document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=143403>Question 143403</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 50-word response to the following: How do you know when an equation has infinitely many solutions? How do you know when an equation has no solution?\r<BR>\n" );
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document.write( "Will someplease explain this to me in just plain old english so that I understand it? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #104368 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=104368\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First of all I don't think you meant to say \"an equation.\"  I think you meant \"a system of equations.\"\r<BR>\n" );
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document.write( "Every equation in two variables has a corresponding straight line graph in the coordinate plane.  Therefore, a system of two equations represents two lines in the plane, and there are three possible situations:\r<BR>\n" );
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document.write( "1.  The two lines intersect in a single point.\r<BR>\n" );
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document.write( "2.  The two lines are parallel and do not intersect at all.\r<BR>\n" );
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document.write( "3.  The two lines are actually the same line and intersect in every point on the line.\r<BR>\n" );
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document.write( "The solution set of a system of equations is the set of ordered pairs (points on the plane) that satisfy both equations simultaneously.  The coordinates of a point satisfy both equations if and only if the coordinates form an ordered pair that represents a point of intersection of the two graphs represented by the equations.\r<BR>\n" );
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document.write( "So, for situation 1 above, you have exactly 1 ordered pair in the solution set.  For situation 2, you have an empty solution set, i.e. no solutions.  And for situation 3, you have a solution set with an infinite number of elements.\r<BR>\n" );
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document.write( "In algebraic terms, given two equations in two variables and you use the elimination method to solve the system you will obtain one of three results:\r<BR>\n" );
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document.write( "1.  You will get a single pair of values representing the two variables (Situation 1 above)\r<BR>\n" );
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document.write( "2.  Your equations will reduce to an absurdity, something like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6=0 ALIGN=MIDDLE  ALT=\"6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, meaning that there is no solution (Situation 2 above), or\r<BR>\n" );
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document.write( "3.  Your equations will reduce to a trivial identity, something like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=0 ALIGN=MIDDLE  ALT=\"0=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, meaning that there are an infinite number of solutions (Situation 3 above)\r<BR>\n" );
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document.write( "Example:<BR>\n" );
document.write( "Situation 1:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=6 ALIGN=MIDDLE  ALT=\"x%2By=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Add the two equations:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B0y=8 ALIGN=MIDDLE  ALT=\"2x%2B0y=8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-y=2 ALIGN=MIDDLE  ALT=\"4-y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2 ALIGN=MIDDLE  ALT=\"y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Solution set: {(4,2)}\r<BR>\n" );
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document.write( "Situation 2:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-2y=6 ALIGN=MIDDLE  ALT=\"2x-2y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Multiply the first equation by -2 and add the result to the 2nd equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%2B2y=-4 ALIGN=MIDDLE  ALT=\"-2x%2B2y=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-2y=6 ALIGN=MIDDLE  ALT=\"2x-2y=6\"   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=0x%2B0y=2 ALIGN=MIDDLE  ALT=\"0x%2B0y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=2 ALIGN=MIDDLE  ALT=\"0=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Absurd result, therefore no solution.\r<BR>\n" );
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document.write( "Situation 3:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-y=2 ALIGN=MIDDLE  ALT=\"x-y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-2y=4 ALIGN=MIDDLE  ALT=\"2x-2y=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Multiply the first equation by -2 and add the result to the 2nd equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%2B2y=-4 ALIGN=MIDDLE  ALT=\"-2x%2B2y=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-2y=4 ALIGN=MIDDLE  ALT=\"2x-2y=4\"   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=0x%2B0y=0 ALIGN=MIDDLE  ALT=\"0x%2B0y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=0 ALIGN=MIDDLE  ALT=\"0=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Trivial identity, therefore infinite solutions.\r<BR>\n" );
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document.write( "Here's another, perhaps simpler way to do it.\r<BR>\n" );
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document.write( "Put both of your equations into slope-intercept form by solving for y (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=mx%2Bb ALIGN=MIDDLE  ALT=\"y=mx%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" >), remembering to reduce to lowest terms.\r<BR>\n" );
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document.write( "1.  If the slopes (m) are different, then you have situation 1, a single solution.\r<BR>\n" );
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document.write( "2.  If the slopes are the same but the intercepts are the different, then you have situation 2, no solution -- they are different but parallel lines.\r<BR>\n" );
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document.write( "3.  If the slopes are the same AND the intercepts are the same, then you have situation 3, infinite solutions -- they are the same line. </SPAN>\n" );
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