document.write( "Question 1011499: Brainstorming !! I send a famer to buy 100 animals with $100.00 dollars,how many off each he can buy? Ducks $10 each, pigs $3.00 each, chicks .50¢ each??
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Algebra.Com's Answer #627156 by Edwin McCravy(20060)\"\" \"About 
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Brainstorming !! I send a famer to buy 100 animals with
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document.write( "This is a number theory problem, three unknowns but only\r\n" );
document.write( "2 equations with the stipulation that all 3 variables can\r\n" );
document.write( "only be non-negative integers.\r\n" );
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document.write( "D + P + C = 100\r\n" );
document.write( "$10.00D + $3.00P + $0.5C = $100.00\r\n" );
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document.write( "Multiply the second equation by 10 and remove the dollar signs:\r\n" );
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document.write( "100D + 30P + 5C = 1000\r\n" );
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document.write( "Divide it through by 5\r\n" );
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document.write( "1)  20D + 6P + C = 200\r\n" );
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document.write( "Solve the first equation for D = 100 - P - C\r\n" );
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document.write( "Substitute that into\r\n" );
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document.write( "20D + 6P + C = 200\r\n" );
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document.write( "20(100 - P - C) + 6P + C = 200\r\n" );
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document.write( "2000 - 20P - 20C + 6P + C = 200\r\n" );
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document.write( "2000 - 14P - 19C = 200\r\n" );
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document.write( "1800 = 14P + 19C\r\n" );
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document.write( "2)  14P + 19C = 1800\r\n" );
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document.write( "The smallest coefficient in absolute value is 14,\r\n" );
document.write( "so we write 19 and 1800 in terms of their nearest\r\n" );
document.write( "multiple of 14.  [We divide 1800/14 = and get\r\n" );
document.write( "approximately 128.6, so the nearest multiple of\r\n" );
document.write( "14 to 1800 is 128*14 = 1792, so 1800 = 1792 + 8\r\n" );
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document.write( "14P + (14 + 5)C = 1792 + 8\r\n" );
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document.write( "14P + 14C + 5C = 1792 + 8\r\n" );
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document.write( "We divide each term by 14\r\n" );
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document.write( "P + C + 5C/14 = 128 + 8/14\r\n" );
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document.write( "Get the fractions on the left and the other terms \r\n" );
document.write( "on the right:\r\n" );
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document.write( "5C/14 -8/14 = 128 - P - C\r\n" );
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document.write( "Since the right side is an integer, so is the left side.\r\n" );
document.write( "Let that integer by A, Then\r\n" );
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document.write( "3)  5C/14 - 8/14 = A\r\n" );
document.write( "4)  128 - P - C = A\r\n" );
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document.write( "Clear 3) of fractions\r\n" );
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document.write( "5C - 8 = 14A\r\n" );
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document.write( "5)  5C - 14A = 8\r\n" );
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document.write( "The smallest coefficient in absolute value is 5,\r\n" );
document.write( "so we write 14 and 8 in terms of their nearest\r\n" );
document.write( "multiple of 5.\r\n" );
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document.write( "5C - (15-1)A = 10-2\r\n" );
document.write( "5C - 15A + A = 10-2\r\n" );
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document.write( "Divide through by 5:\r\n" );
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document.write( "C - 3A + A/5 = 2-2/5  \r\n" );
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document.write( "Get the fractions on the left and the other terms \r\n" );
document.write( "on the right:\r\n" );
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document.write( "A/5 + 2/5 = 3A - C + 2\r\n" );
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document.write( "Since the right side is an integer, so is the left side.\r\n" );
document.write( "Let that integer by B, Then\r\n" );
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document.write( "6)  A/5 + 2/5 = B\r\n" );
document.write( "7)  3A - C + 2 = B\r\n" );
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document.write( "Clear 6) of fractions:\r\n" );
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document.write( "A + 2 = 5B \r\n" );
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document.write( "A = 5B-2\r\n" );
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document.write( "Substitute 5B-2 for A in 7)\r\n" );
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document.write( "3(5B-2) - C + 2 = B\r\n" );
document.write( "15B - 6 - C + 2 = B\r\n" );
document.write( "15B - 4 - C = B\r\n" );
document.write( "14B - 4 - C = 0 \r\n" );
document.write( "14B - 4 = C\r\n" );
document.write( "C = 14B - 4\r\n" );
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document.write( "Substitute 12B - 4 for C and 5B-2 for A in 4)  \r\n" );
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document.write( "128 - P - C = A\r\n" );
document.write( "128 - P - (14B-4) = 5B - 2\r\n" );
document.write( "128 - P - 14B + 4 = 5B - 2\r\n" );
document.write( "132 - P - 14B = 5B - 2\r\n" );
document.write( "134 - 19B = P\r\n" );
document.write( "P = 134 - 19B\r\n" );
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document.write( "Substitute 134-19B for P, 14B-4 for C, and 5B-2 for A in 1)\r\n" );
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document.write( "1)  20D + 6P + C = 200\r\n" );
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document.write( "    20D + 6(134-19B) + 14B-4 = 200\r\n" );
document.write( "    20D + 804 - 114B + 14B - 4 = 200\r\n" );
document.write( "    20D + 800 - 100B = 200\r\n" );
document.write( "    20D = 100B - 600\r\n" );
document.write( "      D = 5B - 30\r\n" );
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document.write( "So \r\n" );
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document.write( "D = 5B - 30\r\n" );
document.write( "P = 134 - 19B\r\n" );
document.write( "C = 14B - 4\r\n" );
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document.write( "Each of those must be non-negative integers, so\r\n" );
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document.write( "5B - 30 >= 0     134 - 19B >= 0           14B - 4 >= 0      \r\n" );
document.write( "     5B >= 30          -19B >= -134            14B >= 4\r\n" );
document.write( "      B >= 6             B <= 7.0526            B >= 2/7\r\n" );
document.write( "                         B <= 7\r\n" );
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document.write( "Therefore B is 6 or 7\r\n" );
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document.write( "If B=6, then\r\n" );
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document.write( "First answer:\r\n" );
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document.write( "D = 5B - 30 = 5(6)-30 = 30 - 30 = 0 ducks\r\n" );
document.write( "P = 134 - 19B = 134-19(6) = 134 - 114 = 20 pigs \r\n" );
document.write( "C = 14B - 4 = 14(6)-4 = 84-4 = 80 chickens\r\n" );
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document.write( "If B=7, then\r\n" );
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document.write( "Second answer:\r\n" );
document.write( "D = 5B - 30 = 5(7)-30 = 35 - 30 = 5 ducks\r\n" );
document.write( "P = 134 - 19B = 134-19(7) = 134 - 133 = 1 pigs \r\n" );
document.write( "C = 14B - 4 = 14(7)-4 = 98-4 = 94 chickens\r\n" );
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document.write( "Edwin
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