Question 813725
Direction__________speed____________time________________distance
UPstream___________x-9______________y+5_________________55
DOWNstream_________x+9______________y___________________55


x is as you have assigned, the speed of the boat in still water.
y is the time in hours to travel downstream the 55 miles.


Basic idea is Rate*Time=Distance.  This gives two equations:

{{{(x-9)(y+5)=55}}} and {{{(x+9)*y=55}}}, neither linear.


{{{xy-9y+5x-45=55}}} and {{{xy+9y=55}}}.
If we have one more equation, this might be easier.  Trying to equate the distances, {{{(x-9)(y+5)=y(x+9)}}}
{{{xy-9y+5x-40=xy+9y}}}
{{{-9y+5x-40=9y}}}
{{{-18y+5x=40}}}
Continue solving this for x (or y if someone wants):
{{{5x=40+18y}}}
{{{x=8+(18/5)y}}}
Try substituting this into the simplified downstream equation:
{{{xy+9y=55}}}
{{{(8+(18/5)y)y+9y=55}}}
{{{8y+(18/5)y^2+9y-55=0}}}
multiply both sides by 25,
{{{25*17y+18y^2-25*55=0}}}
{{{18y^2+425y-1375=0}}}


General Solution to Quadratic Equation:
{{{y=(-425+- sqrt(425^2-4*18*(1375)))/(2*18)}}}
{{{y=(-425+sqrt(279625))/(2*18)}}}

279625=25*5*2237

{{{y=(-425+5*sqrt(5*2237))/36}}}
{{{y=103.7958/36}}}
{{{highlight(y=2.88)}}}  HOURS
This means that y+5 to go upstream was 7.88 hours.


Remember in this analysis, we found, {{{x=8+(18/5)y}}}  ?
Now we can calculate the boat's speed in still water.
{{{highlight(x=8+(18/5)*(2.88)=highlight(18.4))}}} miles per hour.