Question 806686
Let liters of lime juice = {{{ a }}}
Let liters of lemon juice = {{{ b }}}
Let liters of water = {{{ c }}}
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(1) {{{ a/b = 3/6 }}}
(2) {{{ b/c = 6/7 }}}
(3) {{{ a + b + c = 8 }}}
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(1) {{{ 6*a = 3*b }}}
(1) {{{ a = (1/2)*b }}}
and
(2) {{{ b/c = 6/7 }}}
(2) {{{ 7*b = 6*c }}}
(2) {{{ c = (7/6)*b }}}
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Substitute (1) and (2) into (3)
(3) {{{ a + b + c = 8 }}}
(3) {{{ (1/2)*b + b + (7/6)*b = 8 }}}
Multiply both sides by {{{ 6 }}}
(3) {{{ 3b + 6b + 7b = 48 }}}
(3) {{{ 16b = 48 }}}
(3) {{{ b = 3 }}}
and, since
(1) {{{ a = (1/2)*b }}}
(1) {{{ a = 1.5 }}}
and
(2) {{{ c = (7/6)*b }}}
(2) {{{ c = (7/6)*3 }}}
(2) {{{ c = 3.5 }}}
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There are 1.5 liters of lime juice 
There are 3 liters of lemon juice
There are 3.5 liters of water
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check:
(3) {{{ a + b + c = 8 }}}
(3) {{{ 1.5 + 3 + 3.5 = 8 }}}
(3) {{{ 8 = 8 }}}