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We are to find $ \operatorname{Re}(z \overline{w}) $.
Note that $ z \overline{w} + \overline{z} w = 2 \operatorname{Re}(z \overline{w}) $, so from (2):
\operatorname{Re}(z \overline{w}) = 10 \Rightarrow \operatorname{Re}(z \overline{w}) = 5
But we verify consistency using identities.
Recall the identity:
|z + w|^2 = |z|^2 + |w|^2 + 2 \operatorname{Re}(z \overline{w})
Compute $ |z + w|^2 $:
z + w = 4 + 2i \Rightarrow |z + w|^2 = 4^2 + 2^2 = 16 + 4 = 20
Now use the identity:
= |z|^2 + |w|^2 + 2 \operatorname{Re}(z \overline{w}) = 26 + 2 \cdot 5 = 26 + 10 = 36? \quad ext{Wait, contradiction.}