document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=173440>Question 173440</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Explain when a system of equations will have no solution and when it will have an infinite number of solutions. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #128297 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=128297\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The solution set of a linear equation in two variables is a set of ordered pairs such that the coordinates of each pair make the equation a true statement.  The solution set of such an equation, if graphed in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E2 ALIGN=MIDDLE  ALT=\"R%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, will describe a straight line.\r<BR>\n" );
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document.write( "If you have a <i><b>system</i></b> of two equations, there are three possibilities.\r<BR>\n" );
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document.write( "1.  <b>Consistent system</b>:  There is a single ordered pair that is an element of both solution sets.    If both equations are graphed in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E2 ALIGN=MIDDLE  ALT=\"R%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the result will be two intersecting lines and the point of intersection will be defined by the single ordered pair that is an element of both solution sets, and this single ordered pair is the solution set of the <i><b>system</i></b>.\r<BR>\n" );
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document.write( "2.  <b>Inconsistent system</b>:  There is no ordered pair that is an element of both solution sets, in other words, the solution set of the <i><b>system</i></b> is the empty set.  If both equations are graphed in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E2 ALIGN=MIDDLE  ALT=\"R%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the result will be two parallel lines.\r<BR>\n" );
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document.write( "3.  <b>Undetermined system</b>:  Every element of the solution set of one of the equations is also an element of the solution set of the other equation.  In other words, the solution set for the <i><b>system</i></b> is identical to the solution set for either of the lines and therefore has an infinite number of elements. If both equations are graphed in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E2 ALIGN=MIDDLE  ALT=\"R%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the result will be that both equations graph to the same line.\r<BR>\n" );
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document.write( "All of these definitions can be extended to systems of equations that are comprised of any number of equations with a like number of variables, except that once you have more than three variables/equations, the graphical analogue becomes impossible to visualize -- we can draw pictures of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E2 ALIGN=MIDDLE  ALT=\"R%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E3 ALIGN=MIDDLE  ALT=\"R%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E4 ALIGN=MIDDLE  ALT=\"R%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and beyond can't be done.\r<BR>\n" );
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