document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=985240>Question 985240</A>: <I><SPAN STYLE=\"word-break: break-all;\"> True or False: \r<BR>\n" );
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document.write( "The system:<BR>\n" );
document.write( "ax+by=c<BR>\n" );
document.write( "dx+ey=f<BR>\n" );
document.write( "is a consistent system of independent equations (with variables x and y). When c and f are replaced by two new numbers g and h, the new system of equations is:<BR>\n" );
document.write( "ax+by=g<BR>\n" );
document.write( "dx+ey=h <BR>\n" );
document.write( "Is the new system also a consistent system of independent equations?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #606060 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52810)</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=606060\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "If the system \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28ax%2Bby=c%2C%0D%0Adx%2Bey=f%29 ALIGN=MIDDLE  ALT=\"system%28ax%2Bby=c%2C%0D%0Adx%2Bey=f%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)\r<BR>\n" );
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document.write( "is a consistent system of independent equations &nbsp;(with variables &nbsp;x&nbsp; and &nbsp;y), &nbsp;then the system \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28ax%2Bby=g%2C%0D%0Adx%2Bey=h%29 ALIGN=MIDDLE  ALT=\"system%28ax%2Bby=g%2C%0D%0Adx%2Bey=h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2)\r<BR>\n" );
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document.write( "is a consistent system of independent equations too.\r<BR>\n" );
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document.write( "<B><U>Algebraic explanation</U></B><BLOCKQUOTE>The system &nbsp;(1)&nbsp; is a consistent system of independent equations if and only if its determinant &nbsp;det <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C+a%2C+b%2C+d%2C+e%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C+a%2C+b%2C+d%2C+e%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; is non-zero.\r<BR>\n" );
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document.write( "Therefore, &nbsp;if the system &nbsp;(1)&nbsp; is a consistent system of independent equations, &nbsp;then the system &nbsp;(2)&nbsp; is too, &nbsp;because both the systems have the same matrix.</BLOCKQUOTE> \r<BR>\n" );
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document.write( "<B><U>Geometric explanation</U></B><BLOCKQUOTE>The system &nbsp;(1)&nbsp; is a consistent system of independent equations if and only if two straight lines &nbsp;ax + by = 0 &nbsp;and &nbsp;dx + ey = 0 are distinct and non-parallel \r<BR>\n" );
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document.write( "(then they intersect in some unique point).\r<BR>\n" );
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document.write( "Therefore, &nbsp;if the system &nbsp;(1)&nbsp; is a consistent system of independent equations, &nbsp;then the system &nbsp;(2)&nbsp; is too, &nbsp;because the system &nbsp;(2)&nbsp; represents \r<BR>\n" );
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document.write( "the same straight lines as the system &nbsp;(1).</BLOCKQUOTE> \r<BR>\n" );
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document.write( " </SPAN>\n" );
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