ГДЗ по алгебре 8 класс Макарычев ФГОС Задание 706

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\[\textbf{а)}\ \left\{ \begin{matrix} y - 2x = 2\ \\ 5x^{2} - y = 1 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} y = 2x + 2\ \ \ \ \ \ \ \ \ \\ 5x^{2} - 2x - 2 = 0 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} y = 2x + 2\ \ \ \ \ \ \ \ \ \\ 5x^{2} - 2x - 3 = 0 \\ \end{matrix} \right.\ \]

\[5x^{2} - 2x - 3 = 0\]

\[D_{1} = 1 + 15 = 16\]

\[x_{1} = \frac{1 + 4}{5} = 1;\]

\[x_{2} = \frac{1 - 4}{5} = - \frac{3}{5} = - 0,6.\]

\[1)\ \left\{ \begin{matrix} x_{1} = 1 \\ y_{1} = 4 \\ \end{matrix} \right.\ ;\ \ 2)\ \left\{ \begin{matrix} x_{2} = - 0,6 \\ y_{2} = 0,8\ \ \ \\ \end{matrix} \right.\ \]

\[Ответ:( - 0,6;0,8);(1;4).\]

\[\textbf{б)}\ \left\{ \begin{matrix} x - 2y^{2} = 2 \\ 3x + y = 7\ \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x = 2y^{2} + 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 3 \cdot \left( 2y^{2} + 2 \right) + y = 7 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x = 2y^{2} + 2\ \ \ \ \ \ \ \ \\ 6y^{2} + y - 1 = 0 \\ \end{matrix} \right.\ \]

\[6y^{2} + y - 1 = 0\]

\[D = 1 + 6 \cdot 4 = 25\]

\[y_{1} = \frac{- 1 - 5}{12} = - \frac{6}{12} =\]

\[= - \frac{1}{2} = - 0,5;\]

\[y_{2} = \frac{- 1 + 5}{12} = \frac{4}{12} = \frac{1}{3}.\]

\[1)\ \left\{ \begin{matrix} y_{1} = - 0,5 \\ x_{1} = 2,5\ \ \ \\ \end{matrix} \right.\ ;\ \ 2)\ \left\{ \begin{matrix} y_{2} = \frac{1}{3}\text{\ \ \ } \\ x_{2} = 2\frac{2}{9} \\ \end{matrix} \right.\ .\]

\[Ответ:(2,5; - 0,5);\ \ \left( 2\frac{2}{9};\frac{1}{3} \right).\]

\[\textbf{в)}\ \left\{ \begin{matrix} x^{2} - 3y^{2} = 52 \\ y - x = 14\ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} y = x + 14\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - 3 \cdot (x + 14)^{2} - 52 = 0 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} y = x + 14\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ - 2x^{2} - 84x - 640 = 0 \\ \end{matrix} \right.\ \]

\[2x^{2} + 84x + 640 = 0\ \ \ \ |\ :2\]

\[x^{2} + 42x + 320 = 0\]

\[x_{1} + x_{2} = - 42;\ \ x_{1} \cdot x_{2} = 320\]

\[x_{1} = - 32;\ \ x_{2} = - 10;\]

\[1)\ x_{1} = - 32;\ \ \ \ \ y_{1} = - 18;\]

\[2)\ x_{2} = - 10;\ \ \ \ y_{2} = 4.\]

\[Ответ:( - 32; - 18);\ \ ( - 10;4).\]

\[\textbf{г)}\ \left\{ \begin{matrix} 3x^{2} + 2y^{2} = 11 \\ x + 2y = 3\ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x = 3 - 2y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 3 \cdot (3 - 2y)^{2} + 2y^{2} = 11 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x = 3 - 2y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 14y² - 36y + 16 = 0 \\ \end{matrix} \right.\ \]

\[7y^{2} - 18y + 8 = 0\]

\[D_{1} = 81 - 56 = 25\]

\[y_{1} = \frac{9 + 5}{7} = 2;\ \ \ \]

\[y_{2} = \frac{9 - 5}{7} = \frac{4}{7}.\]

\[1)\ \ y_{1} = 2;\ \ x_{1} = - 1;\]

\[2)\ y_{2} = \frac{4}{7};\ \ x_{2} = 1\frac{6}{7}.\]

\[Ответ:( - 1;2);\ \ \left( 1\frac{6}{7};\frac{4}{7} \right).\]

\[\textbf{д)}\ \left\{ \begin{matrix} x^{2} + y^{2} = 100 \\ 3x = 4y\ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} x = \frac{4}{3}\text{y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \frac{16}{9}y^{2} + y^{2} = 100 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} y^{2} = 36 \\ x = \frac{3}{4}\text{y\ \ } \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} y_{1} = 6 \\ x_{1} = 8 \\ \end{matrix} \right.\ \text{\ \ \ }или\ \left\{ \begin{matrix} y_{2} = - 6 \\ x_{2} = - 8. \\ \end{matrix} \right.\ \]

\[Ответ:( - 8; - 6);\ \ (8;6).\]

\[\textbf{е)}\ \left\{ \begin{matrix} 2x^{2} - y^{2} = 32 \\ 2x - y = 8\ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} y = 2x - 8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 2x^{2} - \left( 4x^{2} - 32x + 64 \right) - 32 = 0 \\ \end{matrix} \right.\ \Longrightarrow\]

\[\Longrightarrow \left\{ \begin{matrix} y = 2x - 8\ \ \ \ \ \ \ \ \ \ \ \ \ \\ x² - 16x + 48 = 0 \\ \end{matrix} \right.\ \ \]

\[x^{2} - 16x + 48 = 0\]

\[D_{1} = 64 - 48 = 16\]

\[x_{1} = 8 + 4 = 12;\ \ \ \]

\[x_{2} = 8 - 4 = 4.\]

\[1)\ x_{1} = 12;\ \ \ y_{1} = 16;\]

\[2)\ x_{2} = 4;\ \ \ \ \ \ y_{2} = 0.\]

\[Ответ:(4;0);(12;16).\]