Let x be the number of rounds in which she scored 3 and y the number in which she scored 5.
But she only scored in 10 of them so x+y=10.
Her total score was 34 so 3x+5y=34.
From the first equation y=10-x so we can substitute this in the other equation:
3x+5(10-x)=34, 3x+50-5x=34, 50-34=5x-3x, 16=2x, so x=8 and y=10-8=2.
So she scored 3 in 8 rounds and 5 in 2 rounds.
(There is enough information to find out the same for her friend, who scored 3 of each.)