document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=812509>Question 812509</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am having problems figuring out matrices and row operations. If anyone can help with this it would be a huge help.\r<BR>\n" );
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document.write( "Use matricies to find the general solution of the following system, if a solution exists. <BR>\n" );
document.write( "2x - y + z = 8<BR>\n" );
document.write( "3x + y - 6z = -28<BR>\n" );
document.write( "x - y + 2z = 12 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #489200 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20060)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=489200\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "The other tutor doesn't know about\r\n" );
document.write( "solving dependent systems of equations\r\n" );
document.write( "with Gaussian elimination.  Here's how\r\n" );
document.write( "to solve that using matrices and \r\n" );
document.write( "Gaussian elimination:\r\n" );
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document.write( "2x - y + z = 8\r\n" );
document.write( "3x + y - 6z = -28\r\n" );
document.write( "x - y + 2z = 12\r\n" );
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document.write( "Line up the terms vertically:\r\n" );
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document.write( "2x -  y +  z =   8\r\n" );
document.write( "3x +  y - 6z = -28\r\n" );
document.write( " x -  y + 2z =  12\r\n" );
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document.write( "Fill in all invisible 1's for\r\n" );
document.write( "coefficients:\r\n" );
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document.write( "2x - 1y + 1z =   8\r\n" );
document.write( "3x + 1y - 6z = -28\r\n" );
document.write( "1x - 1y + 2z =  12\r\n" );
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document.write( "Erase all the letters and\r\n" );
document.write( "equal signs:\r\n" );
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document.write( " 2  - 1  + 1    8\r\n" );
document.write( " 3  + 1  - 6  -28\r\n" );
document.write( " 1  - 1  + 2   12 \r\n" );
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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" );
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document.write( " 2  -1   1    8\r\n" );
document.write( " 3   1  -6  -28\r\n" );
document.write( " 1  -1   2   12 \r\n" );
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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  -1   1  |   8]\r\n" );
document.write( "[3   1  -6  | -28]\r\n" );
document.write( "[1  -1   2  |  12]\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 three lower left\r\n" );
document.write( "corner positions, \r\n" );
document.write( "underneath the diagonal:\r\n" );
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document.write( "[2  -1   1  |   8]\r\n" );
document.write( "[3   1  -6  | -28]\r\n" );
document.write( "[1  -1   2  |  12]\r\n" );
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document.write( "It's easier when there is a 1\r\n" );
document.write( "in the upper left corner, the\r\n" );
document.write( "1st number on the diagonal, so\r\n" );
document.write( "we will swap row 1 and row 3,\r\n" );
document.write( "so we'll have a 1 on the first\r\n" );
document.write( "diagonal element:\r\n" );
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document.write( "[1  -1   2  |  12]\r\n" );
document.write( "[3   1  -6  | -28]\r\n" );
document.write( "[2  -1   1  |   8]\r\n" );
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document.write( "To get a 0 where the 3 is,\r\n" );
document.write( "mentally multiply each of the\r\n" );
document.write( "numbers in the top row by -3 \r\n" );
document.write( "and add them to 1 times the \r\n" );
document.write( "middle row. It makes it easy \r\n" );
document.write( "if you write the numbers to \r\n" );
document.write( "multiply the two rows by left \r\n" );
document.write( "of the matrix beside the rows:\r\n" );
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document.write( "-3[1  -1   2  |  12]\r\n" );
document.write( " 1[3   1  -6  | -28]\r\n" );
document.write( "  [2  -1   1  |   8]\r\n" );
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document.write( "We get:\r\n" );
document.write( " \r\n" );
document.write( "  [1  -1   2  |  12]\r\n" );
document.write( "  [0   4 -12  | -64]\r\n" );
document.write( "  [2  -1   1  |   8]\r\n" );
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document.write( "To get a 0 where the 2 is in the\r\n" );
document.write( "bottom left corner, mentally multiply each of the\r\n" );
document.write( "numbers in the top row by -2 \r\n" );
document.write( "and add them to 1 times the \r\n" );
document.write( "bottom row:\r\n" );
document.write( "\r\n" );
document.write( "-2[1  -1   2  |  12]\r\n" );
document.write( "  [0   4 -12  | -64]\r\n" );
document.write( " 1[2  -1   1  |   8]\r\n" );
document.write( " \r\n" );
document.write( "  [1  -1   2  |  12]\r\n" );
document.write( "  [0   4 -12  | -64]\r\n" );
document.write( "  [0   1  -3  | -16]\r\n" );
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document.write( "It's easier when there is a 1\r\n" );
document.write( "on the diagonal, so\r\n" );
document.write( "we will swap row 2 and row 3,\r\n" );
document.write( "so we'll have a 1 on the middle\r\n" );
document.write( "diagonal element: \r\n" );
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document.write( "  [1  -1   2  |  12]\r\n" );
document.write( "  [0   1  -3  | -16]\r\n" );
document.write( "  [0   4 -12  | -64]\r\n" );
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document.write( "To get a 0 where the 4 is,\r\n" );
document.write( "multiply the middle row by -4\r\n" );
document.write( "and add to 1 times the \r\n" );
document.write( "bottom row:\r\n" );
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document.write( "  [1  -1   2  |  12]\r\n" );
document.write( "-4[0   1  -3  | -16]\r\n" );
document.write( " 1[0   4 -12  | -64]\r\n" );
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document.write( "We get\r\n" );
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document.write( "  [1  -1   2  |  12]\r\n" );
document.write( "-4[0   1  -3  | -16]\r\n" );
document.write( " 1[0   0   0  |   0]\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" );
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document.write( "  [ 1x  -1y   2z =  12]\r\n" );
document.write( "  [ 0x   1y  -3z = -16]\r\n" );
document.write( "  [ 0x   0y   0z =   0]\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( "     x -  y + 2z =  12 \r\n" );
document.write( "          y - 3z = -16 \r\n" );
document.write( "              0z =   0 \r\n" );
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document.write( "The bottom equation 0z = 0 is true\r\n" );
document.write( "for any number we choose for z, so we\r\n" );
document.write( "will represent it by the constant k.\r\n" );
document.write( "[There are infinitely many solutions].\r\n" );
document.write( "So we will set z equal to k\r\n" );
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document.write( "               z = k\r\n" );
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document.write( "Substitute z = k into the middle\r\n" );
document.write( "equation:\r\n" );
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document.write( "          y - 3z = -16\r\n" );
document.write( "          y - 3k = -16\r\n" );
document.write( "               y = 3k - 16 \r\n" );
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document.write( "Substitute y = 3k - 16 and z = k into\r\n" );
document.write( "the top equation:\r\n" );
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document.write( "        x - y + 2z = 12\r\n" );
document.write( "x - (3k - 16) + 2k = 12\r\n" );
document.write( "  x - 3k + 16 + 2k = 12\r\n" );
document.write( "        x - k + 16 = 12\r\n" );
document.write( "                 x = k - 4\r\n" );
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document.write( "So the solutions are given by:\r\n" );
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document.write( "(x, y, z) = (k-4, 3k-16, k)\r\n" );
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document.write( "For example the solution when k=0\r\n" );
document.write( "is (x, y, z) = (-4, -16, 0)\r\n" );
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document.write( "The solution for k=1 is\r\n" );
document.write( "(x, y, z) = (1-4, 3(1)-16, 1) or\r\n" );
document.write( "(x, y, z) = (-3, -13, 1)\r\n" );
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document.write( "etc., etc., etc.\r\n" );
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document.write( "Edwin</PRE>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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