Find all real numbers x, y, z such that x*y*z^3 = 24, x*y^3*z = 54, and x^3*y*z = 6.
x*y*z^3 = 24 (1)
x*y^3*z = 54 (2)
x^3*y*z = 6 (3)
Divide (1) by (3) and we have that z^2 / x^2 = 4 ⇒ z / x = ±2
So
z = 2x or z = -2x
Divide (2) by (3) and we get that y^2 / x^2 = 9 ⇒ y /x = ±3
So
y = 3x or y = - 3x
Using (3) and substituting,we have that
x^3 *( 3x) ( 2x) = 6 x^3 (-2x) (3x) = 6
x^5 = 1 x^3 = -1
x = 1 x = -1
And we have these real solutions
x y z
1 3 2
1 -3 -2
-1 -3 2
-1 3 -2