document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=967753>Question 967753</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need to use the Gaussian elimination method on the following:<BR>\n" );
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document.write( "For a circuit, the currents i1, i2, and i3 measured in amperes, are determined by the following linear equations:<BR>\n" );
document.write( "-i1 + 4i3 = 3<BR>\n" );
document.write( "5i1 - 3i2 = 26<BR>\n" );
document.write( "-3i1 +6i2 = -21\r<BR>\n" );
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document.write( "Any help would be very helpful.\r<BR>\n" );
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document.write( "Thanks in advance<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 #591483 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52898)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=591483\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "Let's introduce new variables &nbsp;x = i1, &nbsp;y=i2&nbsp; and &nbsp;z=i3 &nbsp;for brevity. <BR>\n" );
document.write( "Then you need to solve the system of three linear equations in three unknowns\r<BR>\n" );
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document.write( "-x &nbsp;&nbsp;+ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;4z = 3,<BR>\n" );
document.write( "5x &nbsp;&nbsp;- 3y &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;= 26,<BR>\n" );
document.write( "-3x + 6y &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;= -21\r<BR>\n" );
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document.write( "by the Gauss's elimination method.\r<BR>\n" );
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document.write( "Notice that your system just contains the sub-system of two linear equations in two unknowns &nbsp;x&nbsp; and &nbsp;y. &nbsp;It is the system consisting of the second and the third equations. \r<BR>\n" );
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document.write( "Its facilitates the solution. \r<BR>\n" );
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document.write( "So, &nbsp;we need to solve this sub-system, &nbsp;which is \r<BR>\n" );
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document.write( "5x &nbsp;&nbsp;- 3y = 26,<BR>\n" );
document.write( "-3x + 6y = -21.\r<BR>\n" );
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document.write( "Multiply the first equation of these two by &nbsp;2&nbsp; and add to the second equation to eliminate &nbsp;y. &nbsp;You will get\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;10x &nbsp;&nbsp;- 6y = 52,<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;-3x + 6y = -21.<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;7x &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = 31.\r<BR>\n" );
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document.write( "It gives you the solution for &nbsp;x: &nbsp;x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=31%2F7 ALIGN=MIDDLE  ALT=\"31%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Next, substitute it to any of the second or the third equation of the original system. &nbsp;It gives you the solution for &nbsp;y. \r<BR>\n" );
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document.write( "Then substitute the found values of &nbsp;x&nbsp; and &nbsp;y&nbsp; into the first equation, &nbsp;and you will get the solution for &nbsp;z. \r<BR>\n" );
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document.write( "Good luck.\r<BR>\n" );
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document.write( "Comment from student: Hi ikleyn, thanks for the response. How would I solve this question using an augmented matrix and using gaus elimination? Thanks again for the help. <BR>\n" );
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document.write( "OK, let's do this. \r<BR>\n" );
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document.write( "Your augmented matrix is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C+4%2C+-1%2C+0%2C+4%2C+3%2C+5%2C+-3%2C+0%2C+26%2C+-3%2C+6%2C+0%2C+-21%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C+4%2C+-1%2C+0%2C+4%2C+3%2C+5%2C+-3%2C+0%2C+26%2C+-3%2C+6%2C+0%2C+-21%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Multiply the first row by &nbsp;5&nbsp; and add to the second row to eliminate &nbsp;x&nbsp; in the second equation. &nbsp;You will get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C+4%2C+-1%2C+0%2C+4%2C+3%2C+0%2C+-3%2C+20%2C+145%2C+-3%2C+6%2C+0%2C+-21%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C+4%2C+-1%2C+0%2C+4%2C+3%2C+0%2C+-3%2C+20%2C+145%2C+-3%2C+6%2C+0%2C+-21%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Next, &nbsp;multiply the first row by &nbsp;-3&nbsp; and add to the third row to eliminate &nbsp;x&nbsp; in the third equation. &nbsp;You will get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C+4%2C+-1%2C+0%2C+4%2C+3%2C+0%2C+-3%2C+20%2C+145%2C+0%2C+6%2C+-12%2C+-30%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "OK, &nbsp;we just made the zeroes in the first column below &nbsp;\"-1\". \r<BR>\n" );
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document.write( "Now, &nbsp;multiply the second row of the last matrix by &nbsp;2&nbsp; and add to the third row to eliminate &nbsp;y&nbsp; in the third equation. &nbsp;You will get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C+4%2C+-1%2C+0%2C+4%2C+3%2C+0%2C+-3%2C+20%2C+145%2C+0%2C+0%2C+28%2C+260%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C+4%2C+-1%2C+0%2C+4%2C+3%2C+0%2C+-3%2C+20%2C+145%2C+0%2C+0%2C+28%2C+260%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Thus you just completed the elimination process: &nbsp;you transformed your original matrix to the upper triangle matrix. \r<BR>\n" );
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document.write( "Hence, &nbsp;the solution for &nbsp;z&nbsp; is &nbsp;z = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=260%2F28 ALIGN=MIDDLE  ALT=\"260%2F28\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=65%2F7 ALIGN=MIDDLE  ALT=\"65%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F7 ALIGN=MIDDLE  ALT=\"2%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Now you should make back-substitution.\r<BR>\n" );
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document.write( "Substitute the found value of &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z ALIGN=MIDDLE  ALT=\"z\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=65%2F7 ALIGN=MIDDLE  ALT=\"65%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; into the equation \r<BR>\n" );
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document.write( "-3y + 20z = 145 &nbsp;&nbsp;(in accordance with the second row of the last matrix). &nbsp;It will give you the solution for &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Then back-substitute the found values of &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; and &nbsp;z&nbsp; into the very first equation and find &nbsp;x. \r<BR>\n" );
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