document.write( "Question 115252: Use Gauss-Jordan elimination method to solve \r
\n" ); document.write( "\n" ); document.write( "2x-5y+5z=17
\n" ); document.write( "x-2y+3z=9
\n" ); document.write( "-x+3y=-4\r
\n" ); document.write( "\n" ); document.write( "I NEED HELP KNOWING HOW TO USE THIS METHOD
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #83831 by Edwin McCravy(20054) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "Use Gauss-Jordan elimination method to solve \r\n" );
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document.write( "2x-5y+5z=17\r\n" );
document.write( "x-2y+3z=9\r\n" );
document.write( "-x+3y=-4\r\n" );
document.write( "\r\n" );
document.write( "Line up the terms vertically:\r\n" );
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document.write( "2x - 5y + 5z = 17\r\n" );
document.write( " x - 2y + 3z =  9\r\n" );
document.write( "-x + 3y      = -4\r\n" );
document.write( "\r\n" );
document.write( "Fill up the \"hole\" in the bottom\r\n" );
document.write( "equation with \"+ 0z:\r\n" );
document.write( "\r\n" );
document.write( "2x - 5y + 5z = 17\r\n" );
document.write( " x - 2y + 3z =  9\r\n" );
document.write( "-x + 3y + 0z = -4 \r\n" );
document.write( "\r\n" );
document.write( "Fill in all invisible 1's for\r\n" );
document.write( "coefficients:\r\n" );
document.write( "\r\n" );
document.write( " 2x - 5y + 5z = 17\r\n" );
document.write( " 1x - 2y + 3z =  9\r\n" );
document.write( "-1x + 3y + 0z = -4\r\n" );
document.write( "\r\n" );
document.write( "Erase all the letters and\r\n" );
document.write( "equal signs:\r\n" );
document.write( "\r\n" );
document.write( " 2  - 5  + 5    17\r\n" );
document.write( " 1  - 2  + 3     9\r\n" );
document.write( "-1  + 3  + 0    -4 \r\n" );
document.write( "\r\n" );
document.write( "Erase the plus signs\r\n" );
document.write( "and move the minus\r\n" );
document.write( "signs close to the\r\n" );
document.write( "numbers as negative\r\n" );
document.write( "signs:\r\n" );
document.write( "\r\n" );
document.write( " 2   -5    5    17\r\n" );
document.write( " 1   -2    3     9\r\n" );
document.write( "-1    3    0    -4 \r\n" );
document.write( "\r\n" );
document.write( "Draw a vertical line\r\n" );
document.write( "where the equal signs\r\n" );
document.write( "were and put brackets\r\n" );
document.write( "around the whole thing.\r\n" );
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document.write( "[ 2   -5    5  | 17]\r\n" );
document.write( "[ 1   -2    3  |  9]\r\n" );
document.write( "[-1    3    0  | -4]\r\n" );
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document.write( "This is called the\r\n" );
document.write( "augmented matrix\r\n" );
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document.write( "The idea is to get 0's\r\n" );
document.write( "in the lower left three\r\n" );
document.write( "positions, where the \r\n" );
document.write( "three red numbers are \r\n" );
document.write( "below:\r\n" );
document.write( "\r\n" );
document.write( "[ 2   -5    5  | 17]\r\n" );
document.write( "[ 1   -2    3  |  9]\r\n" );
document.write( "[-1    3    0  | -4]\r\n" );
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document.write( "To get a 0 where the 1 is,\r\n" );
document.write( "temporarily multiply the top \r\n" );
document.write( "row thru by 1 and the middle\r\n" );
document.write( "row thru by -2.\r\n" );
document.write( "\r\n" );
document.write( " 1[ 2   -5    5  | 17]\r\n" );
document.write( "-2[ 1   -2    3  |  9]\r\n" );
document.write( "  [-1    3    0  | -4]\r\n" );
document.write( "\r\n" );
document.write( "  [ 2   -5    5  | 17]\r\n" );
document.write( "  [-2    4   -6  |-18]\r\n" );
document.write( "  [-1    3    0  | -4]\r\n" );
document.write( "\r\n" );
document.write( "Add the top row to the middle\r\n" );
document.write( "row and leave the top row as\r\n" );
document.write( "it is:\r\n" );
document.write( "\r\n" );
document.write( "  [ 2   -5    5  | 17]\r\n" );
