Example 03 - Simultaneous Non-Linear Equations of Three Unknowns

Add the three equations
$x(x + y + z) + y(x + y + z) + z(x + y + z) = -36 + 27 + 90$
 

Factor (x + y + z) in the left side and do the operation in the right side
$(x + y + z)(x + y + z) = 81$

$(x + y + z)^2 = 81$

$x + y + z = \pm 9$   →   Equation (4)
 

Divide Equation (4) from Equation (1)
$\dfrac{x(x + y + z)}{x + y + z} = \dfrac{-36}{\pm 9}$

$x = \pm 4$           answer
 

Divide Equation (4) from Equation (2)
$\dfrac{y(x + y + z)}{x + y + z} = \dfrac{27}{\pm 9}$

$y = \pm 3$           answer
 

Divide Equation (4) from Equation (1)
$\dfrac{z(x + y + z)}{x + y + z} = \dfrac{90}{\pm 9}$

$z = \pm 10$           answer