Question

tim cac so nguyen duong x,y biet

1/2x+1/2y+1/xy=1/2

Answer 1

\(\dfrac{1}{2x}+\dfrac{1}{2y}+\dfrac{1}{xy}=\dfrac{1}{2}\)

\(\dfrac{y}{2xy}+\dfrac{x}{2xy}+\dfrac{2}{2xy}=\dfrac{xy}{2xy}\)

=> x + y + 2 = xy

x + y - xy = -2

x.( 1 - y ) + y = -2

x.( 1 - y ) - ( 1 - y ) = -2 - 1

( 1 - y ).( x - 1 ) = -3

- ( y - 1 ).( x - 1) = -3

=> ( y - 1 ).( x - 1 ) = 3

=> ( y - 1 ) ; ( x - 1 ) \(\in\) Ư( 3 ) = { 1; -1; 3; -3 }

Ta có bảng sau

y - 1	1	-1	3	-3
y	2	0	4	-2
x - 1	3	-3	1	-1
x	4	-2	2	0

Vậy ( x ; y ) \(\in\) { ( 4 ; 2 ); ( -2 ; 0 ); ( 2; 4 ); ( 0; -2 ) }