Bài 2 (Trang 133 SGK Toán Giải tích 12):

Tìm các số thực x và y biết:

a) $(3\mathrm{x} - 2) + (2\mathrm{y} + 1)\mathrm{i} = (\mathrm{x} + 1) - (\mathrm{y} - 5)\mathrm{i}$  ;

b) $\left(1 - 2\mathrm{x}\right) - \mathrm{i}\sqrt{3} = \sqrt{5} + \left(1 - 3\mathrm{y}\right)\mathrm{i}$  ;

c) $\left(2\mathrm{x} + \mathrm{y}\right) + \left(2\mathrm{y} - \mathrm{x}\right)\mathrm{i} = \left(\mathrm{x} - 2\mathrm{y} + 3\right) + \left(\mathrm{y} + 2\mathrm{x} +\hspace{0.17em}1\right)\mathrm{i}$.

Hướng dẫn giải:

Áp dụng a+ bi = c + di $\left\{\begin{array}{l}\mathrm{a} = \mathrm{c}\\ \mathrm{b} = \mathrm{d}\end{array}\right.$

a) $(3\mathrm{x}-2) + (2\mathrm{y}+1)\mathrm{i} = (\mathrm{x}+1) - (\mathrm{y}-5)\mathrm{i} \iff \left\{\begin{array}{ll}3\mathrm{x}-2=& \mathrm{x}+1\\ 2\mathrm{y} +1 =& -(\mathrm{y}-5)\end{array}\right.\iff \left\{\begin{array}{l}\mathrm{x}=\frac{3}{2}\\ \mathrm{y}=\frac{4}{3}\end{array}\right.$

b) $(1-2\mathrm{x}) - \mathrm{i}\sqrt{3} = \sqrt{5} + (1-3\mathrm{y})\mathrm{i} \iff \{\begin{array}{ll}1-2\mathrm{x}=& \sqrt{5}\\ 1-3\mathrm{y}=& -\sqrt{3}\end{array}\iff \{\begin{array}{ll}\mathrm{x}=& \frac{1-\sqrt{5}}{2}\\ \mathrm{y}=& \frac{1+\sqrt{3}}{3}\end{array}$

c) $(2\mathrm{x}+\mathrm{y}) + (2\mathrm{y}-\mathrm{x})\mathrm{i} = (\mathrm{x}-2\mathrm{y}+3) + (\mathrm{y}+2\mathrm{x}+1)\mathrm{i} \iff \{\begin{array}{l}2\mathrm{x}+\mathrm{y}=\mathrm{x}-2\mathrm{y}+3\\ 2\mathrm{y}-\mathrm{x}=\mathrm{y}+2\mathrm{x}+1\end{array}\iff \left\{\begin{array}{l}\mathrm{x}+3\mathrm{y}=3\\ -3\mathrm{x}+\mathrm{y}=1\end{array}\iff \left\{\begin{array}{l}\mathrm{x}=0\\ \mathrm{y}=1\end{array}\right.\right.$