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МатематикаРешебник по алгебре и начала математического анализа 10 класс Колягин Задание 357
Задание 357
\[\boxed{\mathbf{357}.}\]
\[1)\ \left\{ \begin{matrix} x^{2} + y^{2} = 74 \\ x + y = 12\ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]
\[\left\{ \begin{matrix} x = 12 - y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ (12 - y)^{2} + y^{2} = 74 \\ \end{matrix} \right.\ \]
\[144 - 24y + y^{2} + y^{2} - 74 = 0\]
\[2y^{2} - 24y + 70 = 0\ \ \ \ |\ :2\]
\[y^{2} - 12y + 35 = 0\]
\[D_{1} = 36 - 35 = 1\]
\[y_{1} = 6 + 1 = 7;\ \ \ \ \ \]
\[\ y_{2} = 6 - 1 = 5;\]
\[x_{1} = 12 - 7 = 5;\ \ \]
\[\text{\ \ }x_{2} = 12 - 5 = 7.\]
\[Ответ:(5;7);(7;5).\]
\[2)\ \left\{ \begin{matrix} x^{2} - y^{2} = 32 \\ x - y = 4\ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ }\]
\[\text{\ \ }\left\{ \begin{matrix} (x - y)(x + y) = 32 \\ x - y = 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ }\]
\[\ \left\{ \begin{matrix} 4 \cdot (x + y) = 32\ \ |\ :4 \\ x - y = 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x + y = 8 \\ x - y = 4 \\ \end{matrix} \right.\ ( + )\]
\[2x = 12\]
\[x = 6.\]
\[y = x - 4 = 6 - 4 = 2.\]
\[Ответ:(6;2).\]
\[3)\ \left\{ \begin{matrix} x^{2} + y^{2} = 10 \\ x + y = 4\ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]
\[\ \left\{ \begin{matrix} x = 4 - y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ (4 - y)^{2} + y^{2} = 10 \\ \end{matrix} \right.\ \]
\[16 - 8y + y^{2} + y^{2} - 10 = 0\]
\[2y^{2} - 8y + 6 = 0\ \ \ |\ :2\]
\[y^{2} - 4y + 3 = 0\]
\[D_{1} = 4 - 3 = 1\]
\[y_{1} = 2 + 1 = 3;\ \ \ \ \ \]
\[y_{2} = 2 - 1 = 1;\]
\[x_{1} = 4 - 3 = 1;\ \ \ \ \ \]
\[x_{2} = 4 - 1 = 3.\]
\[Ответ:(1;3);(3;1).\]
\[4)\ \left\{ \begin{matrix} x^{2} - y^{2} = 16 \\ x - y = 1\ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]
\[\ \left\{ \begin{matrix} (x - y)(x + y) = 16 \\ x - y = 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x + y = 16 \\ x - y = 1\ \ \ \\ \end{matrix} \right.\ ( + )\]
\[2x = 17\]
\[x = 8,5.\]
\[y = x - 1 = 8,5 - 1 = 7,5.\]
\[Ответ:(8,5;7,5).\]
