Question 1157072
<pre>
{{{((1 + i)/(1 - i))^2 + 1/(x + iy) = 1 + i}}} 

We simplify what's squared in the the first term

{{{(1 + i)/(1 - i)=((1 + i)/(1 - i))((1 + i)/(1 + i))=(1+2i+i^2)/(1-i^2)=

(1+2i+(-1))/(1-(-1))=2i/(1+1)=2i/2=i}}}

Now we square it to get the complete first term simplified:

{{{matrix(1,2,first,term)=((1 + i)/(1 - i))^2=i^2=-1}}}

Substituting -1 for the first term

{{{-1 + 1/(x + iy) = 1 + i}}}

Add 1 to both sides:

{{{1/(x + iy) = 2 + i}}}

Multiply both sides by (x + iy)

{{{1 = (x+iy)(2 + i)}}}

FOIL out the right side:

{{{1 = 2x+ix+2iy + i^2y}}}

{{{1 = 2x+ix+2iy+(-1)y}}}

{{{1 = 2x+ix+2iy-y}}}

Set the real terms on the left equal to the real terms on the right:

{{{1=2x-y}}}

Set the imaginary terms on the left equal to the imaginary terms on
the right (there aren't any on the left so we use 0):

{{{0=ix+2iy}}}

divide through by i

{{{0=x+2y}}}

Solve the system:

{{{system(1=2x-y,0=x+2y)}}}

Solve the first equation for y

{{{y=2x-1}}}

Substitute in the 2nd equation of the system:

{{{0=x+2(2x-1)}}}

Distribute:

{{{0=x+4x-2}}}

Combine terma:

{{{0=5x-2}}}

Add 2 to both sides:

{{{2=5x}}}

Divide both sides by 5

{{{2/5=x}}}

Substitute for x in

{{{y=2x-1}}}

{{{y=2(2/5)-1}}}

{{{y=4/5-1}}}

Multiple through by 5

{{{5y=4-5}}}

{{{5y=-1}}}

Divide both sides by 5

{{{y=-1/5}}} 

{{{matrix(1,5,

(matrix(1,3,x,",",y)),"",""="","",(matrix(1,3,2/5,",",-1/5)) )}}}

Edwin</pre>