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Set up equations like this (keep in mind that perimeter of a rectangle is all sides added up, so two lengths and two widths):

2l + 2w = 340
l = w + 10

Plug (w + 10) in for l in the first equation and solve for w:

2(w + 10) + 2w = 340
2w + 20 + 2w = 340
4w + 20 = 340
4w = 320
w = 80

The width of the field is 80 yd.  Plug 80 in for w in either equation:

l = 80 + 10
l = 90

The length of the field is 90 yd.