The 55-year-old added: “I just can’t believe this could happen to my beautiful boy. Our hearts are broken.”
A family source told The Sun Ms Duncan was “completely devastated” upon hearing that her son’s body had been recovered.
The 19-year-old apprentice bricklayer disappeared on June 17 on the island of Tenerife.
Mr Slater had been visiting Tenerife with friends to attend the NRG music festival.
He left the festival between 3am and 6am on Sunday, June 16, in the car of two other British men he had met that night, spending it at an Airbnb property they had rented in the island’s west side near Masca.
The cottage’s owner, Ofelia Medina Hernández, saw Mr Slater outside at around 8am the following morning.
After she told him that the next bus to his resort at Los Cristianos, in the south of the island, was two hours away, Mr Slater decided to start walking.
He later phoned his friend, Lucy Mae Law, who had been with him at the music festival, saying he was lost, thirsty, had one per cent charge left on his phone’s battery and had cut his leg on a cactus.
She called police and the apprentice bricklayer’s phone was last recorded at a point north of the cottage he had stayed at.
His body was found close to the location of the last “ping” of Mr Slater’s phone.
Mr Slater’s family, who flew out to Tenerife, continued their search after police said official efforts were ceasing on June 29.
More than half a million people joined a Facebook group where bizarre theories about Mr Slater’s disappearance were posted.
Some commentators suggested that police had pretended to shut down the initial search in the hope that amateur detectives would pack up and go home.
Spanish police previously declined to comment on media reports.