Crocodile System

Solve for positive integers x, y & z :

$x + y - z = 4$

$x^2 - y^2 + z^2 = 4$

$xyz = 6$

#Algebra

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Comments

Usual Method:

There're no integral solutions. As $(x,y,z)$ will only be the 9 permutations of $(1,1,6)$ and $(1,2,3)$ . None of them satisfy.

Brute Force Method:

I also did some nonsense calculations to find the values:

$x+y = 4+z, xy=\dfrac{6}{z}$

Thus I obtained $x-y = \sqrt{\dfrac{z^3+8z^2+16z-24}{z}}$

$(x+y)(x-y) = 4 - z^2 \Rightarrow (x+y)^2(x-y)^2 = (4 - z^2)^2$

$\Rightarrow (4+z)^2 \left(\dfrac{z^3+8z^2+16z-24}{z} \right)= (4 - z^2)^2$

And then visit this.

No integral solutions obtained by Brute Force as well.