anonymous
  • anonymous
Solve the pair of simultaneous equations for the complex numbers z and w. (1+i)z + (2-i)w=3+4i iz + (3+i)w=-1+5i
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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experimentX
  • experimentX
(z+2w)+i(z-2w) = 2 + 4i =>z+2w=2, z-2w=4
experimentX
  • experimentX
from second w=5, z+3w=-1 => w=-2
anonymous
  • anonymous
z=2, w=i are the answers...? How to get there?

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More answers

Callisto
  • Callisto
For (1+i)z + (2-i)w=3+4i (1+i)z + (2-i)w=3+4i z +zi + 2w - wi = 3+4i (z+2w) + (z-w) i = 3+4i So you get z+2w =1 z-w =4
anonymous
  • anonymous
z+2w =1 why?
Callisto
  • Callisto
Like the previous one, compare the real part and the imaginary part to get the 2 equations
Callisto
  • Callisto
Sorry. that should be 3
experimentX
  • experimentX
oops sorry .. i thought w and z were real numbers
Callisto
  • Callisto
z+2w =3
anonymous
  • anonymous
So, how do you get w as i in the final answer?
Callisto
  • Callisto
Wait ... i thought i read that wrong
experimentX
  • experimentX
let z = a+ib and w = c+id
experimentX
  • experimentX
(1+i)z + (2-i)w=3+4i =>(1+i)( a+ib) + (2-i)(c+id)=3+4i => (a-b)+(a+b)i+(2c+d)+(2d-c)i = 3+4i => (a-b+2c+d)+i(a+b+2d-c) = 3+4i => (a-b+2c+d) = 2 and =>(a+b+2d-c) = 4
experimentX
  • experimentX
for other part i( a+ib) + (3+i)(c+id)=-1+5i => -b+ai+(3c-d)+(3d+c)i = -1+5i => (-b+3c-d)+i(a+3d+c) = -1 + 5i => (-b+3c-d) = -1 and => (a+3d+c) = 5
experimentX
  • experimentX
4 unknowns 4 equations .. pretty messy let's ask wolf for answer
experimentX
  • experimentX
there you have your 4 quantities http://www.wolframalpha.com/input/?i=+%28a-b%2B2c%2Bd%29+%3D+2%3B+%28a%2Bb%2B2d-c%29+%3D+4%3B+%28-b%2B3c-d%29+%3D+-1%3B++%28a%2B3d%2Bc%29+%3D+5 put the values in w and z and you have your answer
anonymous
  • anonymous
I got it!!! :D
anonymous
  • anonymous
\[(1+i)z+(2-i)w=3+4i\]\[iz+(3+i)w=-1+5i\] \[z={(-1+5i)-(3+1)w \over i}\] \[(1+i){(-1+5i)-(3+i)w \over i}+(2w-iw)=3+4i\] ---> which simplifies as, after lots of calculation ---> \[w=-3+10i+3 ----> w=i\] (There are 20 steps involved in getting the answer...)
anonymous
  • anonymous
@Callisto @experimentX
experimentX
  • experimentX
yeah ... a better way!!

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