\[\boxed{\text{770\ (770).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[C_{10}^{6} = \frac{10!}{6! \cdot 4!} = \frac{7 \cdot 8 \cdot 9 \cdot 10}{2 \cdot 3 \cdot 4} =\]
\[= 210\ (способов) - выбрать\ \]
\[6\ книг\ из\ 10.\]
\[\boxed{\text{770.}\text{\ }\text{ОК\ ГДЗ\ -\ домашка\ на\ 5}}\]
\[\textbf{а)}\ \left\{ \begin{matrix} x^{2} + y + 8 = xy \\ y - 2x = 0\ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} y = 2x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} + 2x + 8 - x \cdot 2x = 0 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} y = 2x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ {- x}^{2} + 2x + 8 = 0 \\ \end{matrix} \right.\ \]
\[x^{2} - 2x - 8 = 0\]
\[D = 4 + 32 = 36\]
\[x_{1} = \frac{2 + 6}{2} = 4,\ \ \]
\[x_{2} = \frac{2 - 6}{2} = - 2,\]
\[1)\ x_{1} = 4,\ \ y = 8,\]
\[2)\ x_{2} = - 2,\ \ y_{2} = - 4.\]
\[Ответ:(4;8);\ \ ( - 2;\ - 4).\]
\[\textbf{б)}\ \left\{ \begin{matrix} x^{2} - y^{2} = 16 \\ x + y = 8\ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x + y = 8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ (x - y)(x + y) = 16 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x + y = 8\ \ \ \ \ \ \ \\ x - y = 16\ :8 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x + y = 8 \\ x - y = 2 \\ \end{matrix} \right.\ ( + ) \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} 2x = 10\ \ \\ x - y = 2 \\ \end{matrix} \right.\ \Longrightarrow \left\{ \begin{matrix} x = 5 \\ y = 3. \\ \end{matrix} \right.\ \]
\[Ответ:(5;3).\]
\[\textbf{в)}\ \left\{ \begin{matrix} x + y = 5\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - xy + y^{2} = 13 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} y = 5 - x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 3x² - 15x + 12 = 0 \\ \end{matrix} \right.\ \]
\[x^{2} - 5x + 4 = 0\]
\[D = 25 - 16 = 9\]
\[x_{1} = \frac{5 + 3}{2} = 4,\]
\[\text{\ \ }x_{2} = \frac{5 - 3}{2} = 1,\]
\[1)\ x_{1} = 4,\ \ y_{1} = 1,\]
\[2)\ x_{2} = 1,\ \ y_{2} = 4.\]
\[Ответ:(4;1);\ \ \ (1;4).\]
\[\textbf{г)}\ \left\{ \begin{matrix} x^{2} + y^{2} + 3xy = 1 \\ 3y + x = 0\ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x = - 3y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 9y^{2} + y^{2} - 9y^{2} = 1 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x = - 3y \\ y² = 1\ \ \ \ \\ \end{matrix} \right.\ \]
\[1)\ y_{1} = 1,\ \ x_{1} = - 3,\]
\[2)\ y_{2} = - 1,\ \ x_{2} = 3.\]
\[Ответ:( - 3;1);\ \ (3;\ - 1).\]
\[\Longrightarrow \left\{ \begin{matrix} 4x^{2} + 10x - 6y = - 24 \\ 6y = 24 + 21x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} 2y = 8 + 7x\ \ \ \ \\ 4x² - 11x = 0 \\ \end{matrix} \right.\ \]
\[4x^{2} - 11x = 0\]
\[x(4x - 11) = 0\]
\[x_{1} = 0,\ \ x_{2} = \frac{11}{4},\]
\[1)\ x_{1} = 0,\ \ y_{1} = 4,\]
\[2)\ x_{2} = \frac{11}{4} = 2\frac{3}{4},\]
\[\text{\ \ }y_{2} = \frac{109}{8} = 13\frac{5}{8}.\]
\[Ответ:(0;4);\ \ \left( 2\frac{3}{4};13\frac{5}{8} \right).\]
\[\textbf{е)}\ \left\{ \begin{matrix} y^{2} - 6x + y = 0\ \ \ \ \ \ \ \ \ \ \ \\ 2x - \frac{1}{2}y = 1\ \ \ | \cdot 3\ \ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} 6x = 3 + \frac{3}{2}\text{y\ \ \ \ \ \ \ \ \ \ \ \ \ } \\ y^{2} - 3 - \frac{3}{2y} + y = 0 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} 2x = 1 + \frac{1}{2}\text{y\ \ \ \ \ } \\ 2y² - y - 6 = 0 \\ \end{matrix} \right.\ \]
\[2y^{2} - y - 6 = 0\]
\[D = 1 + 48 = 49,\]
\[y_{1} = \frac{1 + 7}{4} = 2,\]
\[\text{\ \ }y_{2} = \frac{1 - 7}{4} = - 1,5;\]
\[1)\ y_{1} = 2,\ \ x_{1} = 1,\]
\[2)\ y_{2} = - 1,5,\ \ x_{2} = \frac{1}{8}.\]
\[Ответ:(1;2);\ \ \left( \frac{1}{8};\ - 1,5 \right).\]