document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1050347>Question 1050347</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, I'm having trouble with this question. I don't know how I'm supposed to work it out, so any help would be great appreciated! Thank you!\r<BR>\n" );
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document.write( "Determine the values for \"a\" for which the following system of linear equations has no solutions, a unique solution, or infinitely many solutions. I need to solve using a matrix, and then row reducing. \r<BR>\n" );
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document.write( "3X1 + aX2 + 6X3 = 6<BR>\n" );
document.write( "-X1 + 3X2 - X3    = 1<BR>\n" );
document.write( "2X1 - 2X2 + 6X3 = 11 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #666359 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=666359\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C3%2Ca%2C6%2C6%2C-1%2C3%2C+-1%2C1%2C2%2C-2%2C6%2C11%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C4%2C3%2Ca%2C6%2C6%2C-1%2C3%2C+-1%2C1%2C2%2C-2%2C6%2C11%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "~<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C0%2Ca%2B9%2C3%2C9%2C-1%2C3%2C+-1%2C1%2C0%2C1%2C1%2C13%2F4%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C4%2C0%2Ca%2B9%2C3%2C9%2C-1%2C3%2C+-1%2C1%2C0%2C1%2C1%2C13%2F4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, after <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3R%5B2%5D%2BR%5B1%5D ALIGN=MIDDLE  ALT=\"3R%5B2%5D%2BR%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2R%5B2%5D%2BR%5B3%5D ALIGN=MIDDLE  ALT=\"2R%5B2%5D%2BR%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "~<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C1%2C-3%2C+1%2C-1%2C0%2C1%2C1%2C13%2F4%2C0%2Ca%2B9%2C3%2C9%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C4%2C1%2C-3%2C+1%2C-1%2C0%2C1%2C1%2C13%2F4%2C0%2Ca%2B9%2C3%2C9%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "~<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C1%2C0%2C4%2C+35%2F4%2C0%2C1%2C1%2C13%2F4%2C0%2C0%2C-a-6%2C-13a%2F4-81%2F4%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >, after <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28a%2B9%29R%5B2%5D%2BR%5B3%5D ALIGN=MIDDLE  ALT=\"-%28a%2B9%29R%5B2%5D%2BR%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  <BR>\n" );
document.write( "(We may assume that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2B9%3C%3E0 ALIGN=MIDDLE  ALT=\"a%2B9%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  If a+9=0, or a = -9, then the system will have a unique solution.)\r<BR>\n" );
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document.write( "Case 1. -a-6 = 0. ===> a = -6.\r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-13a%2F4+-+81%2F4+=+-3%2F4++%3C%3E+0 ALIGN=MIDDLE  ALT=\"-13a%2F4+-+81%2F4+=+-3%2F4++%3C%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "===> System has no solution.\r<BR>\n" );
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document.write( "Case 2. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-a-6+%3C%3E+0 ALIGN=MIDDLE  ALT=\"-a-6+%3C%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C%3E-6 ALIGN=MIDDLE  ALT=\"a%3C%3E-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > ===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C1%2C0%2C4%2C+35%2F4%2C0%2C1%2C1%2C13%2F4%2C0%2C0%2C1%2C%2813a%2F4%2B81%2F4%29%2F%28a%2B6%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >  ===> System has unique solution.\r<BR>\n" );
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document.write( "There is no instance that the system will have infinite number of solutions. </SPAN>\n" );
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