X+y+xy=3 y+z+yz=8 and z+x+zx=15 what is the value of 6xyz
Answers
Answer:
25 if x, y, z are positive
-385 if x, y, z are negative
Step-by-step explanation:
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 ]
Answer:
Step-by-step explanation:
