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Math Help - real numbers x and y

  1. #1
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    real numbers x and y

    if someone could help me with this i would appreciate it.

    given that

    \frac{1}{x+jy}+\frac{1}{1+2j} =1

    find the real numbers x and y

    many thanks
    Last edited by decoy808; February 18th 2010 at 03:58 AM. Reason: no math tags
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  2. #2
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    Quote Originally Posted by decoy808 View Post
    if someone could help me with this i would appreciate it.

    given that

    \frac{1}{x+jy}+\frac{1}{1+2j} =1

    find the real numbers x and y

    many thanks
    "Rationalize" the denominators by multiplying numerator and denominator of each fraction by the complex conjugate of the denominator:
    \frac{1}{x+ jy}\frac{x- jy}{x- jy}+ \frac{1}{1+2j}\frac{1-2j}{1-2j}= 1
    \frac{x- jy}{x^2+y^2}+ \frac{1-2j}{5}= 1

    Get rid of the fractions by multiplying both sides by 5(x^2+y^2)
    5x- 5jy+ (x^2+y^2)- 2(x^2+y^2)j= 5x^2+5y^2

    Finally, set "real part" equal to "real part" and "imaginary part" equal to imaginary part:
    5x+ x^2+y^2= 5x^2+ 5y^2
    and
    -5y- 2x^2- 2y^2= 0

    Solve those two equations for x and y.
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  3. #3
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    Hello, decoy808!

    Given: . \frac{1}{x+yj}+\frac{1}{1+2j} \:=\:1

    find real numbers x and y.

    Multiply through by the LCD:

    . . (x+yj)(1 + 2j)\cdot\frac{1}{x+yj} + (x+yj)(1+2j)\cdot\frac{1}{1+2j} \;=\;(x+yj)(1+2j)\cdot 1

    . . (1 + 2j) + (x+yj) \:=\:(x+yj)(1 + 2j)

    . . 1 + 2j + x + yj \:=\:x + 2xj + yj + 2yj^2

    . . 1 + 2j + x + yj \;=\;x + 2xj + yj - 2y

    . . (1 + x) + (2 + y)j \;=\;(x-2y) + (2x+y)j


    Equate real and imaginary components:

    . . \begin{array}{cccccccccccc}<br />
1 + x \:=\: x-2y & \Rightarrow & 1 \:=\: \text{-}2y & \Rightarrow &\boxed{ y \:=\: \text{-}\tfrac{1}{2}} \\ \\[-3mm]<br />
2+y \L=\: 2x +y & \Rightarrow & 2 \:=\: 2x & \Rightarrow & \boxed{x \:=\: 1} \end{array}



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