The number of pairs of positive integers $(x,y)$ satisfying $3x+5y=47$ is:

$3x=47-5y$ must be positive and a multiple of 3. Testing $y=1$, 2, 3, … gives $x=14(y=1),x=9(y=4)$ and $x=4(y=7)$; every other $y$ up to 9 leaves a non-multiple of 3, and $y$ ≥ 10 makes $47-5y$ negative. So there are 3 pairs: (14, 1), (9, 4) and (4, 7).