Question 839479
Let {{{ d }}} = the distance in miles she cycled
{{{ 50 - d }}} = the distance in miles she ran
Let {{{ t }}} = the time in hours she spent cycled
{{{ 5.5 - t }}} = the time in hours she spent running
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Equation for cycling:
(1) {{{ d = 12t }}}
Equation for running:
(2) {{{ 50 - d = 8*( 5.5 - t ) }}}
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This is 2 equations and 2 unknowns, so it's solvable
(2) {{{ 50 - d = 44 - 8t }}}
Add {{{ d }}} to both sides
(2) {{{ 50 = d + 44 - 8t }}}
Subtract {{{ 8t }}} from both sides
(2) {{{ 50 + 8t = d + 44 }}}
Subtract {{{ 44 }}} from both sides 
(2) {{{ d = 8t + 6 }}}
Substitute (1) into (2)
(2) {{{ 12t = 8t + 6 }}}
Subtract {{{ 8t }}} from both sides
(2) {{{ 4t = 6 }}}
(2) {{{ t = 1.5 }}}
and
{{{ 5.5 - t = 4 }}}
and, since
(1) {{{ d = 12t }}}
(1) {{{ d = 12*1.5 }}}
(1) {{{ d = 18 }}}
and
{{{ 50 - d = 32 }}}
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18 = the distance in miles she cycled
32 = the distance in miles she ran
1.5 = the time in hours she spent cycled
4 = the time in hours she spent running
check:
(2) {{{ 50 - d = 8*( 5.5 - t ) }}} 
(2) {{{ 50 - 18 = 8*( 5.5 - 1.5 ) }}}
(2) {{{ 32 = 8*4 }}}
(2) {{{ 32 = 32 }}}
OK
(1) {{{ d = 12t }}}
(1) {{{ 18 = 12*1.5 }}} 
(1) {{{ 18 = 18 }}}
OK