Question

xy+x+y=5 xy+yz=11 zx+z+x=7 x:y:z:=?

Answer 1

Answer:

25 if x, y, z are positive

-385 if x, y, z are negative

x + y + xy = 3 => 1 + x + y + xy = 4 => ( 1 + x ) ( 1 + y ) = 4

y + z + yz = 8 => 1 + y + z + yz = 9 => ( 1 + y ) ( 1 + z ) = 9

z + x + zx = 15 => 1 + z + x + zx = 16 => ( 1 + z ) ( 1 + x ) = 16

Multiplying the first and third, then dividing by the second:

( 1 + x )² = 4 × 16 / 9 = 64/9 => 1 + x = ±8/3 => x = -1 ± 8/3 = 5/3 or -11/3.

From the first equation:

x + ( 1 + x ) y = 3 => y = ( 3 - x ) / ( 1 + x )

From the third equation:

x + ( 1 + x ) z = 15 => z = ( 15 - x ) / ( 1 + x )

So...

6xyz = 6x ( 3 - x ) ( 15 - x ) / ( 1 + x )²

= 6x ( 3 - x ) ( 15 - x ) / ( 64/9 )

= 27x ( 3 - x ) ( 15 - x ) / 32

= 3x ( 9 - 3x ) ( 45 - 3x ) / 32

Case 1 : x = 5/3

6xyz = 5 × ( 9 - 5 ) × ( 45 - 5 ) / 32

= 5 × 4 × 40 / 32

= 5 × 1 × 5

= 25

[ Although not needed, we also have y = 1/2, z = 5 ]

Case 2 : x = -11/3

6xyz = -11 × ( 9 - -11) × ( 45 - -11 ) / 32

= -11 × 20 × 56 / 32

= -11 × 5 × 7

= -385

[ Although not needed, we also have y = -5/2, z = -7 ]

Step-by-step explanation:

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