document.write( "Question 615967: We're trying to help our soon with his math homework but we're totally stuck one this problem:\r
\n" ); document.write( "\n" ); document.write( "A store has a sale on almonds, pecans and pistachios. One lb of almonds, one lb of pecans and one lb of pistachios cost $12. Two lbs of almonds and three lbs of pecans cost $16. Three lbs of pecans and two lbs pistachios cost $24. Find the price of each kind of nut.\r
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Algebra.Com's Answer #852457 by ikleyn(53298) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "We're trying to help our $\"highlight%28cross%28soon%29%29\"$ son with his math homework but we're totally stuck one this problem:
\n" ); document.write( "A store has a sale on almonds, pecans and pistachios. One lb of almonds, one lb of pecans and one lb of pistachios
\n" ); document.write( "cost $12. Two lbs of almonds and three lbs of pecans cost $16. Three lbs of pecans and two lbs pistachios cost $24.
\n" ); document.write( "Find the price of each kind of nut.\r
\n" ); document.write( "\n" ); document.write( "Thank you in advance for any help.
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\n" ); document.write( "\n" ); document.write( "        The solution by @Theo is tedious and boring, and you're unlikely to learn anything useful from his post.\r
\n" ); document.write( "\n" ); document.write( "        Meanwhile, the problem has a much simpler solution, which teaches you to think and to approach the problem \r
\n" ); document.write( "\n" ); document.write( "        in a new way. This solution is effective and instructive, and can work in many other problems.\r
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document.write( "Let  'a'  be the pounds of almonds,\r\n" );
document.write( "     'b'  be the pounds of pecans, and\r\n" );
document.write( "     'c'  be the pounds of pistachios.\r\n" );
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document.write( "You have 3 equations that need to be solved simultaneously.\r\n" );
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document.write( "     a +  b + c  = 12,     (1)\r\n" );
document.write( "    2a + 3b      = 16,     (2)\r\n" );
document.write( "         3b + 2c = 24.     (3)\r\n" );
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document.write( "This system of equations has a hidden symmetry, which will help us to solve.\r\n" );
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document.write( "Add equations (2) and (3)  (both sides separately).  You will get\r\n" );
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document.write( "    2a + 6b + 2c = 40.     (4)\r\n" );
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document.write( "From equation (4), subtract equation (1), multiplied by 2.  \r\n" );
document.write( "The terms with  'a'  and  'c'  will cancel, and you will get\r\n" );
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document.write( "    6b - 2b = 40 - 2*12,\r\n" );
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document.write( "or\r\n" );
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document.write( "       4b = 16.\r\n" );
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document.write( "Hence,  b = 16/4 = 4.\r\n" );
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document.write( "Now, substitute b=4 into equation (2) and find  'a'\r\n" );
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document.write( "    2a + 3*4 = 16  --->  2a = 16 - 12 = 4  --->  a = 4/2 = 2.\r\n" );
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document.write( "Next, substitute b=4 into equation (3) and find  'c'\r\n" );
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document.write( "    3*4 + 2c = 24  --->  2c = 24 - 12 = 12  --->  c = 12/2 = 6.\r\n" );
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document.write( "At this point, the problem is solved completely.\r\n" );
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document.write( "ANSWER.  a = $2 per pound (almonds);  b = $4 per pound (pecans),  and  c = $6 per pound (pistachios).\r\n" );
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The lesson to learn\n" ); document.write( "\n" ); document.write( " If you have to solve a system of three equations with three unknowns,
\n" ); document.write( " look if there is a hidden symmetry in it that could help.
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