document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=881485>Question 881485</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all solutions of the system<BR>\n" );
document.write( "x+y+z+w=4<BR>\n" );
document.write( "x-2y-z-w=3<BR>\n" );
document.write( "2x-y+z-w=2<BR>\n" );
document.write( "x-y+2z-2w=-7 <BR>\n" );
document.write( "I need to use Gaussian elimination but don't know how with 4 variables. Please help asap!<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #544249 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=544249\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1,1,1,1,4<BR>\n" );
document.write( "1,-2,-1,-1,3<BR>\n" );
document.write( "2,-1,1,-1,2<BR>\n" );
document.write( "1,-1,2,-2,-7\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "add  down (-1) *row 1 to row 2<BR>\n" );
document.write( "1,1,1,1,4<BR>\n" );
document.write( "0,-3,-2,-2,-1<BR>\n" );
document.write( "2,-1,1,-1,2<BR>\n" );
document.write( "1,-1,2,-2,-7\r<BR>\n" );
document.write( "\n" );
document.write( "add  down (-2) *row 1 to row 3<BR>\n" );
document.write( "1,1,1,1,4<BR>\n" );
document.write( "0,-3,-2,-2,-1<BR>\n" );
document.write( "0,-3,-1,-3,-6<BR>\n" );
document.write( "1,-1,2,-2,-7\r<BR>\n" );
document.write( "\n" );
document.write( "add  down (-1) *row 1 to row 4<BR>\n" );
document.write( "1,1,1,1,4<BR>\n" );
document.write( "0,-3,-2,-2,-1<BR>\n" );
document.write( "0,-3,-1,-3,-6<BR>\n" );
document.write( "0,-2,1,-3,-11\r<BR>\n" );
document.write( "\n" );
document.write( "divide row 2 by -3<BR>\n" );
document.write( "1,1,1,1,4<BR>\n" );
document.write( "0,1,-2/-3,-2/-3,-1/-3<BR>\n" );
document.write( "0,-3,-1,-3,-6<BR>\n" );
document.write( "0,-2,1,-3,-11\r<BR>\n" );
document.write( "\n" );
document.write( "add  down (3) *row 2 to row 3<BR>\n" );
document.write( "1,1,1,1,4<BR>\n" );
document.write( "0,1,2/3,2/3,1/3<BR>\n" );
document.write( "0,0,1,-1,-5<BR>\n" );
document.write( "0,-2,1,-3,-11\r<BR>\n" );
document.write( "\n" );
document.write( "add  down (2) *row 2 to row 4<BR>\n" );
document.write( "1,1,1,1,4<BR>\n" );
document.write( "0,1,2/3,2/3,1/3<BR>\n" );
document.write( "0,0,1,-1,-5<BR>\n" );
document.write( "0,0,7/3,-5/3,-31/3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "add  down (-7/3) *row 3 to row 4<BR>\n" );
document.write( "1,1,1,1,4<BR>\n" );
document.write( "0,1,2/3,2/3,1/3<BR>\n" );
document.write( "0,0,1,-1,-5<BR>\n" );
document.write( "0,0,0,6/9,12/9\r<BR>\n" );
document.write( "\n" );
document.write( "divide row 4 by 2/3<BR>\n" );
document.write( "1,1,1,1,4<BR>\n" );
document.write( "0,1,2/3,2/3,1/3<BR>\n" );
document.write( "0,0,1,-1,-5<BR>\n" );
document.write( "0,0,0,1,2\r<BR>\n" );
document.write( "\n" );
document.write( "We now have the value for the last variable.<BR>\n" );
document.write( "We will work our way up and get the other solutions.\r<BR>\n" );
document.write( "\n" );
document.write( "add up  (1) *row 4 to row 3<BR>\n" );
document.write( "1,1,1,1,4<BR>\n" );
document.write( "0,1,2/3,2/3,1/3<BR>\n" );
document.write( "0,0,1,0,-3<BR>\n" );
document.write( "0,0,0,1,2\r<BR>\n" );
document.write( "\n" );
document.write( "add up  (-2/3) *row 4 to row 2<BR>\n" );
document.write( "1,1,1,1,4<BR>\n" );
document.write( "0,1,6/9,0,-1<BR>\n" );
document.write( "0,0,1,0,-3<BR>\n" );
document.write( "0,0,0,1,2\r<BR>\n" );
document.write( "\n" );
document.write( "add up  (-1) *row 4 to row 1<BR>\n" );
document.write( "1,1,1,0,2<BR>\n" );
document.write( "0,1,2/3,0,-1<BR>\n" );
document.write( "0,0,1,0,-3<BR>\n" );
document.write( "0,0,0,1,2\r<BR>\n" );
document.write( "\n" );
document.write( "add up  (-2/3) *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,-3<BR>\n" );
document.write( "0,0,0,1,2\r<BR>\n" );
document.write( "\n" );
document.write( "add up  (-1) *row 3 to row 1<BR>\n" );
document.write( "1,1,0,0,5<BR>\n" );
document.write( "0,1,0,0,1<BR>\n" );
document.write( "0,0,1,0,-3<BR>\n" );
document.write( "0,0,0,1,2\r<BR>\n" );
document.write( "\n" );
document.write( "add up  (-1) *row 2 to row 1<BR>\n" );
document.write( "1,0,0,0,4<BR>\n" );
document.write( "0,1,0,0,1<BR>\n" );
document.write( "0,0,1,0,-3<BR>\n" );
document.write( "0,0,0,1,2\r<BR>\n" );
document.write( "\n" );
document.write( "final<BR>\n" );
document.write( "1,0,0,0,4<BR>\n" );
document.write( "0,1,0,0,1<BR>\n" );
document.write( "0,0,1,0,-3<BR>\n" );
document.write( "0,0,0,1,2<BR>\n" );
document.write( "\"4\",\"1\",\"-3\",\"2\"<BR>\n" );
document.write( "(4,1,-3,2)<BR>\n" );
document.write( "x,y,z,w </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );