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МатематикаРешебник по алгебре 9 класс Макарычев Задание 433
Задание 433
\[\boxed{\text{433\ (433).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
Пояснение.
Решение.
\[\textbf{а)}\ \left\{ \begin{matrix} y - 2x = 2\ \\ 5x^{2} - y = 1 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} y = 2x + 2\ \ \ \ \ \ \ \ \ \\ 5x^{2} - 2x - 2 = 0 \\ \end{matrix} \right.\ \Longrightarrow \ \]
\[\Longrightarrow \left\{ \begin{matrix} y = 2x + 2\ \ \ \ \ \ \ \ \ \\ 5x^{2} - 2x - 3 = 0 \\ \end{matrix} \right.\ \]
\[5x^{2} - 2x - 3 = 0\]
\[D = 1 + 15 = 16\]
\[x_{1,2} = \frac{1 \pm 4}{5} = 1;\ - \frac{3}{5}.\]
\[1)\ \left\{ \begin{matrix} x_{1} = 1 \\ y_{1} = 4 \\ \end{matrix} \right.\ \text{\ \ }\]
\[2)\ \left\{ \begin{matrix} x_{2} = - \frac{3}{5} \\ y_{2} = \frac{4}{5}\text{\ \ \ .} \\ \end{matrix} \right.\ \]
\[\textbf{б)}\ \left\{ \begin{matrix} x - 2y^{2} = 2 \\ 3x + y = 7\ \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x = 2y^{2} + 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 3 \cdot \left( 2y^{2} + 2 \right) + y = 7 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x = 2y^{2} + 2\ \ \ \ \ \ \ \ \\ 6y^{2} + y - 1 = 0 \\ \end{matrix} \right.\ \]
\[6y^{2} + y - 1 = 0\]
\[D = 1 + 6 \cdot 4 = 25\]
\[y_{1,2} = \frac{- 1 \pm 5}{12} = - \frac{1}{2};\frac{1}{3}\text{.\ }\]
\[1)\ \left\{ \begin{matrix} y_{1} = - \frac{1}{2} \\ x_{1} = 2\frac{1}{2} \\ \end{matrix} \right.\ \]
\[2)\ \left\{ \begin{matrix} y_{2} = \frac{1}{3}\text{\ \ \ } \\ x_{2} = 2\frac{2}{9} \\ \end{matrix} \right.\ .\]
\[\textbf{в)}\ \left\{ \begin{matrix} x^{2} - 3y^{2} = 52 \\ y - x = 14\ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} y = x + 14\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - 3 \cdot (x + 14)^{2} - 52 = 0 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} y = x + 14\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ - 2x^{2} - 84x - 640 = 0 \\ \end{matrix} \right.\ \]
\[2x^{2} + 84x + 640 = 0\ \ \ \ \ |\ :2\]
\[x^{2} + 42x + 320 = 0\]
\[x_{1} + x_{2} = - 42;\ \ \ x_{1} \cdot x_{2} = 320\]
\[x_{1} = - 32;\ \ \ \ x_{2} = - 10.\]
\[1)\ x_{1} = - 32:\ \ \]
\[y_{1} = - 18.\]
\[2)\ x_{2} = - 10:\ \ \]
\[y_{2} = 4.\]
\[\textbf{г)}\ \left\{ \begin{matrix} 3x^{2} + 2y^{2} = 11 \\ x + 2y = 3\ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x = 3 - 2y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 3 \cdot (3 - 2y)^{2} + 2y^{2} = 11 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x = 3 - 2y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 14y^{2} - 36y + 16 = 0 \\ \end{matrix} \right.\ \]
\[7y^{2} - 18y + 8 = 0\]
\[D_{1} = 81 - 56 = 25\]
\[y_{1,2} = \frac{9 \pm 5}{7} = 2;\frac{4}{7}\text{.\ }\]
\[1)\ \ y_{1} = 2:\ \ \]
\[x_{1} = - 1.\]
\[2)\ y_{2} = \frac{4}{7}:\]
\[x_{2} = 1\frac{6}{7}.\]
