document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203611>Question 1203611</A>: <I><SPAN STYLE=\"word-break: break-all;\"> x+4y-6z=-1<BR>\n" );
document.write( "2x-y+2z=-7<BR>\n" );
document.write( "-x+2y-4z=5 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #839301 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=839301\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll show how to solve this system using matrix row reduction.<BR>\n" );
document.write( "The goal is to get the matrix into reduced row echelon form (RREF).\r<BR>\n" );
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document.write( "We have this given system<BR>\n" );
document.write( "x+4y-6z=-1<BR>\n" );
document.write( "2x-y+2z=-7<BR>\n" );
document.write( "-x+2y-4z=5\r<BR>\n" );
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document.write( "which is the same as<BR>\n" );
document.write( "1x+4y-6z=-1<BR>\n" );
document.write( "2x-1y+2z=-7<BR>\n" );
document.write( "-1x+2y-4z=5\r<BR>\n" );
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document.write( "It forms this matrix<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >4</TD><TD >-6</TD><TD >-1</TD></tr><tr><TD >2</TD><TD >-1</TD><TD >2</TD><TD >-7</TD></tr><tr><TD >-1</TD><TD >2</TD><TD >-4</TD><TD >5</TD></tr>\n" );
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document.write( "The last column represents the right hand values -1, -7 and 5. <BR>\n" );
document.write( "The rest of the matrix represents the coefficients.\r<BR>\n" );
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document.write( "Normally a matrix does not have separating grid lines, which is unfortunate. But I'll use grid lines to help separate the values. It should hopefully make things look a bit cleaner.\r<BR>\n" );
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document.write( "-------------------------------------\r<BR>\n" );
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document.write( "Here are the steps for RREF.\r<BR>\n" );
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document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >4</TD><TD >-6</TD><TD >-1</TD></tr><tr><TD >2</TD><TD >-1</TD><TD >2</TD><TD >-7</TD></tr><tr><TD >-1</TD><TD >2</TD><TD >-4</TD><TD >5</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >4</TD><TD >-6</TD><TD >-1</TD><TD ></TD></tr><tr><TD >0</TD><TD >-9</TD><TD >14</TD><TD >-5</TD><TD >R2  -  2*R1 --> R2</TD></tr><tr><TD >-1</TD><TD >2</TD><TD >-4</TD><TD >5</TD><TD ></TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "Notation like R2 - 2*R1 --> R2 means \"subtract off twice of row 1 from row 2. Then store the results in row 2 (we overwrite row 2)\".\r<BR>\n" );
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document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >4</TD><TD >-6</TD><TD >-1</TD><TD ></TD></tr><tr><TD >0</TD><TD >-9</TD><TD >14</TD><TD >-5</TD><TD ></TD></tr><tr><TD >0</TD><TD >6</TD><TD >-10</TD><TD >4</TD><TD >R3+R1 --> R3</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >4</TD><TD >-6</TD><TD >-1</TD><TD ></TD></tr><tr><TD >0</TD><TD >1</TD><TD >-14/9</TD><TD >5/9</TD><TD >(-1/9)*R2 -> R2</TD></tr><tr><TD >0</TD><TD >6</TD><TD >-10</TD><TD >4</TD><TD ></TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >4</TD><TD >-6</TD><TD >-1</TD><TD ></TD></tr><tr><TD >0</TD><TD >1</TD><TD >-14/9</TD><TD >5/9</TD><TD ></TD></tr><tr><TD >0</TD><TD >0</TD><TD >-2/3</TD><TD >2/3</TD><TD >R3  -  6*R2 -> R3</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >4</TD><TD >-6</TD><TD >-1</TD><TD ></TD></tr><tr><TD >0</TD><TD >1</TD><TD >-14/9</TD><TD >5/9</TD><TD ></TD></tr><tr><TD >0</TD><TD >0</TD><TD >1</TD><TD >-1</TD><TD >(-3/2)*R3 -> R3</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "The matrix is in row echelon form (REF) but not RREF. This is because we have a lower triangular region of zeros below the main diagonal pivot entries. \r<BR>\n" );
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document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >4</TD><TD >-6</TD><TD >-1</TD><TD ></TD></tr><tr><TD >0</TD><TD >1</TD><TD >0</TD><TD >-1</TD><TD >R2 + (14/9)*R3 -> R2</TD></tr><tr><TD >0</TD><TD >0</TD><TD >1</TD><TD >-1</TD><TD ></TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >0</TD><TD >-6</TD><TD >3</TD><TD >R1  -  4*R2 -> R1</TD></tr><tr><TD >0</TD><TD >1</TD><TD >0</TD><TD >-1</TD><TD ></TD></tr><tr><TD >0</TD><TD >0</TD><TD >1</TD><TD >-1</TD><TD ></TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >0</TD><TD >0</TD><TD >-3</TD><TD >R1 + 6*R3 -> R1</TD></tr><tr><TD >0</TD><TD >1</TD><TD >0</TD><TD >-1</TD><TD ></TD></tr><tr><TD >0</TD><TD >0</TD><TD >1</TD><TD >-1</TD><TD ></TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "For more practice with RREF, here is a very useful tool<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"http://www.math.odu.edu/~bogacki/lat/\">http://www.math.odu.edu/~bogacki/lat/</a><BR>\n" );
document.write( "It is called \"linear algebra toolkit\". It is a collection of matrix solvers that show step by step solutions. \r<BR>\n" );
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document.write( "-------------------------------------\r<BR>\n" );
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document.write( "To summarize:<BR>\n" );
document.write( "We started with this 3x4 matrix<BR>\n" );
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document.write( "<tr><TD >1</TD><TD >4</TD><TD >-6</TD><TD >-1</TD></tr><tr><TD >2</TD><TD >-1</TD><TD >2</TD><TD >-7</TD></tr><tr><TD >-1</TD><TD >2</TD><TD >-4</TD><TD >5</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "and ended up with this reduced row echelon form (RREF)<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >0</TD><TD >0</TD><TD ><font color=red>-3</font></TD></tr><tr><TD >0</TD><TD >1</TD><TD >0</TD><TD ><font color=red>-1</font></TD></tr><tr><TD >0</TD><TD >0</TD><TD >1</TD><TD ><font color=red>-1</font></TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "The original 3x3 sub-block has morphed into the 3x3 identity matrix which has the main diagonal of all \"1\"s, and the rest are \"0\"s.<BR>\n" );
document.write( "The right hand side of this RREF matrix are the solution values.\r<BR>\n" );
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document.write( "Answer:<BR>\n" );
document.write( "<font color=red>x = -3<BR>\n" );
document.write( "y = -1<BR>\n" );
document.write( "z = -1</font><BR>\n" );
document.write( "I'll let the student check each equation. <BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );