Question 452270
m - rate of Matt
c - rate of Connor
:
t1- time for Matt to walk the course
t2- time for Connor to walk the course
:
Distance passed by Matt is 12 mi    -> {{{m*t1=12}}} -> {{{t1=12/m}}}
Distance passed by Connor is 12 mi  -> {{{c*t2=12}}} -> {{{t2=12/c}}}
:
Matt walks 1mi/h faster than Connor -> {{{m=c+1}}}
Matt finishes 1 hour before Connor  -> {{{t1=t2-1}}} -> {{{12/m=12/c - 1}}}
:
Substitute m from the first equation into the second
:
{{{12/(c+1)=12/c - 1}}}, multiply whole equation by c(c+1)
{{{12c=12(c+1)-c(c+1)}}}, distribute multiplication
{{{12c=12c+12-c^2-c}}}, move everything to the left side
{{{12c-12c-12+c^2+c=0}}}, combine 
{{{c^2+c-12=0}}}, this is quadratic equation that can be factorized
{{{(c-3)(c+4)=0}}}, solutions are c=3 and c=-4
:
Only acceptable solution is positive number so rate for Connor is 3mi/h
:
Time he needs to finish the course is {{{t2=12/c}}} = {{{12/3}}} = 4 hours