What is the answer to |z|-3z=1-3i ?

The answer to |z|-3z=1-3i is z=i

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

|z|-3z=1-3i

∣ z ∣ - 3 z = 1 - 3 i

z = i

Solution steps

∣ z ∣ - 3 z = 1 - 3 i

Substitute

z = x + y i

∣ (x + y i) ∣ - 3 (x + y i) = 1 - 3 i

Expand

∣ x + y i ∣ - 3 (x + y i) : (x^{2} + y^{2} - 3 x) - 3 i y

(x^{2} + y^{2} - 3 x) - 3 i y = 1 - 3 i

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[x^{2} + y^{2} - 3 x = 1 - 3 y = - 3]

[x^{2} + y^{2} - 3 x = 1 - 3 y = - 3] : (x = 0, y = 1)

Substitute back

z = x + y i

z = i

z^{2} = 5 - 12 i

z^{2} - i z + 2 = 0

z^{2} - (5 + 4 i) z + 20 i = 0

4 (x - y i) = 8 + 2 y i + 10 i

(1 + 2 i) x + (3 - 5 i) y = 1 - 3 i