3X+7Y = 188 ; What integer solution (X, Y) yields the smallest possitive difference (Y-X)?

Answer 1

3X + 7Y = 188

Calculate X :
X = (188 - 7Y) / 3

X = 188 / 3 - 7Y / 3

X = 62 + 2 / 3 - 2Y - Y / 3

X = 62 - 2Y - (Y - 2) / 3

X is an integer, so all these terms are integers.
So let (Y - 2) / 3 = J, where J is an integer.
Rearranging to get Y, gives :
Y = 3J + 2.

Substituting this into the last equation for X, gives :
X = 58 - 7J

These are parametric equations for X and Y,
in which it can be seen that J must be an
integer from 0 to 8, if X and Y are positive.

Now calculate Y - X :

Y - X = 3J + 2 - (58 - 7J)
= 10J - 56

This has to be greater than zero :
10J - 56 > 0
10J > 56
J > 5.6

The smallest J greater than 5.6 is J = 6.

Substituting this value into the parametrics gives :
X = 16 and Y = 20,
for the smallest difference, Y - X = 4.

Answer 2

3X+7Y=188 Or 7Y=188 - 3X Or Y=188/7 - 3X/7. So Y- X= 188/7- 3X/7 - X =188/7 - 10X/7. This should be the smallest positive for some integer solution (X,Y) For this to be the smallest positive, we have: 188/7 - 10X/7>0 Or 188-10X>0. Or X < 18.8. For (Y -X) to be to be smallest positive, Y should be the smallest and X should be maximum.. then minimum value of Y=188/7 -3X/7 =(188- 56.4)/7 =18.8 Therefore, the integer solution should be such that X is less or equal to 18 and Y is greater than or equal to 19. If we put X=18 in 3X+7Y=188, and solve for Y, it does not lead to an integer. If we put X=17 in 3X+7Y=188, and solve for Y, it does not lead to an integer. If we put X=16 in 3X+7Y=188, and solve for Y, it does lead to an integer value Y=20 which is greater than 19 as desired. Therefore, the integer solution (X,Y) with smallest positive difference is (16,20) Answer=(16,20) and the difference Y - X = 4 .

Answer 3

You can work to the correct answer starting from the incorrect (2,26), because increasing X by 7 and decreasing Y by 3 leaves the result unchanged, e.g.

3*(2+14) + 7*(26 - 6) = 48 + 140 = 188

So (16, 20) could be the answer, differing only by 4.

In fact it is the answer, because the next pair, (23, 17) differ by -6.

Answer 4

(0,Y=188)