5m+y=22,
10y-2m=16, so 5y-m=8 and m=5y-8.
Substitute for m in the other equation:
5(5y-8)+y=22, 25y-40+y=22, 26y=62, y=62/26=31/13.
m=5y-8=155/13-8=(155-104)/13=51/13.
CHECK:
5m+y=255/13+31/12=286/13=22 OK
10y-2m=310/13-102/13=208/13=16 OK
This looks like an awkward solution so perhaps there’s an error in the question. Let’s suppose that the second equation had been 10m-2y=16, so 5m-y=8, then y=5m-8 and substituting into the first equation we get:
5m+5m-8=22, so 10m=30 and m=3, making y=15-8=7.