Question 681960
You need an equation for each trip
let {{{ t }}} = time for GA to KY flight in hrs
{{{ t - .5 }}} = time for KY to GA flight in hrs
Let {{{ r }}} = rate going from KY to GA in km/hr
{{{ r - 50 }}} = rate going from GA to KY in km/hr
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Equation for GA to KY:
(1) {{{ 500 = ( r - 50 )*t }}}
Equation for KY to GA:
(2) {{{ 500 = r*( t - .5 ) }}}
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(1) {{{ 500 = r*t - 50t }}}
(1) {{{ r*t = 500 + 50t }}}
(1) {{{ r = 500/t + 50 }}}
Substitute (1) into (2)
(2) {{{ 500 = r*( t - .5 ) }}}
(2) {{{ 500 = ( 500/t + 50 )*( t - .5 ) }}}
(2) {{{ 500 = 500 + 50t - 250/t - 25 }}}
(2) {{{ 250/t = 50t - 25 }}}
(2) {{{ 250 = 50t^2 - 25t }}}
(2) {{{ 2t^2 - t - 10 = 0 }}}
Use quadratic formula
{{{ t = (-b +- sqrt( b^2 - 4*a*c )) / (2*a) }}}
{{{ a = 2 }}}
{{{ b = -1 }}}
{{{ c = -10 }}}
{{{ t = (-(-1) +- sqrt( (-1)^2 - 4*2*(-10) )) / (2*2) }}}
{{{ t = ( 1 +- sqrt( 1 + 80 )) / 4 }}}
{{{ t = ( 1 + 9 ) / 4 }}}
{{{ t = 5/2 }}}
and
(1) {{{ r = 500/t + 50 }}}
(1) {{{ r = 500/(5/2) + 50 }}}
(1) {{{ r = 200 + 50 }}}
(1) {{{ r = 250 }}} 
and
{{{ r - 50 = 250 - 50 }}}
{{{ r - 50 = 200 }}}
The rate from GA to KY is 200 km/hr
The rate from KY to GA is 250 km/hr