Question:
Tina the tourist goes on a trip. She starts at the origin and drives north (in the positive $y$ direction) for $10$ units. Then she turns east (the positive $x$ direction) and as she's turning her camera flies out the window and lands exactly at $(0,10)$. She then drives $9$ units east, turns and drives $8$ units north.  She continues this pattern of turning and driving one unit less than after the previous turn, until stopping after driving $1$ unit east. She reaches for her camera only to find it missing! She activates the GPS homing device on her camera and drives back to it in a straight line. What is the equation of this line? Express your answer as $ax+by=c$, where $a$, $b$, and $c$ are integers, $a>0$, and $a$ is as small as possible.

Answer:
We know one point on the line: the camera is at $(0,10)$. To find another point on the line we can determine where Tina was when she noticed her camera was missing. She travels a total of $10+8+6+4+2$ units north from the origin, so her ending $y$-coordinate is $30$. She travels $9+7+5+3+1$ units east, so her ending $x$-coordinate is $25$. So we must find the equation of the line through $(0,10)$ and $(25,30)$. The slope of the line is $\frac{30-10}{25-0}=\frac{20}{25}=\frac{4}{5}$. We can use point-slope form to find that the equation of the line is $(y-10)=\frac{4}{5}(x-0)$, or $5(y-10)=4x$. Simplifying this gives $5y-50=4x$, so in the form requested, $\boxed{4x-5y=-50}$.