# Example 02 - Simultaneous Non-Linear Equations of Three Unknowns

**Problem**

Find the value of *x*, *y*, and *z* from the following equations.

$xy = -3$ → Equation (1)

$yz = 12$ → Equation (2)

$xz = -4$ → Equation (3)

**Solution**

Multiply the three equations

$(xy)(yz)(xz) = -3(12)(-4)$

$(xy)(yz)(xz) = -3(12)(-4)$

$x^2 y^2 z^2 = 144$

$(xyz)^2 = 144$

$xyz = \pm 12$ → Equation (4)

Equation (4) divided by Equation (2)

$\dfrac{xyz}{yz} = \dfrac{\pm 12}{12}$

$x = \pm 1$ *answer*

Equation (4) divided by Equation (3)

$\dfrac{xyz}{xz} = \dfrac{\pm 12}{-4}$

$y = \pm 3$ *answer*

Equation (4) divided by Equation (1)

$\dfrac{xyz}{xy} = \dfrac{\pm 12}{-3}$

$z = \pm 4$ *answer*

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