document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=576079>Question 576079</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help solve the following.  The instructions read \"Solve each system by triangularizing the augmented matrix and using back-substitution.  If the system is linearly dependent, give the solution in terms of a parameter.  If the system has coincident dependence, answer in set notation.  <BR>\n" );
document.write( "x + 3y +5z = 20<BR>\n" );
document.write( "2x + 3y +4z = 16<BR>\n" );
document.write( "x + 2y +3z = 12<BR>\n" );
document.write( "I listed the matrix as <BR>\n" );
document.write( "1  3  5  20<BR>\n" );
document.write( "2  2  4  16<BR>\n" );
document.write( "1  2  3  12<BR>\n" );
document.write( "I used the row operation -2R1 + R2 > R2 to get the matrix to<BR>\n" );
document.write( "1  3  5  20<BR>\n" );
document.write( "0 -3 -6 -24<BR>\n" );
document.write( "1  2  3  12<BR>\n" );
document.write( "I then used row operation -1R1 + R3 > R3 to get the matrix to<BR>\n" );
document.write( "1  3  5  20<BR>\n" );
document.write( "0 -3 -6 -24<BR>\n" );
document.write( "0 -1 -2  -8<BR>\n" );
document.write( "I then used -1/3R2 + R3 > R3 to get the matrix to<BR>\n" );
document.write( "1  3  5  20<BR>\n" );
document.write( "0 -3 -6 -24<BR>\n" );
document.write( "0  0  0  0<BR>\n" );
document.write( "I know the system is linearly dependent but do understand how to solve from here.  Please provide detail.  Thank you.  (also please let me know if I have made a mistake so far.  I was given the answer of (p-4, -2P+8, p) but do not know how to come up with the answer with what I have so far. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #369716 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=369716\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You're doing great. I'm going to finish up.\r<BR>\n" );
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document.write( "Row 2 has the values 0 -3 -6 -24, which means that -3y -6z = -24. Solve for y to get\r<BR>\n" );
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document.write( "-3y -6z = -24\r<BR>\n" );
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document.write( "-3y = 6z -24\r<BR>\n" );
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document.write( "y = (6z)/(-3) -24/(-3)\r<BR>\n" );
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document.write( "y = -2z + 8\r<BR>\n" );
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document.write( "Now if you let z = P (ie let z be the free variable), then y = -2P + 8. So that explains why the second coordinate is -2P+8 and the third is P\r<BR>\n" );
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document.write( "Now that you know that z = P and y = -2P+8, you can use this to find x in terms of the free variable P (or z). So use the first equation (you can use any equation that has x in it)\r<BR>\n" );
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document.write( "x + 3y +5z = 20\r<BR>\n" );
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document.write( "x + 3(-2P+8) +5P = 20\r<BR>\n" );
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document.write( "x -6P+24 +5P = 20\r<BR>\n" );
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document.write( "x -P+24 = 20\r<BR>\n" );
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document.write( "x -P = 20-24\r<BR>\n" );
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document.write( "x -P = -4\r<BR>\n" );
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document.write( "x = P-4\r<BR>\n" );
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document.write( "So the solutions are\r<BR>\n" );
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document.write( "x = P-4, y = -2P+8, and z = P\r<BR>\n" );
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document.write( "which form the ordered triple\r<BR>\n" );
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document.write( "(P-4, -2P+8, P) </SPAN>\n" );
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