Question 934770
r, rate of plane in calm air
w, speed of the wind.   Let w=25 mph.
t, time of travel with the wind
v, quantity of time in excess of t.  Let v=2 hours.
q, quantity of distance against the wind.  Let q=375 miles.
Also, q is the quantity of distance flown with the wind.



______________________speed___________time______________distance
AGNST_________________(r-w)___________t+v________________q
WITH__________________(r+w)____________t_________________q


The idea in this way is to keep everything in symbols, and save all computations for later.  This is not always necessary.  The only two unknown variables are r and t.  Use the travel rate equation to form two equations for against and for with, the wind.


{{{system((r-w)(t+v)=q,(r+w)*t=q)}}}


Solve the system for r, and its value.  This just requires several arithmetic, or algebra steps.


{{{system(rt-wt+rv-wv=q,rt+wt=q)}}},  but the second equation here was better in its original form.  You want to substitute for t.


The WITH equation is also {{{t=q/(r+w)}}}.  Substitute into "first" equation, the AGAINST one.


{{{r(q/(r+w))-w(q/(r+w))+rv-wv=q}}}


{{{rq-wq+rv(r+w)-wv(r+w)=q(r+w)}}}


{{{rq-wq+vr^2+rw-wvr-w^2v=qr+qw}}}


{{{qr+vr^2+wr-wvr-w^2v-qr-qw=0}}}


{{{vr^2+qr+wr-wvr-qr-w^2v-qw=0}}}


{{{vr^2+(q+w-wv-q)r-w^2v-qw=0}}}


{{{highlight_green(vr^2+(w-wv)r-(w^2v+qw)=0)}}}
You could keep going completely in symbols, but now would be a good place for the next step to plug in the known values, AND THEN finish the solving of this quadratic equation for r.  Note, the question asks for r, the rate of the plane in calm air, and r is the only variable UNKNOWN in this quadratic equation.  You would then use general solution ....