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Question
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
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Solution
\[ \left( 1 + i \right)\left( x + iy \right) = 2 - 5i\]
\[ \Rightarrow x + iy + ix + i^2 y = 2 - 5i\]
\[ \Rightarrow x + iy + ix - y = 2 - 5i\]
\[ \Rightarrow \left( x - y \right) + i\left( y + x \right) = 2 - 5i\]
\[\text { Comparing both the sides }, \]
\[x - y = 2 . . . (1) \]
\[x + y = - 5 . . . (2)\]
\[\text { Adding equations (1) and (2) }, \]
\[2x = - 3\]
\[ \Rightarrow x = \frac{- 3}{2}\]
\[\text { Substituting the value of x in equation (1) }, \]
\[\frac{- 3}{2} - y = 2\]
\[ \Rightarrow y = \frac{- 3}{2} - 2\]
\[ \Rightarrow y = \frac{- 7}{2}\]
\[ \therefore x = \frac{- 3}{2} \text { and y } = \frac{- 7}{2}\]
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