$x, y, z$ are positive real numbers which satisfy the following three equations.
$$x + 2y + z = 5(x + y)(y + z)$$ $$x + y + 2z = 7(y + z)(z + x)$$ $$2x + y + z = 6(z + x)(x + y)$$
Find the value of $(24)^3(xyz)$.
Okay , so here I'm probably not supposed to solve these equations , but rather find some "trick" to calculate $xyz$.
First thing that came to my mind was adding all three equations , which gives :
$$6 x^2+18 x y+18 x z+5 y^2+18 y z+7 z^2 = 4 (x+y+ z)$$
Now what? Any hints are apreciated. (This is not class-homework , I'm solving sample questions for a competitive exam )