\[\textbf{д)}\ \left\{ \begin{matrix} x^{2} + y^{2} = 100 \\ 3x = 4y\ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x = \frac{4}{3}\text{y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \frac{16}{9}y^{2} + y^{2} = 100 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} y^{2} = 36 \\ x = \frac{3}{4}\text{y\ \ } \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} y_{1} = 6 \\ x_{1} = 8 \\ \end{matrix} \right.\ \text{\ \ \ }или\ \left\{ \begin{matrix} y_{2} = - 6 \\ x_{2} = - 8. \\ \end{matrix} \right.\ \]
\[\textbf{е)}\ \left\{ \begin{matrix} 2x^{2} - y^{2} = 32 \\ 2x - y = 8\ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} y = 2x - 8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 2x^{2} - \left( 4x^{2} - 32x + 64 \right) - 32 = 0 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} y = 2x - 8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - 16x + 48 = 0 \\ \end{matrix} \right.\ \ \]
\[x^{2} - 16x + 48 = 0\]
\[D_{1} = 64 - 48 = 16\]
\[x_{1} = 8 + 4 = 12;\ \ \ \]
\[x_{2} = 8 - 4 = 4.\]
\[1)\ x_{1} = 12:\]
\[y_{1} = 16.\]
\[2)\ x_{2} = 4:\ \ \]
\[y_{2} = 0.\]
\[Ответ:а)\ (1;4);\left( - \frac{3}{5};\frac{4}{5} \right);\ \ \]
\[\textbf{б)}\ \left( 2\frac{1}{2};\ - \frac{1}{2} \right);\ \ \left( 2\frac{2}{9};\frac{1}{3} \right);\]
\[\textbf{в)}\ ( - 32;\ - 18);( - 10;4);\ \ \]
\[\textbf{г)}\ ( - 1;2);\left( 1\frac{6}{7};\frac{4}{7} \right);\]
\[\textbf{д)}\ (8;6);( - 8;\ - 6);\ \ \ \]
\[\textbf{е)}\ (12;16);(4;0).\]
\(\boxed{\text{433.}\text{\ }\text{ОК\ ГДЗ\ -\ домашка\ на\ 5}}\)
\[Пусть\ \text{x\ }часов - время\ работы\ \]
\[первого\ экскаватора,\ \]
\[а\ y\ часов -\]
\[время\ второго.\ Так\ как\ один\ \]
\[экскаватор\ может\ \]
\[выполнить\ работу\]
\[на\ 4\ часа\ быстрее \Longrightarrow x + 4 = y.\]
\[3\ ч\ 45\ мин = 3\ ч + \frac{3}{4}\ ч =\]
\[= \frac{15}{4}\ ч.\ Вся\ работа = 1.\]
\[Составим\ систему\ уравнений:\]
\[\left\{ \begin{matrix} x + 4 = y\ \ \ \ \ \ \ \ \ \ \ \ \\ \frac{15}{4} \cdot \left( \frac{1}{x} + \frac{1}{y} \right) = 1 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x + 4 = y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \frac{15}{4} \cdot \left( \frac{1}{x} + \frac{1}{x + 4} \right) = 1 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x + 4 = y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \frac{15}{4} \cdot \left( \frac{x + 4 + x}{x(x + 4)} \right) = 1 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x + 4 = y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 15 \cdot (2x + 4) = 4x(x + 4) \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x + 4 = y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 4x^{2} - 14x - 60 = 0 \\ \end{matrix} \right.\ \]
\[2x^{2} - 7x - 30 = 0\]
\[D = 49 + 4 \cdot 2 \cdot 30 = 289 = 17^{2}.\]
\[x_{1,2} = \frac{7 \pm 17}{4};\ \ x = 6;\ \ x = - 2,5.\]
\[Так\ как\ x > 0:\]
\[x = 6 \Longrightarrow y = 10.\]
\[Ответ:\ 6\ ч\ и\ 10\ ч.\]
