearing man who always assists anyone who needs help."
The attacks began in late spring, with all but four of the 18 occurring the Flint area. The others were in Leesburg, Va., and Toledo, Ohio. In one case, the attacker used a hammer.
The youngest victim was 15; the oldest 67. At least 15 victims were black, although there's no evidence that race played a role, Leyton said. A motive was not known.
On Friday, Israeli police said Abuelazam was believed to have stabbed a close acquaintance during an argument in a parked car in central Israel about six months ago. The commander said police dropped the case because the victim refused to cooperate with investigators.
The Associated Press contributed to this story.