Question 1154014
Let {{{ a }}} = gallons of 20% solution needed
Let {{{ b }}} = gallons of 45% solution needed
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(1) {{{ a + b = 10 }}}
(2) {{{ ( .2a + .45b ) / 10 = .25 }}}
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(2) {{{ .2a + .45b = 2.5 }}}
(2) {{{ 20a + 45b = 250 }}}
(2) {{{ 4a + 9b = 50 }}}
Multiply both sides of (1) by {{{4}}}
and subtract (1) from (2)
(2) {{{ 4a + 9b = 50 }}}
(1) {{{ -4a - 4b = -40 }}}
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{{{ 5b = 10 }}}
{{{ b = 2 }}}
and
(1) {{{ a + 2 = 10 }}}
(1) {{{ a = 8 }}}
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8  gallons of 20% solution are needed
2  gallons of 45% solution are needed
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check:
(2) {{{ ( .2a + .45b ) / 10 = .25 }}}
(2) {{{ ( .2*8 + .45*2 ) / 10 = .25 }}}
(2) {{{ ( 1.6 + .9 ) / 10 = .25 }}}
(2) {{{ 2.5 = 10*.25 }}}
(2) {{{ 2.5 = 2.5 }}}
OK