Peter picked one pepper more than Paul. Pat picked one pepper more than Pam. Peter and Paul picked 10 more peppers than Pat and Pam. Peter, Paul, Pat and Pam picked 60 peppers. How many peppers did Peter pick?

a = Peter, b = Paul, c = Pat, d = Pam
u = a + b, v = c + d

Peter picked one pepper more than Paul -----: a - b = 1 [1]
Pat picked one pepper more than Pam -------: c - d = 1 [2]
Peter & Paul picked 10 more than Pat & Pam : u - v = 10 [3]
Peter, Paul, Pat & Pam picked 60 peppers ---: u + v = 60 [4]

[3] + [4]: 2u = 70 ; u = 35 ; so:
a + b = 35
a - b = 1 [1] ; add 'em up:
========
2a = 36
So a = 18, b = 17

By default, c + d = 60 - 18 - 17 = 25
So (using [2]) c = 13 and d = 12

Peter=18, Paul=17, Pat=13, Pam = 12

For the "impatients(!)" who want something short:
a = Peter, b = Pat

2a - 1 + 2b - 1 = 60 : 2a + 2b = 62 : a + b = 31 [1]

2a - 1 - 2b + 1 = 10 : 2a - 2b = 10 : a - b = 5 [2]

[1] + [2] : 2a = 36 : a = 18 = Peter; so Paul = 17

Pat and Pam = 13 and 12 (by default).

Very nicely done, Denis!