Brute forcing simultaneous equations involving Complex Numbers

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 $z = -4+2i$

About

KS Teng

KS has been teaching H2/H1 Mathematics and IB mathematics for almost two decades. Having taught students from all various Junior Colleges, KS adapts to students' abilities and help them better understand the topics. As someone who loves to teach mathematics and sees it as a truly useful tool in life, KS seeks to enable students to appreciate math. Therefore, his tuition mission is to motivate and cultivate students to be independent and confident thinkers.

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Brute forcing simultaneous equations involving Complex Numbers

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