Question 129829
Let's use the formula d=r*t


Since "a canoeist travels 82 miles at a certain speed", this means that our first equation is 


{{{82=rt[1]}}} Notice how I plugged in {{{d=82}}} and {{{t=t[1]}}}



Also, since the "canoeist travels 19 miles on the second part of the trip at a speed 5 mph slower", the second equation is 


{{{19=(r-5)t[2]}}} Notice how I plugged in {{{d=19}}} and {{{t=t[2]}}}. Also, I replaced r with {{{r-5}}} (since the canoeist traveled 5 mph slower)



Finally, because the "total time for the trip is 3 hrs", the third equation is 



{{{t[1]+t[2]=3}}}




So we have the three equations



{{{82=rt[1]}}} 


{{{19=(r-5)t[2]}}}


{{{t[1]+t[2]=3}}}



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{{{82=rt[1]}}}  Start with the first equation



{{{82/r=t[1]}}}  Divide both sides by r  to isolate {{{t[1]}}}



{{{t[1]=82/r}}}  Rearrange the equation




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{{{19=(r-5)t[2]}}} Start with the second equation



{{{19/(r-5)=t[2]}}}  Divide both sides by {{{r-5}}} to isolate {{{t[2]}}}



{{{t[2]=19/(r-5)}}}  Rearrange the equation




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Now let's focus on the third equation {{{t[1]+t[2]=3}}}


{{{82/r+19/(r-5)=3}}} Plug in {{{t[1]=82/r}}} and {{{t[2]=19/(r-5)}}} into the third equation




{{{r(r-5)(82/r+19/(r-5))=r(r-5)(3)}}} Multiply both sides by the LCD {{{r(r-5)}}}




{{{82(r-5)+19r=3r(r-5)}}} Distribute and multiply 



{{{82r-410+19r=3r^2-15r}}} Distribute again 




{{{82r-410+19r-3r^2+15r=0}}}  Subtract 3r^2 from both sides.  Add 15r to both sides. 



{{{-3r^2+116r-410=0}}} Combine like terms



So after using the quadratic formula or a calculator, we get the solutions


{{{r=3.93492}}} or {{{r=34.73175}}}



However, since you cannot take 5 mph away from 3.93 mph (you'll get a negative answer if you do), this means that our only answer is 


{{{r=34.73175}}}



So the speed on the first part of the trip was about 34.73 mph