document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=81532>Question 81532</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0321279085/algebrahomeworkh>college algebra with modeling and visualization</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Can you please help me do this problem. I really just don't know where to start.It is solutions to linear systems using matrices.\r<BR>\n" );
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document.write( "12x-4y-7z=8<BR>\n" );
document.write( "-8x-6y+9z=7<BR>\n" );
document.write( "34x+6y-2z=5\r<BR>\n" );
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document.write( " Can you show me how to do it step by step please. Thank you for your time </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #58384 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20054)</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=58384\">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;\"><font size = 5 color = \"red\"><B>\r\n" );
document.write( "Well, how about that! It looks as though LongJonSilver \r\n" );
document.write( "started to do it, found it was too messy and quit. He\r\n" );
document.write( "suggested Cramer's rule, but that's the determinant\r\n" );
document.write( "method, not the matrix method.  OK, I'm game.  \r\n" );
document.write( "I'll do it by matrices\r\n" );
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document.write( "Solution by Edwin McCravy:\r\n" );
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document.write( "Erase all the letters, plus signs,\r\n" );
document.write( "and equal signs and put a line \r\n" );
document.write( "where the equal signs were: \r\n" );
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document.write( "[12 -4 -7 |8]\r\n" );
document.write( "[-8 -6  9 |7]\r\n" );
document.write( "[34  6 -2 |5]\r\n" );
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document.write( "The plan is seven-fold, in this order:\r\n" );
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document.write( "1. Use the 1st row to get a 0 where the -8 is in the 2nd row\r\n" );
document.write( "2. Use the 1st row to get a 0 where the 34 is in the 3rd row\r\n" );
document.write( "3. Use the new 2nd row to get a 0 where the 6 is in the 3rd row\r\n" );
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document.write( "Your matrix will then look like this:\r\n" );
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document.write( "    [ #  #  # | #]\r\n" );
document.write( "    [ 0  #  # | #]\r\n" );
document.write( "    [ 0  0  # | #]\r\n" );
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document.write( "There will be numbers where the #'s are. This form is called\r\n" );
document.write( "the reduced echelon form.\r\n" );
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document.write( "4. Rewrite the matrix as three equations.\r\n" );
document.write( "5. Solve the 3rd equation for z\r\n" );
document.write( "6. Substitute that value for z into the 2nd equation and solve for y\r\n" );
document.write( "7. Substitute both the value of z and the value for y in the 1st\r\n" );
document.write( "   equation and solve for x. \r\n" );
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document.write( "Here we go:\r\n" );
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document.write( "[12  -4  -7 | 8]\r\n" );
document.write( "[-8  -6   9 | 7]\r\n" );
document.write( "[34   6  -2 | 5]\r\n" );
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document.write( "To get a zero where the -8 is, to the side, \r\n" );
document.write( "multiply the 1st row by 2 and the \r\n" );
document.write( "2nd row by 3, and add them vertically \r\n" );
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document.write( "        1st row x2     24  -8 -14 | 16\r\n" );
document.write( "        2nd row x3    -24 -18  27 | 21\r\n" );
document.write( "                      ----------------\r\n" );
document.write( "        new 2nd row     0 -26  13  | 37\r\n" );
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document.write( "Replace only the 2nd row by that bottom line\r\n" );
document.write( "in the matrix\r\n" );
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document.write( "[12  -4  -7 |  8]\r\n" );
document.write( "[ 0 -26  13 | 37]\r\n" );
document.write( "[34   6  -2 |  5]\r\n" );
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document.write( "To get a zero where the 34 is, to the side, \r\n" );
document.write( "multiply the 1st row by -17 and the \r\n" );
document.write( "3rd row by 6, and add them vertically \r\n" );
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document.write( "        1st row x -17    -204  68 119 | -136\r\n" );
document.write( "        2nd row x 6       204  36 -12 |   30 \r\n" );
document.write( "                      ----------------------\r\n" );
document.write( "        new 3rd row         0 104 107 | -106\r\n" );
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document.write( "Replace only the 3rd row by that bottom line\r\n" );
document.write( "in the matrix\r\n" );
