Question

Solve for x and y: 1/x - 1/y = 1/3, 1/x2 + 1/y2 = 5/9.

Answer 1

1/x2 + 1/y2 -2/xy=9/81
therefore 1/xy =18/81
(1/x+1/y)^2 = 1
therefore 1/x + 1/y = 1 or -1

Answer 2

(1/x) - (1/y) = (1/3) ... (1/x²) + (1/y²) = (5/9)

(1/x) = A ... (1/y) = B

A - B = (1/3)

A² + B² = (5/9)

( A - B )² = (1/9)

A² - 2AB + B² = (1/9)

(5/9) -2AB = (1/9)

(4/9) = 2AB

( A + B )² = A² + 2AB + B²

( A + B )² = (5/9) + (4/9) =1

(1) when A + B = 1

A + B = 1

A - B = (1/3)

2A = (4/3) ... A = (2/3) ... B = (1/3)

( x , y ) = ( 3/2 , 3 ) valid

(2) when A + B = -1

A + B = -1

A - B = (1/3)

2A = (-2/3) ... A = (-1/3) ... B = (-2/3)

( x , y ) = ( -3 , -3/2 ) valid

(1/x) + (-1/y) = (1/3) ... (1/x²) + (1/y²) = (5/9)

α=(1/x)...β=(-1/y)

α+β=(1/3)...α²+β²=(5/9)

αβ=[(α+β)²-(α²+β²)]/2

 　=[(1/9)-(5/9)]/2=(-2/9)

t²-(1/3)t-(2/9)=0

[ t-(2/3) ][t+(1/3)]=0

(1) α=(2/3) , β=(-1/3)

 (x,y)=( 3/2 , 3 )

(2) α=(-1/3) , β=(2/3)

 (x,y)=( -3 , -3/2 )