What is z if z/2 + 3=z'/3 - 2? z complex number
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As z is a complex number let it be x + yi. z' = x - yi.
If z/2 + 3=z'/3 - 2
=> (x + yi)/2 + 3 = (x - yi)/3 - 2
=> (x + yi)/2 - (x -yi)/3 = -5
=> x/2 - x/3 - i( y/2 - y/3) = -5
=> x/2 - x/3 = -5
=> (3x - 2x)/6 = -5
=> x = -30
y = 0 as there is no imaginary component on the opposite side.
Therefore z = -30
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We'll write the rectangular form of the complex number z:
z = a + bi
z' is the conjugate of z:
z' = a - bi
To determine z, we'll have to determine it's coefficients:
(a + bi)/2 + 3 = (a - bi)/3 - 2
We'll multiply by 6 both sides:
3(a + bi) + 18 = 2(a - bi) - 12
We'll remove the brackets:
3a + 3bi + 18 = 2a - 2bi - 12
We'll move all terms to the left side:
3a - 2a + 3bi + 2bi + 18 + 12 = 0
a + 5bi + 30 = 0
The real part of the complex number from the left side is:
Re(z) = a + 30
The real part of the complex number from the right side is:
Re(z) = 0
Comparing, we'll get:
a + 30 = 0
a = -30
The imaginary part of the complex number from the left side is:
Im(z) = 5b
The imaginary part of the complex number from the right side is:
Im(z) = 5b
Comparing, we'll get:
5b = 0
b = 0
The complex number z is: z = -30 + 0*i
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