document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202812>Question 1202812</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the system of linear equations, using the Gauss-Jordan elimination method. (If there is no solution, enter NO SOLUTION. If there are infinitely many solutions, express your answer in terms of the parameters t and/or s.)\r<BR>\n" );
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document.write( "x	 + 	2y	 + 	z	 = 	−2<BR>\n" );
document.write( "−2x	 − 	3y	 − 	z	 = 	3<BR>\n" );
document.write( "4x	 + 	8y	 + 	4z	 = 	−8\r<BR>\n" );
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document.write( "(x, y, z) = \r<BR>\n" );
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document.write( "I'm very confused to how I would start to do this. Please give steps then a final answer so that I can check my work. Please and thank you.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #837963 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=837963\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "You can't check your work by looking at my solution, because we can easily take very different paths to the answer.<br><BR>\n" );
document.write( "Here is one way to get to the solution to the system of equations.<br><BR>\n" );
document.write( "The original matrix, directly from the system of equations:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C1%2C2%2C1%2C-2%2C-2%2C-3%2C-1%2C3%2C4%2C8%2C4%2C-8%29 ALIGN=MIDDLE  ALT=\"matrix%283%2C4%2C1%2C2%2C1%2C-2%2C-2%2C-3%2C-1%2C3%2C4%2C8%2C4%2C-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "I can see immediately that this system is not going to have a unique solution, because the 3rd row of the matrix is exactly 4 times the 1st row.  One of the operations we can perform is to multiply one row by a constant, so the first thing I would do in this example is multiply the 3rd row by 1/4:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C1%2C2%2C1%2C-2%2C-2%2C-3%2C-1%2C3%2C1%2C2%2C1%2C-2%29 ALIGN=MIDDLE  ALT=\"matrix%283%2C4%2C1%2C2%2C1%2C-2%2C-2%2C-3%2C-1%2C3%2C1%2C2%2C1%2C-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now I am ready to start on the standard sequence of steps to find the solution.<br><BR>\n" );
document.write( "First step: get \"1\" in row 1 column 1 if it is not already there.  Row 1 column 1 is already 1, so we don't need to do anything here.<br><BR>\n" );
document.write( "Next step: Use the \"1\" in row 1 column 1 to get \"0\" in every other row in column 1 by multiplying the first row by a constant and adding it to another row.<br><BR>\n" );
document.write( "Get \"0\" in row 2 column 1 by multiplying row 1 by 2 and adding it to row 2; get a \"0\" in row 3 column 1 by multiplying row 1 by -1 and adding it to row 3.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C1%2C2%2C1%2C-2%2C0%2C1%2C1%2C-1%2C0%2C0%2C0%2C0%29 ALIGN=MIDDLE  ALT=\"matrix%283%2C4%2C1%2C2%2C1%2C-2%2C0%2C1%2C1%2C-1%2C0%2C0%2C0%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Note the row of all \"0\" in row 3 is because in the previous matrix rows 3 and 1 were identical.<br><BR>\n" );
document.write( "Next step: get \"1\" in row 2 column 2 if it is not already there.  Again that \"1\" is already there, so there is no work to do here.<br><BR>\n" );
document.write( "Next step: Use the \"1\" in row 2 column 2 to get \"0\" in every other row in column 2 by multiplying the second row by a constant and adding it to another row.<br><BR>\n" );
document.write( "Get a \"0\" in row 1 column 2 by multiplying row 2 by -2 and adding it to row 1.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C1%2C0%2C-1%2C0%2C0%2C1%2C1%2C-1%2C0%2C0%2C0%2C0%29 ALIGN=MIDDLE  ALT=\"matrix%283%2C4%2C1%2C0%2C-1%2C0%2C0%2C1%2C1%2C-1%2C0%2C0%2C0%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In a general problem, the next step would be to get a \"1\" in row 3 column 3.  But since the third row is all \"0\", we have gone as far as we can performing operations on the matrix.<br><BR>\n" );
document.write( "The matrix we finish with tells us....<br><BR>\n" );
document.write( "row 1: x-z = 0<BR>\n" );
document.write( "row 2: y+z = -1<br><BR>\n" );
document.write( "From this we get an infinite set of solutions using a parameter t:<br><BR>\n" );
document.write( "(1) x = t<BR>\n" );
document.write( "(2) x-z = 0; t-z = 0; so z = t<BR>\n" );
document.write( "(3) y+z = -1; y+t = -1; so y = -t-1<br><BR>\n" );
document.write( "The family of solutions is defined by (x,y,z) = (t,-t-1,t) where t is any constant.<br><BR>\n" );
document.write( "To check the solution, we substitute these parametric expressions for x, y, and z in each of the original equations.<br><BR>\n" );
document.write( "(1) x+2y+z = (t)+(-2t-2)+t = -2  correct<BR>\n" );
document.write( "(2) -2x-3y-z = (-2t)+(3t+3)-t = 3  correct<BR>\n" );
document.write( "(3) 4x+8y+4z = (4t)+(-8t-8)+(4t) = -8  correct<br><BR>\n" );
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