# Example 05 - Simultaneous Non-Linear Equations of Three Unknowns

**Problem**

Solve for *x*, *y*, and *z* from the following simultaneous equations.

$x^2 - yz = 3$ ← Equation (1)

$y^2 - xz = 4$ ← Equation (2)

$z^2 - xy = 5$ ← Equation (3)

**Solution**

*z*, Equation (2) by

*x*, and Equation (3) by

*y*

$x^2z - yz^2 = 3z$

$xy^2 - x^2z = 4x$

$yz^2 - xy^2 = 5y$

Add the above results

$4x + 5y + 3z = 0$ ← Equation (4)

Multiply Equation (1) by *y*, Equation (2) by *z*, and Equation (3) by *x*

$x^2y - y^2z = 3y$

$y^2z - xz^2 = 4z$

$xz^2 - x^2y = 5x$

Add the above results

$5x + 3y + 4z = 0$ ← Equation (5)

Eliminate *z* from Equations (4) and (5)

$x + 11y = 0$

$y = -\frac{1}{11}x$

Eliminate *y* from Equations (4) and (5)

$-13x - 11z = 0$

$z = -\frac{13}{11}x$

Substitute *y* = -(1/11)*x* and *z* = -(13/11)*x* to Equation (1)

$x^2 - \left( -\frac{1}{11}x \right)\left( -\frac{13}{11}x \right) = 3$

$x^2 - \frac{13}{121}x^2 = 3$

$\frac{108}{121}x^2 = 3$

$x^2 = \frac{121}{36}$

$x = \pm \frac{11}{6}$

$y = -\frac{1}{11}(\pm \frac{11}{6}) = \mp \frac{1}{6}$

$z = -\frac{13}{11}(\pm \frac{11}{6}) = \mp \frac{13}{6}$