Question 696133
Let {{{ a }}} = gallons of 20% solution needed
Let {{{ b }}} = gallons of 50% solution needed
given:
(1) {{{ a + b = 10 }}}
(2) {{{ ( .2a + .5b ) / 10 = .3 }}}
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(2) {{{ .2a + .5b = 3 }}}
(2) {{{ 2a + 5b = 30 }}}
Multiply both sides of (1) by {{{ 2 }}} and
subtract (1) from (2)
(2) {{{ 2a + 5b = 30 }}}
(1) {{{ -2a - 2b = -20 }}}
{{{ 3b = 10 }}}
{{{ b = 10/3 }}} 
and, since
(1) {{{ a + b = 10 }}}
(1) {{{ a + 10/3 = 10 }}}
(1) {{{ a = 30/3 - 10/3 }}}
(1) {{{ a = 20/3 }}}
3 and 1/3 gallons of 20% solution are needed
6 and 2/3 gallons of 50% solution are needed
check answer:
(2) {{{ ( .2a + .5b ) / 10 = .3 }}}
(2) {{{ ( (2/10)*(20/3) + (1/2)*(10/3) ) / 10 = 3/10 }}}
(2) {{{ ( 40/30 + 10/6 ) / 10 = 3/10 }}}
(2) {{{ 40/30 + 50/30 = 3 }}}
(2) {{{ 90/30 = 3 }}}
(2) {{{ 3 = 3 }}}
OK