Explicație pas cu pas:
a)
[tex]x(x + y + z) + y(x + y + z) + z(x + y + z) = 20 + 30 + 50 \\[/tex]
[tex](x + y + z)(x + y + z) = 100 \\ {(x + y + z)}^{2} = {10}^{2}[/tex]
I.
[tex]x + y + z = - 10[/tex]
[tex]x \cdot ( - 10) = 20 \implies x = - 2 \\y \cdot ( - 10) = 30 \implies y = - 3 \\z \cdot ( - 10) = 50 \implies z = - 5[/tex]
II.
[tex]x + y + z = 10[/tex]
[tex]x \cdot 10 = 20 \implies x = 2 \\y \cdot 10 = 30 \implies y = 3 \\z \cdot 10 = 50 \implies z = 5[/tex]
=> (x,y,z) ∈ {(-2; -3; -5); (2; 3; 5)}
b)
x, y, z > 0
[tex]\begin{cases} x \sqrt{yz} = 4 \\y \sqrt{zx} = 9 \\z \sqrt{xy} = 16\end{cases} \iff \begin{cases} {x}^{2}yz = {2}^{4} \\ {y}^{2}xz = {3}^{4} \\ {z}^{2}xy = {2}^{8} \end{cases}[/tex]
[tex]\begin{cases} yz = \frac{ {2}^{4} }{ {x}^{2} } \\ xy \cdot \frac{ {2}^{4} }{ {x}^{2} } = {3}^{4} \\ xz\cdot \frac{ {2}^{4} }{ {x}^{2} } = {2}^{8} \end{cases} \iff \begin{cases} yz = \frac{ {2}^{4} }{ {x}^{2} } \\ y = \frac{x \cdot {3}^{4}}{{2}^{4}} \\ z = x \cdot {2}^{4} \end{cases}[/tex]
[tex]\begin{cases} {x}^{4} = {( \frac{2}{3}) }^{4} \\ y = \frac{x \cdot {3}^{4}}{{2}^{4}} \\ z = x \cdot {2}^{4} \end{cases} \iff \begin{cases} x = \frac{2}{3} \\ y = \frac{27}{8} \\ z = \frac{32}{3} \end{cases}[/tex]
