What is the answer to z=-7+i4z+3w ?

The answer to z=-7+i4z+3w is z=(-7+3w)/(17)+(4(-7+3w))/(17)i

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z=-7+i4z+3w

z = - 7 + i 4 z + 3 w

z = \frac{- 7 + 3 w}{17} + \frac{4 ( - 7 + 3 w )}{17} i

Solution steps

z = - 7 + i 4 z + 3 w

Substitute

z = x + y i

x + y i = - 7 + i 4 (x + y i) + 3 w

Simplify

- 7 + i 4 (x + y i) + 3 w : (- 7 - 4 y + 3 w) + 4 i x

x + y i = (- 7 - 4 y + 3 w) + 4 i x

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

[x = - 7 - 4 y + 3 w y = 4 x]

[x = - 7 - 4 y + 3 w y = 4 x] : x = \frac{- 7 + 3 w}{17}, y = \frac{4 ( - 7 + 3 w )}{17}

Substitute back

z = x + y i

z = \frac{- 7 + 3 w}{17} + \frac{4 ( - 7 + 3 w )}{17} i

x^{2} + (8 - i) x + 17 - 7 i = 0

(4 i - x) (- x) = 4

7 + 4 i = x + y i

z - 5 = i (10 + 4^{-} z)

r - 8 = r - i

What is the answer to z=-7+i4z+3w ?
The answer to z=-7+i4z+3w is z=(-7+3w)/(17)+(4(-7+3w))/(17)i