xy xz yz
_____ = 1 _______ = 2 _______ = 3
x + y x + z y + z
We can convert these to this system:
x + y = xy ⇒ xy - y = x ⇒ y ( x - 1) = x ⇒ y = x / (x - 1) (1)
2x + 2z = xz ⇒ xz - 2z = 2x ⇒ z ( x - 2) = 2x ⇒ z = 2x / (x - 2) (2)
3y + 3z = yz (3)
Sub (1) and (2) into 3
3x / (x -1) + 3 (2x) /(x - 2) = (x * 2x) / [ (x - 1)(x - 2)]
Multiply through by (x -1) (x - 2)
3x ( x -2) + 6x ( x -1) = 2x^2 simplify
3x^2 - 6x + 6x^2 - 6x = 2x^2
7x^2 - 12x = 0 factor
x ( 7x - 12) = 0
The first factor x = 0 provides no solution
Setting the second factor to 0 and solving for x produces x = 12/7
And y = (12/7) / (12/7 -1) = (12/7) / (5/7) = 12/5
And z = 2 (12/7)/ (12/7 - 2) = (24/7) / (-2/7) = 24/ -2 = -12
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