If the mid-point of the line segment joining the points $A(3, 4)$ and $B(k, 6)$ is $P(x, y)$ and $x + y - 10 = 0$, find the value of $k$.

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Step 1: $P(x, y)$ is the mid-point.
$	herefore (x, y) = \left(\frac{3+k}{2}, \frac{4+6}{2}ight)$
$x = \frac{3+k}{2}$, $y = 5$
Step 2: $x + y - 10 = 0$
$\frac{3+k}{2} + 5 - 10 = 0$
Step 3: $\frac{3+k}{2} = 5 \Rightarrow 3 + k = 10 \Rightarrow k = 7$

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