Can someone solve this complex number equation for x and y

[anjunawoop]

If x and y are real quantities what are the solutions for x and y

(ix)(1+iy)=(3x+i4)/(x+3y)

I have tried grouping the equation into two equal complex numbers but have failed to find a solution which isolates x and y from the two quite long polynomial equations.

Does anybody know how to do this.

 

[vela]

Your approach sounds fine. Show us what you did so far.

 

[anjunawoop]

(xi)/(1 + yi) = (3x + 4i)/(x + 3y)

First rationalized the left hand fraction.

(xi)/(1 + yi) * (1 - yi)/(1 - yi) = (xi -yxi^2)/(1 + y^2)

= (yx + xi)/(1 + y^2)
Grouped real and imaginary terms

[ yx/(1 + y^2) ] + [ x/(1 + y^2) ]i

Then grouped real and imaginary terms on the right hand fraction

(3x + 4i)/(x + 3y) = [ 3x/(x + 3y) ] + [ 4/(x + 3y) ]i

Set the two complex numbers equal

[ yx/(1 + y^2) ] + [ x/(1 +y^2) ]i = [ 3x/(x + 3y) ] + [ 4/(x +3y) ]i

Take the two imaginary parts to be identical
Take the two real parts to be identical

[ yx/(1 + y^2) ] = [ 3x/(x +3y) ]
[ x/(1 + y^2) ] = [ 4/(x + 3y) ]

Expressing both fractions as polynomials

(yx)(x + 3y) = (3x)(1 + y^2)
(x)(x + 3y) = (4)(1 + y^2)

Then multiplying out both equations in x and y

1. yx^2 + 3y^2x = 3x + 3xy^2

2. x^2 + 3yx = 4 +4y^2

Grouping terms on one side to form two equations that equal zero.

1. yx^2 + 3y^2x - 3x - 3xy^2 = 0

2. x^2 +3yx - 4 - 4y^2 = 0

Now trying to find some way of dealing with these terms.

 

[I like Serena]

Hmm, let's just cross multiply and simplify:

(xi)/(1 + yi) = (3x + 4i)/(x + 3y)
(xi)(x + 3y) = (3x + 4i)(1 + yi)
x²i + 3xyi = 3x + 4i + 3xyi - 4y
x²i = 3x + 4i - 4y
(4y-3x) + (x² - 4)i = 0

That doesn't look so bad does it?
I suspect you made a mistake somewhere, but I haven't figured out where yet.

 

[vela]

Expressing both fractions as polynomials

(yx)(x + 3y) = (3x)(1 + y^2)
(x)(x + 3y) = (4)(1 + y^2)

Note that you can write these two equations as
\begin{align*}
\frac{x(x+3y)}{1+y^2} &= \frac{3x}{y} \\
\frac{x(x+3y)}{1+y^2} &= 4
\end{align*}So you have y=(3/4)x. You can plug use that to eliminate y in one of the equations and solve for x.

Or you can do it the way ILS did it.