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

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}.