Question

Andre went ___ miles before his tire went flat :-)

Answer 1

m = miles before the tire went flat

so he left the house zipping his way through at 9mph, then heck what the dickens? a flat tire "m" miles later, man!!! so he turned around back home at 3mph, he had gone "m" miles over, so going back home will be the same "m" back.

We know the trip in total took 8 hours, so if say he took "h" hours on the way over, then on the way back is just "8 - h" hours, namely the slack from 8 and "h".

`{\Large \begin{array}{llll} \underset{distance}{d}=\underset{rate}{r} \stackrel{time}{t} \end{array}} \\\\[-0.35em] ~\dotfill\\\\ \begin{array}{lcccl} &\stackrel{miles}{distance}&\stackrel{mph}{rate}&\stackrel{hours}{time}\\ \cline{2-4}&\\ \textit{On his way over}&m&9&t\\ \textit{On his way back}&m&3&8-t \end{array}\hspace{5em} \begin{cases} m=(9)(t)\\\\ m=(3)(8-t) \end{cases} \\\\[-0.35em] ~\dotfill`

`\stackrel{\textit{using the 1st equation}}{m=9t}\implies \cfrac{m}{9}=t \\\\\\ \stackrel{\textit{substituting on the 2nd equation}}{m=3\left( 8-\cfrac{m}{9} \right)}\implies m=24-\cfrac{m}{3}\implies m+\cfrac{m}{3}=24 \\\\\\ \cfrac{4m}{3}=24\implies 4m=72\implies m=\cfrac{72}{4}\implies \boxed{m=18}`