x/y = 2/3 ( equation 1)

2 is subtracted from x and 8 from y so it becomes reciprocal then

x-2/y-8 = 3/2(equation 2)

now simplify equation 1

x/y = 2/3

3x = 2y ( cross multiply)

3x -2y = 0

equation 2

x-2/y-8 = 3/2

2x - 4 =3y - 24

2x - 3y = -24 +4

2x - 3y = -20

now subtract 1 from 2

3x - 2y = 0

2x - 3y = -20

(-) (+) (+)

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x + y = 20

y = 20 - x

substitute y in equation 1

3x - 2(20 - x) = 0

3x - 40 + 2x = 0

3x + 2x = 40

5x = 40

x = 40/5

x = 8

substitute x = 8 in equation 2

2x - 3y = -20

2(8) - 3y = -20

16 - 3y = -20

-3y = -20 -16

-3y = -36

y = -36/-3

y = 12

so the no.s are 8 and 12

verification also there

x/y = 8 / 12 which is same as 2/3 ( cancelled by 4)

x - 2 / y - 8 = 3 / 2

8 - 2 / 12 - 8

6 / 4

3 / 2 ( cancelled by 2)

*VERIFIED*