Example 05 - Simultaneous Non-Linear Equations of Three Unknowns

Multiply Equation (1) by 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)

4 * Equation (4) - 3 * Equation (5)
$x + 11y = 0$

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

 

Eliminate y from Equations (4) and (5)

3 * Equation (4) - 5 * Equation (5)
$-13x - 11z = 0$

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

 

Substitute y = -(1/11)x and z = -(13/11)x to Equation (1)
$x^2 - \left( -\frac{1}{11}x \right)\left( -\frac{13}{11}x \right) = 3$

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

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

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

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

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

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