It happened around 12:40 p.m. at 5200 Flower St. near Exposition Park.
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A spokesperson with the Los Angeles Police Department told Eyewitness News the suspect hit a curb and drove into two tents.
"His car, according to witnesses, launched somewhat airborne ... when his car comes back down and lands in the street, he appears to lose control," said the spokesperson. "He ends up running off the roadway, going up over the curb, and into a homeless encampment."
Police said the man killed was inside one of the tents and the woman was in another tent next to it.
The LAPD spokesperson said the woman was sleeping when she was killed.
Police say the suspect tried to run away but was apprehended shortly after. Investigators don't believe the suspect knew the victims.
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"Just tragic and real unfortunate," said the LAPD spokesperson. "These poor victims are really just minding their own business, you know? At home, basically in their tents. One's asleep and one's just resting."
Meanwhile, a witness told ABC7 she saw the suspect getting chased by police. She said the crash happened when he hit the corner.
Police said the suspect sustained minor injuries. He's likely facing manslaughter charges and possible murder charges, according to investigators.
This incident remains under investigation.