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Question
Find the real value of x and y, if
\[(x + iy)(2 - 3i) = 4 + i\]
Solution
\[ \left( x + iy \right)\left( 2 - 3i \right) = 4 + i\]
\[2x - 3ix + 2iy - 3 i^2 y = 4 + i\]
\[2x + 3y + i\left( - 3x + 2y \right) = 4 + i\]
\[\text{Comparing both the sides:} \]
\[2x + 3y = 4 . . . . (1) \]
\[ - 3x + 2y = 1 . . . . (2)\]
\[\text { Multiplying equation (1) by 3 and equation (2) by 2 }: \]
\[ 6x + 9y = 12 . . . (3)\]
\[ - 6x + 4y = 2 . . . (4)\]
\[\text { Adding equations (3) and (4) }: \]
\[13y = 14\]
\[y = \frac{14}{13}\]
\[\text { Substituting the value of y in equation (1):} \]
\[2x + 3 \times \frac{14}{13} = 4\]
\[ \Rightarrow 2x = 4 - \frac{42}{13}\]
\[ \Rightarrow 2x = \frac{10}{13}\]
\[ \Rightarrow x = \frac{5}{13}\]
\[ \therefore x = \frac{5}{13}\text { and } y = \frac{14}{13} \]
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