Example 04 - Simultaneous Non-Linear Equations of Three Unknowns

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