document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=152209>Question 152209</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve the system using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination: \r<BR>\n" );
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document.write( "w+x+y+z=5<BR>\n" );
document.write( "w+2x-y-2z=-1<BR>\n" );
document.write( "w-3x-3y-z=-1<BR>\n" );
document.write( "2w-x+2y-z=-2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #535533 by </B><A HREF=/tutors/aboutme.mpl?userid=richwmiller><B>richwmiller(17219)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richwmiller><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=richwmiller><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=535533\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> w+x+y+z=5<BR>\n" );
document.write( "w+2x-y-2z=-1<BR>\n" );
document.write( "w-3x-3y-z=-1<BR>\n" );
document.write( "2w-x+2y-z=-2\r<BR>\n" );
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document.write( "1,1,1,1,5<BR>\n" );
document.write( "1,2,-1,-2,-1<BR>\n" );
document.write( "1,-3,-3,-1,-1<BR>\n" );
document.write( "2,-1,2,-1,-2\r<BR>\n" );
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document.write( "add  down (-1/1) *row 1 to row 2<BR>\n" );
document.write( "1,1,1,1,5<BR>\n" );
document.write( "0,1,-2,-3,-6<BR>\n" );
document.write( "1,-3,-3,-1,-1<BR>\n" );
document.write( "2,-1,2,-1,-2\r<BR>\n" );
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document.write( "add  down (-1/1) *row 1 to row 3<BR>\n" );
document.write( "1,1,1,1,5<BR>\n" );
document.write( "0,1,-2,-3,-6<BR>\n" );
document.write( "0,-4,-4,-2,-6<BR>\n" );
document.write( "2,-1,2,-1,-2\r<BR>\n" );
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document.write( "add  down (-2/1) *row 1 to row 4<BR>\n" );
document.write( "1,1,1,1,5<BR>\n" );
document.write( "0,1,-2,-3,-6<BR>\n" );
document.write( "0,-4,-4,-2,-6<BR>\n" );
document.write( "0,-3,0,-3,-12\r<BR>\n" );
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document.write( "add  down (4/1) *row 2 to row 3<BR>\n" );
document.write( "1,1,1,1,5<BR>\n" );
document.write( "0,1,-2,-3,-6<BR>\n" );
document.write( "0,0,-12,-14,-30<BR>\n" );
document.write( "0,-3,0,-3,-12\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "add  down (3/1) *row 2 to row 4<BR>\n" );
document.write( "1,1,1,1,5<BR>\n" );
document.write( "0,1,-2,-3,-6<BR>\n" );
document.write( "0,0,-12,-14,-30<BR>\n" );
document.write( "0,0,-6,-12,-30\r<BR>\n" );
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document.write( "divide row 3 by -12/1<BR>\n" );
document.write( "1,1,1,1,5<BR>\n" );
document.write( "0,1,-2,-3,-6<BR>\n" );
document.write( "0,0,1,-14/-12,-30/-12<BR>\n" );
document.write( "0,0,-6,-12,-30\r<BR>\n" );
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document.write( "add  down (6/1) *row 3 to row 4<BR>\n" );
document.write( "1,1,1,1,5<BR>\n" );
document.write( "0,1,-2,-3,-6<BR>\n" );
document.write( "0,0,1,7/6,5/2<BR>\n" );
document.write( "0,0,0,-5,-15\r<BR>\n" );
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document.write( "divide row 4 by -5/1<BR>\n" );
document.write( "1,1,1,1,5<BR>\n" );
document.write( "0,1,-2,-3,-6<BR>\n" );
document.write( "0,0,1,7/6,5/2<BR>\n" );
document.write( "0,0,0,1,3\r<BR>\n" );
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document.write( "This is where you would start back substitution.<BR>\n" );
document.write( "We now have z=3<BR>\n" );
document.write( "We continue with the matrix solution\r<BR>\n" );
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document.write( "add  up  (-7/6) *row 4 to row 3<BR>\n" );
document.write( "1,1,1,1,5<BR>\n" );
document.write( "0,1,-2,-3,-6<BR>\n" );
document.write( "0,0,1,0,-1<BR>\n" );
document.write( "0,0,0,1,3<BR>\n" );
document.write( "now we have y= -1\r<BR>\n" );
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document.write( "add  up  (3/1) *row 4 to row 2<BR>\n" );
document.write( "1,1,1,1,5<BR>\n" );
document.write( "0,1,-2,0,3<BR>\n" );
document.write( "0,0,1,0,-1<BR>\n" );
document.write( "0,0,0,1,3\r<BR>\n" );
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document.write( "add  up  (-1/1) *row 4 to row 1<BR>\n" );
document.write( "row 1 col 2<BR>\n" );
document.write( "1,1,1,0,2<BR>\n" );
document.write( "0,1,-2,0,3<BR>\n" );
document.write( "0,0,1,0,-1<BR>\n" );
document.write( "0,0,0,1,3\r<BR>\n" );
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document.write( "add  up  (2/1) *row 3 to row 2<BR>\n" );
document.write( "1,1,1,0,2<BR>\n" );
document.write( "0,1,0,0,1<BR>\n" );
document.write( "0,0,1,0,-1<BR>\n" );
document.write( "0,0,0,1,3<BR>\n" );
document.write( "now we have  z=3 y=-1 x=1\r<BR>\n" );
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document.write( "add  up  (-1/1) *row 3 to row 1<BR>\n" );
document.write( "1,1,0,0,3<BR>\n" );
document.write( "0,1,0,0,1<BR>\n" );
document.write( "0,0,1,0,-1<BR>\n" );
document.write( "0,0,0,1,3\r<BR>\n" );
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document.write( "add  up  (-1/1) *row 2 to row 1<BR>\n" );
document.write( "1,0,0,0,2<BR>\n" );
document.write( "0,1,0,0,1<BR>\n" );
document.write( "0,0,1,0,-1<BR>\n" );
document.write( "0,0,0,1,3\r<BR>\n" );
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document.write( "now we have all four solutions<BR>\n" );
document.write( "w=2 x=1 y=-1 z=3<BR>\n" );
document.write( "(2,1,-1,3)\r<BR>\n" );
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document.write( "w+x+y+z=5<BR>\n" );
document.write( "check<BR>\n" );
document.write( "2+1-1-3=5<BR>\n" );
document.write( "ok\r<BR>\n" );
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document.write( "w+2x-y-2z=-1<BR>\n" );
document.write( "2+2-(-1)-6=-1<BR>\n" );
document.write( "5-6=-1<BR>\n" );
document.write( "ok<BR>\n" );
document.write( "w-3x-3y-z=-1<BR>\n" );
document.write( "2-3-3(-1)-3=-1<BR>\n" );
document.write( "-1+3-3=-1<BR>\n" );
document.write( "ok<BR>\n" );
document.write( "2w-x+2y-z=-2<BR>\n" );
document.write( "2*2-1+2(-1)-3=-2<BR>\n" );
document.write( "4-1-2-3=-2<BR>\n" );
document.write( "3-2-3=-2<BR>\n" );
document.write( "ok\r<BR>\n" );
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document.write( "(2,1,-1,3) works in all 4 equations </SPAN>\n" );
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