# How to solve the simultaneous equations ∣z + 1∣ = ∣z − 1∣, ∣z + 2∣ = ∣z − 3∣?

## Apr 16, 2018

No solution

#### Explanation:

$\left\mid z \right\mid = \sqrt{z \overline{z}}$ then

$\left\mid z + 1 \right\mid = \sqrt{\left(z + 1\right) \left(\overline{z} + 1\right)} = \left\mid z - 1 \right\mid = \sqrt{\left(z - 1\right) \left(\overline{z} - 1\right)}$

and

$\left\mid z + 2 \right\mid = \sqrt{\left(z + 2\right) \left(\overline{z} + 2\right)} = \left\mid \overline{z} - 3 \right\mid = \sqrt{\left(\overline{z} - 3\right) \left(z - 3\right)}$

or

$\left\{\begin{matrix}\overline{z} z + z + \overline{z} + 1 = \overline{z} z - \left(\overline{z} + z\right) + 1 \\ z \overline{z} + 2 \left(z + \overline{z}\right) + 4 = \overline{z} z - 3 \left(\overline{z} + z\right) + 9\end{matrix}\right.$

or

$\left\{\begin{matrix}z + \overline{z} = - \left(\overline{z} + z\right) \\ 2 \left(z + \overline{z}\right) = - 3 \left(\overline{z} + z\right) + 5\end{matrix}\right.$

or

$\left\{\begin{matrix}z + \overline{z} = 0 \\ z + \overline{z} = 1\end{matrix}\right.$

so no solution is possible.