document.write( "  [ 0   -1   -1  | -1]\r\n" );
document.write( "  [-1    3    0  | -4]\r\n" );
document.write( "\r\n" );
document.write( "To get a 0 where the -1 at the\r\n" );
document.write( "bottom left is, multiply the\r\n" );
document.write( "top row by 1 and the bottom row\r\n" );
document.write( "by 2:\r\n" );
document.write( "\r\n" );
document.write( " 1[ 2   -5    5  | 17]\r\n" );
document.write( "  [ 0   -1   -1  | -1]\r\n" );
document.write( " 2[-1    3    0  | -4]\r\n" );
document.write( "\r\n" );
document.write( "  [ 2   -5    5  | 17]\r\n" );
document.write( "  [ 0   -1   -1  | -1]\r\n" );
document.write( "  [-2    6    0  | -8]\r\n" );
document.write( "\r\n" );
document.write( "Add the top row to the bottom\r\n" );
document.write( "row, leaving the top row as it\r\n" );
document.write( "is:\r\n" );
document.write( "\r\n" );
document.write( "  [ 2   -5    5  | 17]\r\n" );
document.write( "  [ 0   -1   -1  | -1]\r\n" );
document.write( "  [ 0    1    5  |  9]\r\n" );
document.write( "\r\n" );
document.write( "To get a 0 where the 1 is,\r\n" );
document.write( "multiply the middle row by 1\r\n" );
document.write( "and add to the bottom row:\r\n" );
document.write( "\r\n" );
document.write( "  [ 2   -5    5  | 17]\r\n" );
document.write( " 1[ 0   -1   -1  | -1]\r\n" );
document.write( " 1[ 0    1    5  |  9]\r\n" );
document.write( "\r\n" );
document.write( "  [ 2   -5    5  | 17]\r\n" );
document.write( "  [ 0   -1   -1  | -1]\r\n" );
document.write( "  [ 0    0    4  |  8]\r\n" );
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document.write( "Now that there are 0's\r\n" );
document.write( "in those three positions,\r\n" );
document.write( "we rewrite the augmented\r\n" );
document.write( "matrix as a system of\r\n" );
document.write( "equations, by putting the\r\n" );
document.write( "variables and equal signs\r\n" );
document.write( "back in:\r\n" );
document.write( "\r\n" );
document.write( "  [ 2x  -5y   5z = 17]\r\n" );
document.write( "  [ 0x  -1y  -1z = -1]\r\n" );
document.write( "  [ 0x   0y   4z =  8]\r\n" );
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document.write( "Erase the brackets, the\r\n" );
document.write( "terms with 0 coefficients,\r\n" );
document.write( "the 1's, and move the negative signs\r\n" );
document.write( "left as minus signs:\r\n" );
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document.write( "    2x - 5y + 5z = 17 \r\n" );
document.write( "         -y -  z = -1 \r\n" );
document.write( "              4z =  8 \r\n" );
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document.write( "Solve the bottom equation for z:\r\n" );
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document.write( "              4z = 8\r\n" );
document.write( "               z = 2\r\n" );
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document.write( "Substitute z = 2 into the middle\r\n" );
document.write( "equation:\r\n" );
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document.write( "         -y -  z = -1\r\n" );
document.write( "         -y -  2 = -1\r\n" );
document.write( "              -y =  1\r\n" );
document.write( "               y = -1\r\n" );
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document.write( "Substitute y = -1 and z = 2 into\r\n" );
document.write( "the top equation:\r\n" );
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document.write( "     2x - 5y + 5z = 17\r\n" );
document.write( "2x - 5(-1) + 5(2) = 17\r\n" );
document.write( "      2x + 5 + 10 = 17\r\n" );
document.write( "          2x + 15 = 17\r\n" );
document.write( "               2x =  2\r\n" );
document.write( "                x =  1\r\n" );
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document.write( "So the solution is\r\n" );
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document.write( "(x, y, z) = (1, -1, 2)\r\n" );
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document.write( "Edwin
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