Question 983517
r, the speed in still-water
c, speed of the river current
d=12
u=2, time with the current
b=4, time against the current


Basic uniform rates for travel, RT=D to relate speed, time, distance


"Paddling at the same pace", must mean kayaker still uses his own speed r, but that the speed of current, c, changes the effective rate according to which direction he paddles.


<pre>
                           speed    time    distance
With current, up           r+c      u       d
Against current, back      r-c      b       d

</pre>


Unknown variables are r and c; but you are interested in answering for solving r.


{{{system((r+c)*u=d,(r-c)*b=d)}}}


{{{system(ur+uc=d,br-bc=d)}}}


Elimination Method can be used  (but you could use substitution instead; your choice). Best to first eliminate r.  I will instead use substitution.


{{{uc=d-ur}}}
{{{c=(d-ur)/u}}}
-
{{{br-b(d-ur)/u=d}}}, subst into second equation.  Notice this equation uses only the unknown variable r, to be found as asked.
{{{bru-b(d-ur)=d*u}}}
{{{bru-bd+bur=du}}}
{{{bru+bur-bd=du}}}
{{{bur+bur=du+bd}}}
{{{r(bu+bu)=du+bd}}}
{{{highlight(r=(du+bd)/(2bu))}}}


Substitute for d, u, b,  and evaluate r.


{{{r=12(2+4)/(2*4*2)}}}

{{{r=(6*6)/(2*4)}}}

{{{r=(3*2*3)/(2*2)}}}

{{{r=9/2}}}
{{{highlight(r=4&1/2)}}}