Question 1030960
Let {{{ a }}} = number of servings of dish A allowed
Let {{{ b }}} = number of servings of dish B allowed
Let {{{ c }}} = number of servings of dish C allowed
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Grams of fat allowed:
(1) {{{ 1a + 2b + 2c = 15 }}}
Grams of carbs allowed:
(2) {{{ 2a + 1b + 4c = 24 }}}
Grams of protein allowed:
(3) {{{ 4a + 3b + 3c = 30 }}}
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Multiply both sides of (1) by {{{ 2 }}}
and subtract (2) from (1)
(1) {{{ 2a + 4b + 4c = 30 }}}
(2) {{{ -2a - 1b - 4c = -24 }}}
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{{{ 3b = 6 }}}
{{{ b = 2 }}}
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Plug this result into (1) and (3)
(1) {{{ 1a + 2*2 + 2c = 15 }}}
(1) {{{ 1a + 2c = 11 }}}
and
(3) {{{ 4a + 3*2 + 3c = 30 }}}
(3) {{{ 4a + 3c = 24 }}}
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Multiply both sides of (1) by {{{ 4 }}}
and subtract (3) from (1)
(1) {{{ 4a + 8c = 44 }}}
(2) {{{ -4a - 3c = -24 }}}
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{{{ 5c = 20 }}}
{{{ c = 4 }}}
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Plug these results back into (1)
(1) {{{ 1a + 2c = 11 }}}
(1) {{{ 1a + 2*4 = 11 }}}
(1) {{{ a = 3 }}}
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3 servings of dish A are allowed
2 servings of dish B are allowed
4 servings of dish C are allowed
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check:
(1) {{{ 1a + 2b + 2c = 15 }}}
(1) {{{ 1*3 + 2*2 + 2*4 = 15 }}}
(1) {{{ 3 + 4 + 8 = 15 }}}
(1) {{{ 15 = 15 }}}
OK -you can check (2) and (3)