complex numbers

We can use the fact that for any complex number a, (a)(cong(a)) = |a|^2.

$|5z + w|^2 = (5z + w)(5\overline{z} + \overline{w}) = 121$

$25|z|^2 + |w|^2 + 5(\overline{z}w + \overline{w}z) = 121$

$5(\overline{z}w + \overline{w}z) = 47$

$|3z + 4w|^2 = (3z + 4w)(3\overline{z} + 4\overline{w})$

$=793 + 12(\overline{w}z + \overline{z}w)$

$=793 + 564/5$

Take the square root of that value and you will the answer.