Question

24. If xy = a, xz = b, and yz = c and none of these quantities is 0. then x+ y + z =
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Answer 1

Answer:

x+y+z = √{ac/b + ab/c + bc/a +2(a+b+c)}

Step-by-step explanation:

xy = a , yz = b , zx = c

ab= y²xz = y²c ; y² = ab/c

bc = z²xy = z²a ; z² = bc/a

ac = x²yz = x²b ,x² = ac/b

x+y+z =?

multiply by x , x² + xy + yz

multiply by y , y² + xy + yz

multiply by z , z ² + xz + yz

add all these

x²+y²+z²+ 2xy +2yz+2xz = (x+y+z)²

put values

ac/b + ab/c + bc/a +2(a+b+c) = (x+y+z)²

x+y+z = √{ac/b + ab/c + bc/a +2(a+b+c)}

Answer 2

xy = a,. xyz = az,. z = xyz/a

xz = b,. xyz = by,. y = xyz/b

yz = c,. xyz = cx,. x = xyz/c

also abc = xy*xz*yz = (xyz)^2

now

x + y + z

= (xyz/a) + (xyz/b) + (xyz/c)

= (xyz) ( 1/a + 1/b + 1/c)

=√ (abc). (ab + bc + ca)/abc

= (ab + bc + ca)/√abc

or

(ab/c)^(1/2) + (bc/a)^(1/2) + (ab/c)^(1/2)