Explicație pas cu pas:
x, y, z trei numere raționale
x şi y sunt direct proporţionale cu 5, respectiv 6
[tex]\frac{x}{5} = \frac{y}{6} = > x = \frac{5y}{6} \\ [/tex]
y şi z sunt invers proportionale cu 1/2 respectiv 1/7
[tex]\frac{y}{ \frac{1}{2} } = \frac{z}{ \frac{1}{7} } < = > 2y = 7z = > y = \frac{7z}{2} \\ [/tex]
[tex]x = \frac{5y}{6} = \frac{5}{6} \times \frac{7z}{2} = > x = \frac{35z}{12} \\ [/tex]
n ∈ N*
a)
[tex]\frac{x}{10} = \frac{y}{12} = \frac{z}{n} \\ \frac{35z}{12 \times 10} = \frac{7z}{2 \times 12} = \frac{z}{n} \\ \frac{7z}{24} = \frac{7z}{24} = \frac{z}{n} = > n = \frac{24}{7} \not \in \mathbb{ {N}^{*} }[/tex]
b)
[tex]y = \frac{6x}{5} = \frac{7z}{2} \\12x = 10y = 35z = k \\ x = \frac{k}{12};y = \frac{k}{10};z = \frac{k}{35}[/tex]
[tex]3x + y - z = \frac{3k}{12} + \frac{k}{10} - \frac{k}{35} \\ = \frac{35k + 14k - 4k}{140} = \frac{9k}{28} [/tex]
