9. Let x and y be positive integers such that x y + y x − 7 x − 7 y + 7 xy = 7 .
Determine x+y.
【Solution 1】
Let a = x , b = y and c = 7 . Then a b
2
+ b a
2
− ca − + cb abc = c
2
. Hence
= + − − + −
2
2
2
a b b a ca cb abc c
0
= + + − + +
( ) ( )
ab a b c c a b c
= − + +
( )( )
ab c a b c
Since a + + = b c x + y + 7 > 0 , we must have ab − = c 0 or xy = 7 .
Since x and y are positive integers, (x, y)=(1, 7) or (7, 1). In either case, x+y=8.
【Solution 2】
We know x y + y x + 7 xy = + 7 7 x + 7 y .
That is, xy ( y + x + 7 ) = 7 ( 7 + x + y ) .
Hence xy = 7 , this imply xy = 7 .
ANS: 8
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