document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=358829>Question 358829</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do I determine whether a system of a linear equation is independent, dependent, or inconsistent. ''Without'' Graphing.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #256045 by </B><A HREF=/tutors/aboutme.mpl?userid=neatmath><B>neatmath(302)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=neatmath><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=neatmath><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=256045\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Dependent linear equations are ones which when simplified are the same equations.  Example: <br>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=5 ALIGN=MIDDLE  ALT=\"x%2By=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B2y=10 ALIGN=MIDDLE  ALT=\"2x%2B2y=10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "Notice that both equations will have the same slope, and will also be the same line!  When this occurs, you will have an infinite number of solutions.<br>\r<BR>\n" );
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document.write( "Inconsistent linear equations are ones which when simplified are parallel lines, but are NOT the same line.  Example:<br>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x%2B1 ALIGN=MIDDLE  ALT=\"y=3x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x-7 ALIGN=MIDDLE  ALT=\"y=3x-7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "Notice that they do have the same slope, but have different y-intercepts, and thus, are totally different lines.  When this occurs, you will have no solutions!<br>\r<BR>\n" );
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document.write( "Independent linear equations are neither dependent nor inconsistent.  Example:<br>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x%2B1 ALIGN=MIDDLE  ALT=\"y=3x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-5x-7 ALIGN=MIDDLE  ALT=\"y=-5x-7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "When this occurs you will have one and only one solution.<br>\r<BR>\n" );
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document.write( "Simply put:<br>\r<BR>\n" );
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document.write( "Same slope, same y-intercept are dependent linear equations.<br>\r<BR>\n" );
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document.write( "Same slope, different y-intercept are inconsistent linear equations.<br>\r<BR>\n" );
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document.write( "Different slopes are independent linear equations.<br>\r<BR>\n" );
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document.write( "So check the slope, then the y-intercept and you will find out what kind of system of linear equations you are dealing with.<br>\r<BR>\n" );
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document.write( "I hope this helps!<br> </SPAN>\n" );
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