Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 676

\[1)\ \left\{ \begin{matrix} z_{1} + 2z_{2} = 1 + i\ \ \ \\ 3z_{1} + iz_{2} = 2 - 3i \\ \end{matrix} \right.\ \]

\[z_{1} + 2z_{2} = 1 + i\]

\[z_{1} = 1 + i - 2z_{2}.\]

\[3z_{1} + iz_{2} = 2 - 3i\]

\[3\left( 1 + i - 2z_{2} \right) + iz_{2} = 2 - 3i\]

\[3 + 3i - 6z_{2} + iz_{2} = 2 - 3i\]

\[z_{2}(i - 6) = - 1 - 6i =\]

\[= \frac{- 1 - 6i}{i - 6} = \frac{1 + 6i}{6 - i} =\]

\[= \frac{(1 + 6i)(6 + i)}{(6 - i)(6 + i)} =\]

\[= \frac{6 + i + 36i - 6}{36 + 1} = \frac{37i}{37} = i;\]

\[z_{1} = 1 + i - 2i = 1 - i.\]

\[Ответ:\ \ z_{1} = 1 - i;\ z_{2} = i.\]

\[2)\ \left\{ \begin{matrix} z^{2} + |z| = 0 \\ \overline{z} = - 4z\ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[z^{2} + |z| = 0\]

\[(x + yi)^{2} + \sqrt{x^{2} + y^{2}} = 0\]

\[x + yi = 0;\text{\ \ \ }\]

\[x^{2} + y^{2} = 0\]

\[x = y = 0.\]

\[\overline{z} = - 4z\]

\[x - yi = - 4(x + yi)\]

\[x - yi = - 4x - 4yi\]

\[5x + 3yi = 0\]

\[5x = 0;\ \ \ 3y = 0;\]

\[x = y = 0.\]

\[Ответ:\ \ z = 0.\]