What is the answer to 1/z =1-i ?

The answer to 1/z =1-i is z= 1/2+1/2 i

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1/z =1-i

\frac{1}{z} = 1 - i

z = \frac{1}{2} + \frac{1}{2} i

Solution steps

\frac{1}{z} = 1 - i

Substitute

z = x + y i

\frac{1}{x + y i} = 1 - i

Multiply both sides by

x + y i

\frac{1}{x + y i} (x + y i) = 1 \cdot (x + y i) - i (x + y i)

Expand

1 = (x + y) + i (- x + y)

Rewrite

1

in standard complex form:

1 + 0 i

1 + 0 i = (x + y) + i (- x + y)

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

[1 = x + y 0 = - x + y]

[1 = x + y 0 = - x + y] : x = \frac{1}{2}, y = \frac{1}{2}

Substitute back

z = x + y i

z = \frac{1}{2} + \frac{1}{2} i

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

(x i)^{2} + 125.6637 x i + 1184 = 0

x + i y + 2 x - 2 i = \frac{2 - i}{1 + 3 i}

r (t) = (3 - 3 t) i + 4 t j, 0 \leq t \leq 1

\frac{a - 6 i}{- 2 + 3 i} = b - 7