# Seeking the triples

Number Theory Level 4

$\frac{1}{x} + \frac{2}{y} - \frac{4}{z} = 1$

Find the sum $$S$$ of all triples $$(x, y, z)$$ for $$x, y, z$$ positive integers, $$x \neq 1$$ and $$y \neq 2$$ such that they satisfy the above expression.

Clarification:

If there are 3 triples $$(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3)$$ that satisfy the expression, then

$$S = x_1 + y_1 + z_1 + x_2 + y_2 + z_2 + x_3 + y_3 + z_3$$

×