Add the three equations
$x(y + z) + y(x + z) + z(x + y) = 12 + 6 + 10$
$(xy + xz) + (xy + yz) + (xz + yz) = 28$
$2xy + 2yz + 2xz = 28$
$xy + yz + xz = 14$ → Equation (4)
Equation (4) - Equation (3)
$(xy + yz + xz) - z(x + y) = 14 - 10$
$(xy + yz + xz) - (xz + yz) = 4$
$xy = 4$ → Equation (5)
Equation (4) - Equation (1)
$(xy + yz + xz) - x(y + z) = 14 - 12$
$(xy + yz + xz) - (xy + xz) = 2$
$yz = 2$ → Equation (6)
Equation (4) - Equation (2)
$(xy + yz + xz) - y(x + z) = 14 - 6$
$(xy + yz + xz) - (xy + yz) = 8$
$xz = 8$ → Equation (7)
Multiply equations (5), (6), and (7)
$(xy)(yz)(xz) = 4(2)(8)$
$x^2 y^2 z^2 = 64$
$(xyz)^2 = 64$
$xyz = \pm 8$ → Equation (8)
Divide Equation (6) from Equation (8)
$\dfrac{xyz}{yz} = \dfrac{\pm 8}{2}$
$x = \pm 4$ answer
Divide Equation (7) from Equation (8)
$\dfrac{xyz}{xz} = \dfrac{\pm 8}{8}$
$y = \pm 1$ answer
Divide Equation (5) from Equation (8)
$\dfrac{xyz}{xy} = \dfrac{\pm 8}{4}$
$z = \pm 2$ answer
