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Judge the identity
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$\dfrac{ x }{ 1+3i } + \dfrac{ y }{ 1-3i } = \dfrac{ 9 }{ 2+i }$
$x = 21 , y = 15$
Find the unknown when the unknown is a rational number
$\color{#FF6800}{ \dfrac { x } { 1 + 3 i } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { y } { 1 - 3 i } } = \dfrac { 9 } { 2 + i }$
$ $ Organize the expression $ $
$\color{#FF6800}{ \dfrac { x } { 10 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 i x } { 10 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { y } { 10 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 3 i y } { 10 } } = \dfrac { 9 } { 2 + i }$
$\dfrac { x } { 10 } - \dfrac { 3 i x } { 10 } + \dfrac { y } { 10 } + \dfrac { 3 i y } { 10 } = \color{#FF6800}{ \dfrac { 9 } { 2 + i } }$
$ $ Organize the expression $ $
$\dfrac { x } { 10 } - \dfrac { 3 i x } { 10 } + \dfrac { y } { 10 } + \dfrac { 3 i y } { 10 } = \color{#FF6800}{ \dfrac { 18 } { 5 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 9 i } { 5 } }$
$\dfrac { x } { 10 } - \dfrac { \color{#FF6800}{ 3 } \color{#FF6800}{ i } \color{#FF6800}{ x } } { 10 } + \dfrac { y } { 10 } + \dfrac { 3 i y } { 10 } = \dfrac { 18 } { 5 } - \dfrac { 9 i } { 5 }$
$ $ Organize the coefficients $ $
$\dfrac { x } { 10 } - \dfrac { \color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ i } } { 10 } + \dfrac { y } { 10 } + \dfrac { 3 i y } { 10 } = \dfrac { 18 } { 5 } - \dfrac { 9 i } { 5 }$
$\dfrac { x } { 10 } - \dfrac { 3 x i } { 10 } + \dfrac { y } { 10 } + \dfrac { \color{#FF6800}{ 3 } \color{#FF6800}{ i } \color{#FF6800}{ y } } { 10 } = \dfrac { 18 } { 5 } - \dfrac { 9 i } { 5 }$
$ $ Organize the coefficients $ $
$\dfrac { x } { 10 } - \dfrac { 3 x i } { 10 } + \dfrac { y } { 10 } + \dfrac { \color{#FF6800}{ 3 } \color{#FF6800}{ y } \color{#FF6800}{ i } } { 10 } = \dfrac { 18 } { 5 } - \dfrac { 9 i } { 5 }$
$\color{#FF6800}{ \dfrac { x } { 10 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 x i } { 10 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { y } { 10 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 3 y i } { 10 } } = \dfrac { 18 } { 5 } - \dfrac { 9 i } { 5 }$
$ $ Calculate the similar terms $ $
$\color{#FF6800}{ \dfrac { x } { 10 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { y } { 10 } } \color{#FF6800}{ + } \left ( \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 x } { 10 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 3 y } { 10 } } \right ) \color{#FF6800}{ i } = \dfrac { 18 } { 5 } - \dfrac { 9 i } { 5 }$
$\color{#FF6800}{ \dfrac { x } { 10 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { y } { 10 } } \color{#FF6800}{ + } \left ( \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 x } { 10 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 3 y } { 10 } } \right ) \color{#FF6800}{ i } = \color{#FF6800}{ \dfrac { 18 } { 5 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 9 i } { 5 } }$
$ $ If the real number part and the imaginary number part are the same, the two complex numbers are the same $ $
$\begin{cases} \color{#FF6800}{ \dfrac { x } { 10 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { y } { 10 } } = \color{#FF6800}{ \dfrac { 18 } { 5 } } \\ \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 x } { 10 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 3 y } { 10 } } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 9 } { 5 } } \end{cases}$
$\begin{cases} \color{#FF6800}{ \dfrac { x } { 10 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { y } { 10 } } = \color{#FF6800}{ \dfrac { 18 } { 5 } } \\ \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 x } { 10 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 3 y } { 10 } } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 9 } { 5 } } \end{cases}$
$ $ Solve the system of equations $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ 21 } , \color{#FF6800}{ y } = \color{#FF6800}{ 15 }$
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