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МатематикаРешебник по алгебре 9 класс Мерзляк Задание 460
Задание 460
\[\boxed{\text{460\ (460).\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[1)\left\{ \begin{matrix} \frac{1}{x} - \frac{1}{y} = \frac{1}{12} \\ 2x - y = 2 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ }\]
\[\left\{ \begin{matrix} \frac{1}{x} - \frac{1}{y} = \frac{1}{12}\ \ | \cdot 12xy \\ y = 2x - 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ }\]
\[\left\{ \begin{matrix} 12y - 12x = xy \\ y = 2x - 2\ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ ;\ \ \ \]
\[x \neq 0;\ \ \ y \neq 0\ \ \ \]
\[12 \cdot (2x - 2) - 12x =\]
\[= x \cdot (2x - 2)\]
\[24x - 24 - 12x - 2x^{2} + 2x = 0\]
\[\left\{ \begin{matrix} - 2x^{2} + 14x - 24 = 0\ \ \ |\ \ :( - 2) \\ x \neq 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[x^{2} - 7x + 12 = 0\]
\(x_{1} + x_{2} = 7,\ \ \ \ \ \ \ \ \ \ \ x_{1} = 4\)
\[x_{1}x_{2} = 12,\ \ x_{2} = 3\]
\[\left\{ \begin{matrix} x = 4 \\ y = 6 \\ \end{matrix}\ \right.\ \text{\ \ \ \ \ \ \ }или\ \ \left\{ \begin{matrix} x = 3 \\ y = 4 \\ \end{matrix} \right.\ \]
\[Ответ:(4;6);\ \ (3;4).\]
\[2)\ \left\{ \begin{matrix} \frac{4}{x} + \frac{3}{y} = 1 \\ x + 5y = 3 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\]
\[\left\{ \begin{matrix} \frac{4}{x} + \frac{3}{y} = 1\ \ \\ x = 3 - 5y \\ \end{matrix} \right.\ \]
\[\frac{4^{\backslash y}}{3 - 5y} + \frac{3^{\backslash 3 - 5y}}{y} = 1\]
\[\frac{4y + 9 - 15y - 3y + 5y^{2}}{y(3 - 5y)} = 0\]
\[\left\{ \begin{matrix} 5y^{2} - 14y + 9 = 0 \\ y \neq \frac{3}{5}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ \]
\[5y^{2} - 14y + 9 = 0\]
\[D = 196 - 180 = 16\]
\[y_{1} = \frac{14 + 4}{10} = \frac{9}{5},\ \ \]
\[\ y_{2} = \frac{14 - 4}{10} = 1\]
\[\left\{ \begin{matrix} x = - 2 \\ y = 1\ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ }или\ \ \ \left\{ \begin{matrix} x = - 6\ \ \ \ \ \ \ \ \\ y = \frac{9}{5} = 1,8 \\ \end{matrix} \right.\ \]
\[Ответ:( - 2;\ - 1);\ ( - 6;1,8)\text{.\ }\]
