Let $w_1, w_2, \dots, w_n$ be [complex numbers](https://artofproblemsolving.com/wiki/index.php/Complex_number).  A line $L$ in the [complex plane](https://artofproblemsolving.com/wiki/index.php/Complex_plane) is called a mean [line](https://artofproblemsolving.com/wiki/index.php/Line) for the [points](https://artofproblemsolving.com/wiki/index.php/Point) $w_1, w_2, \dots, w_n$ if $L$ contains points (complex numbers) $z_1, z_2, \dots, z_n$ such that
\[\sum_{k = 1}^n (z_k - w_k) = 0.\]
For the numbers $w_1 = 32 + 170i$, $w_2 = - 7 + 64i$, $w_3 = - 9 + 200i$, $w_4 = 1 + 27i$, and $w_5 = - 14 + 43i$, there is a unique mean line with $y$-intercept 3.  Find the [slope](https://artofproblemsolving.com/wiki/index.php/Slope) of this mean line.
Solution 1
$\sum_{k=1}^5 z_k - \sum_{k=1}^5 w_k = 0$
$\sum_{k=1}^5 z_k = 3 + 504i$
Each $z_k = x_k + y_ki$ lies on the complex line $y = mx + 3$, so we can rewrite this as 
$\sum_{k=1}^5 z_k = \sum_{k=1}^5 x_k + \sum_{k=1}^5 y_ki$
$3 + 504i = \sum_{k=1}^5 x_k + i \sum_{k=1}^5 (mx_k + 3)$
Matching the real parts and the imaginary parts, we get that $\sum_{k=1}^5 x_k = 3$ and $\sum_{k=1}^5 (mx_k + 3) = 504$. Simplifying the second summation, we find that $m\sum_{k=1}^5 x_k = 504 - 3 \cdot 5 = 489$, and substituting, the answer is $m \cdot 3 = 489 \Longrightarrow m = 163$.

Solution 2
We know that 
$\sum_{k=1}^5 w_k = 3 + 504i$
And because the sum of the 5 $z$'s must cancel this out,
$\sum_{k=1}^5 z_k = 3 + 504i$
We write the numbers in the form $a + bi$ and we know that 
$\sum_{k=1}^5 a_k = 3$ and $\sum_{k=1}^5 b_k = 504$
The line is of equation $y=mx+3$. Substituting in the polar coordinates, we have $b_k = ma_k + 3$.
Summing all 5 of the equations given for each $k$, we get 
$504 =  3m + 15$
Solving for $m$, the slope, we get $\boxed{163}$

Solution 3
The mean line for $w_1, . . ., w_5$ must pass through the mean (the center of mass) of these points, which, if we graph them on the complex plane, is $(\frac{3}{5}, \frac{504i}{5})$. Since we now have two points, namely that one and $(0, 3i)$, we can simply find the slope between them, which is $\boxed{163}$ by the good ol' slope formula.