Solve the following equations :
$3 x + y - 2 z = 0, 4 x - y - 3 z = 0,$
$x^{3} + y^{3} + z^{3} = 467.$

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Solution

$3 x + y - 2 z = 0 . . . (1)$

$4 x - y - 3 z = 0 . . . (2)$

$x^{3} + y^{3} + z^{3} = 467 . . . (3)$

Solving $(1)$ and $(2)$ by the method of cross-multiplication, we obtain
$\frac{x}{- 3 - 2} = \frac{y}{- 8 + 9} = \frac{z}{- 3 - 4}$

or $\frac{x}{5} = \frac{y}{- 1} = \frac{z}{7} = k,$ say

$∴ x = 5 k, y = - k, z = 7 k .$

Substituting in $(3),$ we get
$125 k^{3} - k^{3} + 343 k^{3} = 467$

or $467 k^{3} = 467$ or $k = 1.$

$∴$ The solution is $x = 5, y = - 1, z = 7.$

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If x + y + z = xyz. Then

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x + y + z = x y z

\frac{3 x - x^{3}}{1 - 3 x^{2}} + \frac{3 y - y^{3}}{1 - 3 y^{2}} + \frac{3 z - z^{3}}{1 - 3 z^{2}} = \frac{3 x - x^{3}}{1 - 3 x^{2}} \cdot \frac{3 y - y^{3}}{1 - 3 y^{2}} \cdot \frac{3 z - z^{3}}{1 - 3 z^{2}}

x + y + z = 12, x^{2} + y^{2} + z^{2} = 50, x^{3} + y^{3} + z^{3} = 216.

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