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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.

 Apr 4, 2019

Consider where her camer flew out as the 'origin' of her troubles.... she travels  9+ 7+5+3+1 east from there (25 mi)

    and she travels 8+6+4+2 north = 20 mi


y= mx+b    m = slope = rise / run = 20/25 = 4/5

y = 4/5 x + b     and passes through the point   0,10

10 = 4/5 (0) + b

                   b =10


y = mx+ b becomes


y = 4/5 x + 10      or     y - 4/5 x = 10      or   5y - 4x = 50  or      4x-5y=-50    for the form required

 Apr 4, 2019

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