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Second-placed City broke the deadlock inside six minutes with a Silva free kick from the edge of the box.
Bournemouth went close to an equaliser from a free kick of their own but Junior Stanislas’s effort was saved by City keeper Ederson, who pushed it onto the post.
City doubled their advantage when Jesus slalomed past two defenders to score from close range, with Silva collecting his 10th assist of the league campaign.
Substitute David Brooks gave Bournemouth a lifeline with an 88th-minute goal but City held on to collect all three points and extend their winning streak at home to six games.
Pep Guardiola’s City have already secured a second-placed finish behind title winners Liverpool, while Bournemouth remain 18th and three points adrift of the safety zone.