Solve this Word Problem

Question

Bryn paddled her canoe 10 miles upstream, and then paddled back to her starting point.  If the rate of the current was 3 mph and the entire trip took 3 1/2 hours, what was Bryn's rate?

Answer 1

distance=rate*time

d=r*t

10=(r-3)t               (against the current)

10=(r+3)(3.5-t)     (with the current)

10/(r-3)=t from the first equation

substitute this into the second equation

10=(r+3)(3.5-[10/[r-3])

multiply both sides by (r-3)

10(r-3)=(r+3)(r-3)(3.5-[10/(r-3)])

10(r-3)=(r+3)[(3.5)(r-3)-10]

10(r-3)=(r+3)(3.5r-10.5-10)

10(r-3)=(r+3)(3.5r-20.5)

10r-30=3.5r²+10.5r-20.5r-61.5

10r-30=3.5r²-10r-61.5

3.5r²-10r-61.5-10r+30=0 (get all terms on one side)

3.5r²-20r-31.5=0

multiply both sides by 2 to eliminate the decimals

7r²-40r-63=0

7*63=7*7*9 and (7*7)-9=40, the coefficient of the middle term

(7r+9)(r-7)=0

r-7=0

r=7 mph is Bryn's rate

check: 10=(7-3)t, 10=4t, t=2.5 hours upstream

          10=(7+3)(3.5-2.5)

          10=10(1), t=1 hour downstream