tex]...

August 2022 1 13 Report

Про числа x, y, z известно, что
[tex] \frac{ \frac{y}{3} + \frac{z}{6} + x }{z} = 1[/tex]
[tex] \frac{ \frac{x}{3} + \frac{11y}{12} + \frac{z}{6} }{y} = 1[/tex]
Найти
[tex] \frac{z}{x} [/tex]

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Reideen

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1) Преобразуем первую дробь:

$\displaystyle \frac{\frac{y}{3}+\frac{z}{6} +x }{z} =1 \; \Leftrightarrow \; \frac{\frac{2y}{6}+\frac{z}{6} +\frac{6x}{6} }{z} =1 \; \Leftrightarrow \;\\\; \Leftrightarrow \; \frac{\frac{2y+z+6x}{6}}{z}=1 \; \Leftrightarrow \; \frac{2y+z+6x}{6z}=1 \; \Leftrightarrow \; \\\; \Leftrightarrow \; 2y+z+6x=6z \; \Leftrightarrow \; \boxed{2y+6x=5z} \; (1)$

2) Преобразуем вторую дробь:

$\displaystyle \frac{\frac{x}{3}+\frac{11y}{12} +\frac{z}{6} }{y} =1 \; \Leftrightarrow \; \frac{\frac{4x}{12}+\frac{11y}{12} +\frac{2z}{12} }{y} =1 \; \Leftrightarrow \;\\\; \Leftrightarrow \; \frac{\frac{4x+11y+2z}{12} }{y} =1 \; \Leftrightarrow \; \frac{4x+11y+2z}{12y} =1 \; \Leftrightarrow \;\\\; \Leftrightarrow \; 4x+11y+2z=12y \; \Leftrightarrow \; \boxed{4x+2z=y} \; (2)$

3) Подставим второе уравнение в первое и преобразуем полученное уравнение:

$\displaystyle 2\cdot (4x+2z)+6x=5z\\8x+4z+6x=5z\\14x=z\\\boxed{\frac{z}{x} =14}$

Ответ: $\displaystyle \frac{z}{x} =14$

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