document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177247>Question 1177247</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For what values of k' the following system has\r<BR>\n" );
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document.write( "2x+y=0\r<BR>\n" );
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document.write( "x+ 2y + kz = -1\r<BR>\n" );
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document.write( "x k²z = 1\r<BR>\n" );
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document.write( "(a) Unique Solution (b) No solution. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #805757 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=805757\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2By=0 ALIGN=MIDDLE  ALT=\"2x%2By=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ..............1)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B+2y+%2B+kz+=+-1 ALIGN=MIDDLE  ALT=\"x%2B+2y+%2B+kz+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >..............2)<BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2Bk%5E2%2Az+=+1 ALIGN=MIDDLE  ALT=\"x+%2Bk%5E2%2Az+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >..............3)<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2By=0 ALIGN=MIDDLE  ALT=\"2x%2By=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ..............1), solve for <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( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x ALIGN=MIDDLE  ALT=\"y=-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >..............1a)\r<BR>\n" );
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document.write( "go to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B+2y+%2B+kz+=+-1 ALIGN=MIDDLE  ALT=\"x%2B+2y+%2B+kz+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >..............2), substitute<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y ALIGN=MIDDLE  ALT=\"+y\"   DISABLED_style= \"max-width:90%; height:auto;\" > from 1a)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B+2%28-2x%29%2B+kz+=+-1 ALIGN=MIDDLE  ALT=\"x%2B+2%28-2x%29%2B+kz+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.............., solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-4x%2B+kz+=+-1 ALIGN=MIDDLE  ALT=\"x-4x%2B+kz+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%2B+kz+=+-1 ALIGN=MIDDLE  ALT=\"-3x%2B+kz+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=kz%2B1+=+3x ALIGN=MIDDLE  ALT=\"kz%2B1+=+3x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28kz%2B1%29%2F3 ALIGN=MIDDLE  ALT=\"x=%28kz%2B1%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >...........2a)\r<BR>\n" );
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document.write( "go to\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2Bk%5E2%2Az+=+1 ALIGN=MIDDLE  ALT=\"x+%2Bk%5E2%2Az+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >..............3), substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28kz%2B1%29%2F3+%2Bk%5E2%2Az+=+1 ALIGN=MIDDLE  ALT=\"%28kz%2B1%29%2F3+%2Bk%5E2%2Az+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >........., solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z ALIGN=MIDDLE  ALT=\"z\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=kz%2B1+%2B3k%5E2%2Az+=+3 ALIGN=MIDDLE  ALT=\"kz%2B1+%2B3k%5E2%2Az+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=kz%2B1+%2B3k%5E2%2Az+=3-1 ALIGN=MIDDLE  ALT=\"kz%2B1+%2B3k%5E2%2Az+=3-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%28k%2B3k%5E2%29+=2 ALIGN=MIDDLE  ALT=\"z%28k%2B3k%5E2%29+=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=2%2F%28k%2B3k%5E2%29 ALIGN=MIDDLE  ALT=\"z+=2%2F%28k%2B3k%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=2%2Fk%281%2B3k%29 ALIGN=MIDDLE  ALT=\"z+=2%2Fk%281%2B3k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........3a)\r<BR>\n" );
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document.write( "express <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > in terms of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28kz%2B1%29%2F3 ALIGN=MIDDLE  ALT=\"x=%28kz%2B1%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >...........2a), substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z ALIGN=MIDDLE  ALT=\"z\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28k%282%2Fk%281%2B3k%29%29%2B1%29%2F3 ALIGN=MIDDLE  ALT=\"x=%28k%282%2Fk%281%2B3k%29%29%2B1%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28%282%2F%281%2B3k%29%29%2B1%29%2F3 ALIGN=MIDDLE  ALT=\"x=%28%282%2F%281%2B3k%29%29%2B1%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28%282%2F%281%2B3k%29%29%2B%281%2B3k%29%2F%281%2B3k%29%29%2F3 ALIGN=MIDDLE  ALT=\"x=%28%282%2F%281%2B3k%29%29%2B%281%2B3k%29%2F%281%2B3k%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28%282%2B1%2B3k%29%2F%281%2B3k%29%29%2F3 ALIGN=MIDDLE  ALT=\"x=%28%282%2B1%2B3k%29%2F%281%2B3k%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28%283%2B3k%29%2F%281%2B3k%29%29%2F3 ALIGN=MIDDLE  ALT=\"x=%28%283%2B3k%29%2F%281%2B3k%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%283%281%2Bk%29%2F%281%2B3k%29%29%2F3 ALIGN=MIDDLE  ALT=\"x=%283%281%2Bk%29%2F%281%2B3k%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%281%2Bk%29%2F%281%2B3k%29 ALIGN=MIDDLE  ALT=\"x=%281%2Bk%29%2F%281%2B3k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.....................2b)\r<BR>\n" );
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document.write( "go to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x ALIGN=MIDDLE  ALT=\"y=-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >..............1a), substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2%281%2Bk%29%2F%281%2B3k%29 ALIGN=MIDDLE  ALT=\"y=-2%281%2Bk%29%2F%281%2B3k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%282%2B2k%29%2F%281%2B3k%29 ALIGN=MIDDLE  ALT=\"y=-%282%2B2k%29%2F%281%2B3k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >...............1b)\r<BR>\n" );
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document.write( "The solutions to the system of equations are: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%281%2Bk%29%2F%281%2B3k%29 ALIGN=MIDDLE  ALT=\"x=%281%2Bk%29%2F%281%2B3k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2%281%2Bk%29%2F%281%2B3k%29 ALIGN=MIDDLE  ALT=\"y=-2%281%2Bk%29%2F%281%2B3k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=2%2Fk%281%2B3k%29+ ALIGN=MIDDLE  ALT=\"z+=2%2Fk%281%2B3k%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "-> for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%3C%3E0 ALIGN=MIDDLE  ALT=\"k%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%3C%3E-1%2F3 ALIGN=MIDDLE  ALT=\"k%3C%3E-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );