How to answer questions asking the number of points with integral co-ordinates within an area bounded by inequalities? For e.g. between 2x+7y<81, x>0, y>0.

Question

Answer 1

Pick's theorem

Answer 2

Pick's theorem

Answer 3

on y = 1, there are 37 points
y = 2, there are 33 points
y = 3, 29 points
y = 4, 26 points
y = 5, 22 points
y = 6, 19 points
y = 7, 15 points
y = 8, 12 points
y = 9, 8 points
y = 10, 5 points
y = 11, 1 point
Total = 204 points

Answer 4

on y = 1, there are 37 points
y = 2, there are 33 points
y = 3, 29 points
y = 4, 26 points
y = 5, 22 points
y = 6, 19 points
y = 7, 15 points
y = 8, 12 points
y = 9, 8 points
y = 10, 5 points
y = 11, 1 point
Total = 204 points

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How to answer questions asking the number of points with integral co-ordinates within an area bounded by inequalities? For e.g. between 2x+7y<81, x>0, y>0.