document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=460646>Question 460646</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the solution of the system?<BR>\n" );
document.write( "5x - y = 27<BR>\n" );
document.write( "5x +6y = 13\r<BR>\n" );
document.write( "\n" );
document.write( "Is the system consistent or inconsistent?<BR>\n" );
document.write( "Are the equations dependent or independent? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #315988 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=315988\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Multiply the first equation by -1.  Then add the 2 equations.  The x-term will be eliminated and you will be left with a single equation in y.  Solve for y.  Substitute the value you discover for y into either of the two original equations and solve for x.  If at any point you reach a trivial result, such as 0 = 0, then you know that the two equations represent the exact same solution set.  Graphically, they would be the same line.  If at any point you reach an absurd result, such as 0 = 3 for example, then you have a situation where the intersection of the solution sets of the two equations is the null set, meaning the system has no solution.  Graphically, this would be represented by two parallel lines.  If you reach the conclusion that each of the variables is equal to a specific value, then the solution set is the single ordered pair (x,y) where x and y are the values you determined.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A system is consistent if it has one or more solutions.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A system is inconsistent if it has no solutions.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A system is dependent if any solution for one of the equations is a solution for the other, that is to say, there are infinite solutions.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A system is independent if it has exactly one solution.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Hence a system can bed consistent and independent (has one solution -- two lines that intersect in one point)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Consistent and dependent (infinite solutions -- the same line)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Inconsistent (solution set is the null set -- parallel lines)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
document.write( "<div  DISABLED_style= \"text-align:center\"><a  rel=nofollow HREF=\"http://outcampaign.org/\" target=\"_blank\"><IMG STYLE=\"max-width:80%;\" src=\"http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png\" border=\"0\" alt=\"The Out Campaign: Scarlet Letter of Atheism\" width=\"143\" height=\"122\" /></a></div><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );