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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