Solve iz + z = 0

**madbandlt** · June 15th 2011, 08:54 PM

Hey guys, quick question

(b) Find all complex solutions of iz + z = 0 and sketch them in the complex plane.

Well I let z = x + iy, simplified and obtained x - y + i(x - y) = 0

But as for sketching them I'm not entirely sure how to proceed. The answers have up and until the answer I obtained but then immediately proceeds to sketching y = x

Can someone please explain why y = x is the solution?

**Prove It** · June 15th 2011, 09:30 PM

Your simplification is wrong

$\displaystyle \begin{align*} iz + z &= 0 \\ i(x + iy) + x + iy &= 0 + 0i \\ ix - y + x + iy &= 0 + 0i \\ x - y + i(x + y) &= 0 + 0i \end{align*}$

Now equating real and imaginary parts gives

$\displaystyle x - y = 0$ and $\displaystyle x + y = 0$ .

So the solutions are $\displaystyle y = x$ and $\displaystyle y = -x$ .

**chisigma** · June 15th 2011, 11:10 PM

Originally Posted by madbandlt

Hey guys, quick question

(b) Find all complex solutions of iz + z = 0 and sketch them in the complex plane.

Well I let z = x + iy, simplified and obtained x - y + i(x - y) = 0

But as for sketching them I'm not entirely sure how to proceed. The answers have up and until the answer I obtained but then immediately proceeds to sketching y = x

Can someone please explain why y = x is the solution?

The equation i z + z = (1+i) z = 0 is first degree algebrical and its only solution is z=0...

Kind regards

$\chi$ $\sigma$

**mr fantastic** · June 16th 2011, 05:00 PM

Originally Posted by Prove It

Your simplification is wrong

$\displaystyle \begin{align*} iz + z &= 0 \\ i(x + iy) + x + iy &= 0 + 0i \\ ix - y + x + iy &= 0 + 0i \\ x - y + i(x + y) &= 0 + 0i \end{align*}$

Now equating real and imaginary parts gives

$\displaystyle x - y = 0$ and $\displaystyle x + y = 0$ .

So the solutions are $\displaystyle y = x$ and $\displaystyle y = -x$ .

The equations $\displaystyle x - y = 0$ and $\displaystyle x + y = 0$ have to be true simultaneously. Therefore the only solution is x = y = 0.

But the OP has not actually typed the correct question. The correct question is probably the following:

Find all complex solutions of $iz + \overline{z} = 0$ and sketch them in the complex plane.

In which case, you end up with the two equations x - y = 0 and x - y = 0 and the solution is obviously y = x.

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