document.write( "Question 745957: Square or Cube? Find at least one positive integral solution to the equation: (integral means x and y must be integers) (-x)²y - y³ = 105. Please explain your response, because I don't have a clue. Thanks for your help! \n" ); document.write( "
Algebra.Com's Answer #454109 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "    (-x)²y - y³ = 105\r\n" );
document.write( "       x²y - y³ = 105\r\n" );
document.write( "     y(x² - y²) = 105\r\n" );
document.write( "y(x - y)(x + y) = 105\r\n" );
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document.write( "So we have three factors y, x-y, and x+y having\r\n" );
document.write( "product 105.\r\n" );
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document.write( "Since 105 = 1·3·5·7 either one of those factors is 1 \r\n" );
document.write( "or none are.\r\n" );
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document.write( "If none of the three are 1 then\r\n" );
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document.write( "{y, x-y, x+y} = {3,5,7}.  \r\n" );
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document.write( "x+y must be the largest, 7 so\r\n" );
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document.write( "{y, x-y} = {3,5}.\r\n" );
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document.write( "The sum of the two elements of the left set must equal the \r\n" );
document.write( "sum of the two elements on the right set, so\r\n" );
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document.write( "y + x-y = 3+5\r\n" );
document.write( "      x = 8\r\n" );
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document.write( "But that's impossible since the largest factor x+y is only 7 \r\n" );
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document.write( "So one of those factors must be 1.  Let's try y=1\r\n" );
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document.write( "y(x - y)(x + y) = 105\r\n" );
document.write( "1(x - 1)(x + 1) = 105\r\n" );
document.write( "         x² - 1 = 105\r\n" );
document.write( "             x² = 106\r\n" );
document.write( "              x = √106\r\n" );
document.write( "which is not an integer.  So the first factor y cannot be 1.\r\n" );
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document.write( "The largest factor x+y cannot be 1, so the second factor x-y must be 1, \r\n" );
document.write( "x-y = 1, so we have y = x-1,\r\n" );
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document.write( "Substituting that:\r\n" );
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document.write( "      y(x - y)(x + y) = 105\r\n" );
document.write( "  (x - 1)(1)(x + x-1) = 105\r\n" );
document.write( "      (x - 1)(2x - 1) = 105\r\n" );
document.write( "         2x² - 3x + 1 = 105\r\n" );
document.write( "       2x² - 3x - 104 = 0\r\n" );
document.write( "         (x-8)(2x+13) = 0\r\n" );
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document.write( "So x - 8 = 0,  2x + 13 = 0 \r\n" );
document.write( "       x = 8        2x = -13\r\n" );
document.write( "                     x = \"-13%2F2\" (discard)\r\n" );
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document.write( "So we have x = 8, and since y = x-1 = 8-1 = 7\r\n" );
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document.write( "So the only solution with positive integers is x = 8, y = 7.\r\n" );
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document.write( "Checking:\r\n" );
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document.write( "           (-x)²y - y³ = 105\r\n" );
document.write( "           (-8)²7 - 7³ = 105\r\n" );
document.write( "            64·7 - 343 = 105\r\n" );
document.write( "             448 - 343 = 105\r\n" );
document.write( "                   105 = 105\r\n" );
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document.write( "Edwin
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