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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.

the right system of equations to describe the situation would

be on the form:

x1 = 8000 + y1*t

and

x2 = 8000 + y2*t

where x1 and x2 represents the total money of imogene and her

friend respectively at the end of t years.

now for the value of amount earned, y1 and y2:

y1=8000*0.08

y2=2000*√(t-2)