# Verify if the solution of the system of equations (x-5)*y=4 and (x-y)o6=12 has integer solutions given that x*y=x+y-5 and xoy=xy-5(x+y)+30

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It is given that x*y = x + y - 5 and xoy = xy - 5(x + y) + 30

(x - 5)*y = 4

=> x - 5 + y - 5 = 4

=> x + y = 14 ...(1)

(x - y)o6 = 12

=> 6(x - y) - 5(x - y + 6) + 30 = 12

=> 6x - 6y - 5x + 5y - 30 + 30 = 12

=> x - y = 12 ...(2)

(1) + (2)

=> 2x = 26

=> x = 13

y = 1

**This proves that the system has integer solutions with x = 13 and y = 1**

We'll re-write the equations of the system, using the given laws of composition.

(x-5)*y=4, where

x*y=x+y-5

We'll substitute x by (x-5).

(x-5)*y = x - 5 + y - 5

We'll combine like terms:

x + y - 10 = 4

x + y = 10+4

x + y = 14 (1)

Now, we'll re-write the second equation of the system using the second law of composition.

(x-y)o6=12

xoy=xy-5(x+y)+30

We'll substitute x by (x-y) and y by 6:

(x-y)o6 = 6(x-y) - 5(x-y+6) + 30

We'll remove the brackets:

6x - 6y - 5x + 5y - 30 + 30 = 12

We'll eliminate and combine like terms:

x - y = 12 (2)

We'll solve the system formed from (1) and (2). We'll add (1) + (2):

x + y + x - y = 14 + 12

We'll eliminate and combine like terms:

2x = 26

**x = 13**

We'll substitute x in (1):

x + y = 14

13 + y = 14

y = 14 - 13

y = 1

**Since both values are integers, we'll validate them, therefore the solution of the system is: {13 , 1}.**

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