So I have many students coming up to me and saying, “Mr Teng, How do you know which to substitute away or when to introduce $z=x+yi$ when doing simultaneous equations for complex numbers?”
Here is a lesser method that will give you the answers. It is definitely a clearer method that involves less pitfalls. This method is self-explanatory so I’ll let the working do the talking. Say we want to solve the following

$2w + z = 12i ---(1)$

$w^* + 2z = -6 - i ---(2)$

Let $w = a + bi$ (clearly, $w^* = a-bi$) , and $z=c+di$

$2(a+bi) + (c+di) = 12i ---(1)$

$a-bi + 2(c+di) = -6 - i ---(2)$

From (1), we have $(2a+c) + (2b+d)i = 12i$

$\Rightarrow 2a+c = 0 --- (3)$ , and $2b+d = 12 ---(4)$

From (2), we have $(a+2c) + (2d-b)i = -6-i$

$\Rightarrow a+2c = -6 --- (5)$ , and $2d-b = -1 ---(6)$

Solving (3), (4), (5), (6), we find

$a = 2, b =5, c=-4, d=2$

Thus, $w = 2+5i$ and $z = -4+2i$

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