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document.write( "[12  -4   -7 |    8]\r\n" );
document.write( "[ 0 -26   13 |   37]\r\n" );
document.write( "[ 0 104  107 | -106]\r\n" );
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document.write( "To get a zero where the 104 is, to the side, \r\n" );
document.write( "multiply the 2nd row by 4 and the \r\n" );
document.write( "3rd row by 1, (that is, leave it as it is),\r\n" );
document.write( "and add them vertically \r\n" );
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document.write( "        2nt row x 4       0 -104  52 |  148\r\n" );
document.write( "        2nd row x 1       0  104 107 | -106 \r\n" );
document.write( "                      ----------------------\r\n" );
document.write( "        new 3rd row       0    0 159 |   42\r\n" );
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document.write( "Replace only the 3rd row by that bottom line\r\n" );
document.write( "in the matrix\r\n" );
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document.write( "[12  -4   -7 |   8]\r\n" );
document.write( "[ 0 -26   13 |  37]\r\n" );
document.write( "[ 0   0  159 |  42]\r\n" );
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document.write( "This is reduced echelon form.  If your teacher\r\n" );
document.write( "wants you to get it all the way to row-reduced\r\n" );
document.write( "echelon form, you'll have to get 0's where the\r\n" );
document.write( "-4, the -7 and the 4 are, too.  But it can be \r\n" );
document.write( "solved from the reduced echelon form as\r\n" );
document.write( "follows.  \r\n" );
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document.write( "Erase the brackets and the line, and put the \r\n" );
document.write( "letters, plus signs, and equal signs back in:\r\n" );
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document.write( " 12x -  4y -   7z =  8\r\n" );
document.write( "  0x - 26y +  13z = 37\r\n" );
document.write( "  0x +  0y + 159z = 42\r\n" );
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document.write( "Erase the 0 terms\r\n" );
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document.write( " 12x -  4y -   7z =  8\r\n" );
document.write( "      -26y +  13z = 37\r\n" );
document.write( "             159z = 42\r\n" );
document.write( "  \r\n" );
document.write( "Solve the 3rd equation for z\r\n" );
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document.write( "             159z = 42\r\n" );
document.write( "                z = 42/159\r\n" );
document.write( "                z = 14/53\r\n" );
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document.write( "Sunstitute 14/53 for z in the 2nd equation\r\n" );
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document.write( "      -26y +  13z = 37\r\n" );
document.write( " -26y + 13(14/53) = 37\r\n" );
document.write( "    -26y + 182/53 = 37\r\n" );
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document.write( "Clear of fractions by multiplying through by 53\r\n" );
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document.write( "     -1378y + 182 = 1961\r\n" );
document.write( "           -1378y = 1779\r\n" );
document.write( "                y = -1779/1378\r\n" );
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document.write( "Now substitute y = -1779/1378 and z = 14/53 in\r\n" );
document.write( "the 1st equation:\r\n" );
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document.write( "               12x -  4y -   7z =  8  \r\n" );
document.write( " 12x - 4(-1779/1378) - 7(14/53) = 8\r\n" );
document.write( "        12x + 7116/1378 - 98/53 = 8\r\n" );
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document.write( "Reduce the fraction 7116/1378 to 3558/689\r\n" );
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document.write( "         12x + 3558/689 - 98/53 = 8\r\n" );
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document.write( "Clear of fractions by multiplying through by 689\r\n" );
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document.write( "            8268x + 3558 - 1274 = 5512\r\n" );
document.write( "                   8268x + 2284 = 5512\r\n" );
document.write( "                          8268x = 3228\r\n" );
document.write( "                              x = 3228/8268\r\n" );
document.write( "                              x = 269/689\r\n" );
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document.write( "So the solution is\r\n" );
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document.write( "(x, y, z) = (269/689, -1779/1378, 14/53)\r\n" );
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document.write( "and I'll bet anything you copied it wrong, for\r\n" );
document.write( "no respectable teacher would expect you to do a \r\n" );
document.write( "problem that would come out with such a god-awful \r\n" );
document.write( "answer!  No wonder LongJonSilver gave it up.\r\n" );
document.write( "But what I got is correct for the problem you posted.\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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