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I have a system of 3 equations, each equation has 3 variables, $x,y,z$ :

$$x + y = z \\ x_1 + y_1 = z_1 \\x_2 + y_2 = z_2$$

Is $(\frac{x}{x_1} \cdot x_2 )+ (\frac{y}{y_1} \cdot y_2) = (\frac{z}{z_1})\cdot z_2$ true if all the variables are decimal and positive numbers ?

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  • $\begingroup$ @Arnaldo my bad, variables don't equal 0 and are not negative . $\endgroup$ – Anis Souames Mar 22 '17 at 14:42
  • $\begingroup$ So just write "positive numbers" $\endgroup$ – Arnaldo Mar 22 '17 at 14:43
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$$ \frac{xx_2}{x_1} + \frac{yy_2}{y_1} = \frac{zz_2}{z_1} $$ $$\Leftrightarrow \frac{xx_2}{x_1} + \frac{yy_2}{y_1} = \frac{(x+y)(x_2+y_2)}{(x_1+y_1)}$$ $$\Leftrightarrow \left( \frac{x+y}{x_1+y_1} - \frac{x}{x_1} \right)x_2 = \left( \frac{y}{y_1} - \frac{x+y}{x_1+y_1}\right)y_2 $$ $$\Leftrightarrow \frac{(yx_1 - xy_1)x_2}{x_1} = \frac{(yx_1 - y_1x)y_2}{y_1} $$ $$\Leftrightarrow yx_1-xy_1 = 0 \text{ or } \frac{x_2}{x_1} = \frac{y_2}{y_1}$$ $$\Leftrightarrow \frac{x_1}{x} = \frac{y_1}{y} \text{ or } \frac{x_2}{x_1} = \frac{y_2}{y_1}$$ So, it seems your equality holds only in the above two cases.

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