Complex numbers finding a and b

[BOAS]

Hello,

Homework Statement

The complex numbers $z_{1} = \frac{a}{1 + i}$ and $z_{2} = \frac{b}{1+2i}$ where a and b are real, are such that [itex]z_{1} + z_{2} = 1. Find a and b.

Homework Equations

The Attempt at a Solution

This looked like a time for partial fractions to me, so I went down that road;

$\frac{a}{1 + i} + \frac{b}{1+2i} = 1$

$\frac{a(1+2i)}{(1 + i)(1+2i)} + \frac{b(1+i)}{(1+2i)(1+i)} = 1$

$a(1+2i) + b(1+i) = (1+i)(1+2i)$

Expanding the brackets gives me;

$a(1+2i) + b(1+i) = (1+i)(1+2i)$

$a(1+2i) + b(1+i) = -1 + 3i$

∴ $a + b = -1$

And now I'm stuck...

Is this the right approach? And, how do I move forward?

Thanks!

 

[vela]

What about the imaginary part? What equation does that give you?

 

[BOAS]

What about the imaginary part? What equation does that give you?

Aha, woops.

2a + b = 3

Which allows me to work out via simultaneous equations that a = 4 and b = -5.

Thanks.