document.write( "Question 596541: Using the Gauss-Jordan elimination method, solve the following linear system.
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Algebra.Com's Answer #377791 by AnlytcPhil(1806)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "2x - 3y -  z = 0\r\n" );
document.write( "3x + 2y + 2z = 2\r\n" );
document.write( " x + 5y + 3z = 2\r\n" );
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document.write( "I will assume you have had matrices. If not,\r\n" );
document.write( "post again and I will show you how to solve\r\n" );
document.write( "it without matrices:\r\n" );
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document.write( "The augmented matrix is\r\n" );
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document.write( "[2  -3  -1  |  0]\r\n" );
document.write( "[3   2   2  |  2]\r\n" );
document.write( "[1   5   3  |  2]\r\n" );
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document.write( "We need to get 0's in the three lower left\r\n" );
document.write( "positions, that is, in the positions below\r\n" );
document.write( "the upper left to lower-right diagonal:\r\n" );
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document.write( "To get a 0 where the 3 in row 2 column 1 is,\r\n" );
document.write( "we multiply row 1 temporarily by -3 and add it\r\n" );
document.write( "to 2 times row 2. This is easy to do mentally\r\n" );
document.write( "if you write -3 to the left of row 1 and 2 left\r\n" );
document.write( "of row 2:\r\n" );
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document.write( "-3[2  -3  -1  |  0]\r\n" );
document.write( " 2[3   2   2  |  2]\r\n" );
document.write( "  [1   5   3  |  2]\r\n" );
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document.write( "  [2  -3  -1  |  0]\r\n" );
document.write( "  [0  13   7  |  4]\r\n" );
document.write( "  [1   5   3  |  2]\r\n" );
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document.write( "To get a 0 where the 1 in row 3 column 1 is,\r\n" );
document.write( "we multiply row 1 temporarily by -1 and add it\r\n" );
document.write( "to 2 times row 3. This is easy to do mentally\r\n" );
document.write( "if you write -1 to the left of row 1 and 2 left\r\n" );
document.write( "of row 3:\r\n" );
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document.write( " -1[2  -3  -1  |  0]\r\n" );
document.write( "   [0  13   7  |  4]\r\n" );
document.write( "  2[1   5   3  |  2]\r\n" );
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document.write( "   [2  -3  -1  |  0]\r\n" );
document.write( "   [0  13   7  |  4]\r\n" );
document.write( "   [0  13   7  |  4]\r\n" );
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document.write( "To get a 0 where the 13 in row 3 column 2 is,\r\n" );
document.write( "we multiply row 2 temporarily by -1 and add it\r\n" );
document.write( "to 1 times row 3. This is easy to do mentally\r\n" );
document.write( "if you write -1 to the left of row 2 and 1 left\r\n" );
document.write( "of row 3:\r\n" );
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document.write( "-1[2  -3  -1 |  0]\r\n" );
document.write( "  [0  13   7 |  4]\r\n" );
document.write( " 2[0  13   7 |  4]\r\n" );
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document.write( "  [2  -3  -1 |  0]\r\n" );
document.write( "  [0  13   7 |  4]\r\n" );
document.write( "  [0   0   0 |  0]\r\n" );
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document.write( "Now that we have the three zeros, \r\n" );
document.write( "the above is the abbreviation of this system:\r\n" );
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document.write( "   2x -  3y - 1z = 0\r\n" );
document.write( "   0x + 13y + 7z = 4\r\n" );
document.write( "   0x +  0y + 0z = 0\r\n" );
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document.write( "or\r\n" );
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document.write( "   2x -  3y -  z = 0\r\n" );
document.write( "        13y + 7z = 4\r\n" );
document.write( "              0z = 0\r\n" );
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document.write( "The bottom equation,\r\n" );
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document.write( "              0z = 0\r\n" );
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document.write( "has as a solution \"z can equal to any real number\"\r\n" );
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document.write( "So we can select any letter to stand for any real number,\r\n" );
document.write( "say the letter k.\r\n" );
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document.write( "So z = k\r\n" );
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document.write( "We substitute k for z into the middle equation \r\n" );
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document.write( "       13y + 7z = 4\r\n" );
document.write( "       13y + 7k = 4\r\n" );
document.write( "            13y = 4 - 7k\r\n" );
document.write( "              y = \"4%2F13\" - \"7%2F13\"k\r\n" );
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document.write( "We substitute k for z and \"4%2F13\"-\"7%2F13\"k for y into \r\n" );
document.write( "the top equation:\r\n" );
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document.write( "    2x - 3y - z = 0\r\n" );
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document.write( "2x - 3(\"4%2F13\"-\"7%2F13\"k) - k = 0\r\n" );
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document.write( "2x - (\"12%2F13\" + \"21%2F13\"k) - k = 0\r\n" );
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document.write( "Clear of fractions by multiplying through by 13:\r\n" );
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document.write( "26x - 12 + 21k - 13k = 0\r\n" );
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document.write( "       26x - 12 + 8k = 0\r\n" );
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document.write( "                 26x = 12 - 8k\r\n" );
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document.write( "                   x = \"12%2F26\" - \"8k%2F26\"\r\n" );
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document.write( "                   x = \"6%2F13\" - \"4k%2F13\"\r\n" );
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document.write( "Solution:\r\n" );
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document.write( "(x, y, z) = (\"6%2F13\"-\"4k%2F13\", \"4%2F13\"-\"7%2F13\"k, k)    \r\n" );
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document.write( "Edwin
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