document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=924054>Question 924054</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the pair of linear equations 3x + 7y = k, 12x + 2ky = 4k + 1 do not have any solutions if k = ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #560590 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</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=560590\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> given:<BR>\n" );
document.write( "the pair of linear equations <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x+%2B+7y+=+k ALIGN=MIDDLE  ALT=\"3x+%2B+7y+=+k\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x+%2B+2ky+=+4k+%2B+1+ ALIGN=MIDDLE  ALT=\"12x+%2B+2ky+=+4k+%2B+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "find: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > such that the pair of linear equations does not have any solutions\r<BR>\n" );
document.write( "\n" );
document.write( "recall:\r<BR>\n" );
document.write( "\n" );
document.write( "if solving a linear system, in fact there are only three possible cases:\r<BR>\n" );
document.write( "\n" );
document.write( "    No solution<BR>\n" );
document.write( "    One solution<BR>\n" );
document.write( "    Infinitely many solutions\r<BR>\n" );
document.write( "\n" );
document.write( "When there is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=no ALIGN=MIDDLE  ALT=\"no\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+solution ALIGN=MIDDLE  ALT=\"+solution\"   DISABLED_style= \"max-width:90%; height:auto;\" > the equations are called \"inconsistent\" and the lines are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=parallel ALIGN=MIDDLE  ALT=\"parallel\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "One or infinitely many solutions are called \"consistent\"-overlapping lines\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "so, your lines <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=must ALIGN=MIDDLE  ALT=\"must\"   DISABLED_style= \"max-width:90%; height:auto;\" > be parallel, and they will be parallel if they have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=same ALIGN=MIDDLE  ALT=\"same\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+slope ALIGN=MIDDLE  ALT=\"+slope\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "now, solve your line 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" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x+%2B+7y+=+k ALIGN=MIDDLE  ALT=\"3x+%2B+7y+=+k\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7y+=k-3x+ ALIGN=MIDDLE  ALT=\"7y+=k-3x+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=-%283%2F7%29x+%2Bk%2F7 ALIGN=MIDDLE  ALT=\"y+=-%283%2F7%29x+%2Bk%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" > => slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B1%5D=-%283%2F7%29 ALIGN=MIDDLE  ALT=\"m%5B1%5D=-%283%2F7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x+%2B+2ky+=+4k+%2B+1+ ALIGN=MIDDLE  ALT=\"12x+%2B+2ky+=+4k+%2B+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2ky+=+-12x+%2B4k+%2B+1+ ALIGN=MIDDLE  ALT=\"2ky+=+-12x+%2B4k+%2B+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-%2812%2F2k%29x+%2B4k%2F2k+%2B+1%2F2k+ ALIGN=MIDDLE  ALT=\"y+=+-%2812%2F2k%29x+%2B4k%2F2k+%2B+1%2F2k+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-%286%2Fk%29x+%2B2+%2B+1%2F2k+ ALIGN=MIDDLE  ALT=\"y+=+-%286%2Fk%29x+%2B2+%2B+1%2F2k+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "=> slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B2%5D=++-%286%2Fk%29 ALIGN=MIDDLE  ALT=\"m%5B2%5D=++-%286%2Fk%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "now, since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B1%5D=m%5B2%5D ALIGN=MIDDLE  ALT=\"m%5B1%5D=m%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we have\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%283%2F7%29=+-%286%2Fk%29 ALIGN=MIDDLE  ALT=\"-%283%2F7%29=+-%286%2Fk%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > -solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3k=+-6%2A7 ALIGN=MIDDLE  ALT=\"-3k=+-6%2A7\"   DISABLED_style= \"max-width:90%; height:auto;\" > ...both sides multiply by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3k=42 ALIGN=MIDDLE  ALT=\"3k=42\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=+42+%2F3 ALIGN=MIDDLE  ALT=\"k=+42+%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=14 ALIGN=MIDDLE  ALT=\"k=14\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "then your system is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x+%2B+7y+=14 ALIGN=MIDDLE  ALT=\"3x+%2B+7y+=14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x+%2B+2%2A14y+=+4%2A14+%2B+1+ ALIGN=MIDDLE  ALT=\"12x+%2B+2%2A14y+=+4%2A14+%2B+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x+%2B+7y+=14 ALIGN=MIDDLE  ALT=\"3x+%2B+7y+=14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x+%2B+28y+=+57 ALIGN=MIDDLE  ALT=\"12x+%2B+28y+=+57\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "_________________________________\r<BR>\n" );
document.write( "\n" );
document.write( "see the graph: \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-30%2C30%2C+-30%2C+30%2C+-%283%2F7%29x%2B14%2C++-%286%2F14%29x+%2B2+%2B+1%2F28%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );