document.write( "Question 97502This question is from textbook
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\n" ); document.write( "A farmer wants to buy 100 animals for $100.00 (must buy at least one of them)\r
\n" ); document.write( "\n" ); document.write( "Horses-$5.00
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\n" ); document.write( "\n" ); document.write( "How many horses,cows and chicks does the farmer need to have 100 and 100.00? \n" ); document.write( "

Algebra.Com's Answer #801078 by ikleyn(52797) $\"\"$ $\"About$

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\n" ); document.write( "A farmer wants to buy 100 animals for $100.00 (must buy at least one of them)
\n" ); document.write( "Horses-$5.00
\n" ); document.write( "Cows-$3.00
\n" ); document.write( "Chicks-.50
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document.write( "Let x = # of horses\r\n" );
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document.write( "    y = # of cows\r\n" );
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document.write( "    z = # of chicks\r\n" );
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document.write( "Then\r\n" );
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document.write( "     x  +  y +    z = 100     (1)\r\n" );
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document.write( "     5x + 3y + 0.5z = 100     (2)\r\n" );
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document.write( "Multiply equation (2) by 2 (both sides).  Keep equation (1) as is.\r\n" );
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document.write( "     x  +  y +   z = 100      (3)\r\n" );
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document.write( "    10x + 6y +   z = 100      (4)\r\n" );
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document.write( "Subtract equation (3) from equation (4).  You will get\r\n" );
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document.write( "     9x + 5y       = 100      (5)\r\n" );
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document.write( "All equations (1) - (5) are in integer numbers x, y, z. In particular, equation (5) is in integer numbers x and y.    \r\n" );
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document.write( "In equation (5), right side and the term 5y are multiples of 5.  Hence, the term 9x in equation (5) is a multiple of 5.\r\n" );
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document.write( "It gives for x only these two possibilities:  x= 5  or  x= 10.\r\n" );
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document.write( "So, consider both cases separately.\r\n" );
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document.write( "    a)  If  x= 5,  then from (5)  5y = 100 - 9*5 = 55;  hence,  y= 11.\r\n" );
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document.write( "                   Then from (1)  z = 100 - 5 - 11 = 84.\r\n" );
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document.write( "    b)  If  x= 10,  then from (5)  5y = 100 - 9*10 = 10;  hence,  y= 2.\r\n" );
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document.write( "                   Then from (1)  z = 100 - 10 - 2 = 88.\r\n" );
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document.write( "Thus the problem has two solutions in integer numbers:\r\n" );
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document.write( "    1)  5 horses,  11 cows  and  84 chicks,\r\n" );
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document.write( "and\r\n" );
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document.write( "    2)  10 horses,  2 cows  and  88 chicks.\r\n" );
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