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what is the relationship between first differences and slope?

First differences means take the ordered pairs in increasing order for their x coordinates (assuming the x's go by ones), then subtract each y (except the last) from the y that comes after it.  Keep doing that and you get the first differences.  (The second differences would be what you get if you then subtract each of the first differences from the one that came after it (TMI probably but ...)

Say the points were (1,3), (2,7), (3,11) and (4,15).  The first differences are

7-3 = 4

11-7 = 4

15 - 11 = 4  all 4.  This would also be the slope.  So that is the relationship - they are the same.