document.write( "Question 743687: 3 numbers add to 27 and multiply together to make 288. These numbers also fall into an arithmetic sequence. Show your work.\r
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Algebra.Com's Answer #453025 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "3 numbers add to 27 \r\n" );
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document.write( "x + y + z = 27\r\n" );
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document.write( "and multiply together to make 288. \r\n" );
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document.write( "xyz = 288\r\n" );
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document.write( "These numbers also fall into an arithmetic sequence.\r\n" );
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document.write( "That just means that they are equally spaced.  That is,\r\n" );
document.write( "the diffence between the 2nd and 1st equals the difference\r\n" );
document.write( "between the 3rd and 2nd.\r\n" );
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document.write( "y - x = z - y\r\n" );
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document.write( "which simplifies to\r\n" );
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document.write( "-x + 2y - z = 0 \r\n" );
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document.write( "So we have the system:\r\n" );
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document.write( " x +  y + z = 27\r\n" );
document.write( "xyz = 288\r\n" );
document.write( "-x + 2y - z = 0\r\n" );
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document.write( "Adding the 1st and 3rd equations term by term\r\n" );
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document.write( " x +  y + z = 27\r\n" );
document.write( "-x + 2y - z =  0\r\n" );
document.write( "----------------\r\n" );
document.write( "     3y     = 27\r\n" );
document.write( "          y = 9\r\n" );
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document.write( "Substitute that in the first and 2nd equations:\r\n" );
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document.write( " x +  9 + z = 27          x(9)z = 288\r\n" );
document.write( "      x + z = 18             xz = 32\r\n" );
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document.write( "So now we have the system\r\n" );
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document.write( "          x + z = 18             \r\n" );
document.write( "             xz = 32\r\n" );
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document.write( "Solve the first for z:   z = 18 - x \r\n" );
document.write( "Substitute in the second equation:\r\n" );
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document.write( "      x(18 - x) = 32\r\n" );
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document.write( "       18x - x² = 32\r\n" );
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document.write( " -x² + 18x - 32 = 0\r\n" );
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document.write( "  x² - 18x + 32 = 0\r\n" );
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document.write( "(x - 16)(x - 2) = 0\r\n" );
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document.write( "x - 16 = 0;  x - 2 = 0\r\n" );
document.write( "     x = 16;     x = 2\r\n" );
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document.write( "Using x = 16      Using x = 2\r\n" );
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document.write( "z = 18 - x        z = 18 - x      \r\n" );
document.write( "z = 18 - 16       z = 18 - 2\r\n" );
document.write( "z = 2             z = 16\r\n" );
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document.write( "So x and z can switch places.  One is 16 and the other is 2\r\n" );
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document.write( "So the arithmetic sequence can either go:\r\n" );
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document.write( "2, 9, 16  where the common difference is 7\r\n" );
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document.write( "or\r\n" );
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document.write( "16, 9, 2  where the common difference is -7\r\n" );
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document.write( "Edwin
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