document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=120712>Question 120712</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how do you know when an equation has an infinite amount of solutions or no solution? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #88501 by </B><A HREF=/tutors/aboutme.mpl?userid=nabla><B>nabla(475)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nabla><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=88501\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The answer to this question will sometimes vary depending on which area of mathematics you are considering. However, in general:<BR>\n" );
document.write( "if you get a result such as 0=0, this implies a true statement, and as such is the case, any value of a particular variable will work.\r<BR>\n" );
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document.write( "Example: Consider this system of equations:<BR>\n" );
document.write( "          <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2By=1 ALIGN=MIDDLE  ALT=\"2x%2By=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "         <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%2B2y=2 ALIGN=MIDDLE  ALT=\"4x%2B2y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If we solve the first equation for y, we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x%2B1 ALIGN=MIDDLE  ALT=\"y=-2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "now substituting this y into the second equation,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%2B2%28-2x%2B1%29=2 ALIGN=MIDDLE  ALT=\"4x%2B2%28-2x%2B1%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "as you can see, we will get the statement <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2=2 ALIGN=MIDDLE  ALT=\"2=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Thus, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x%2B1 ALIGN=MIDDLE  ALT=\"y=-2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives us a y value for any x that we can choose. Moreover, y is the dependent variable.\r<BR>\n" );
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document.write( "On the other hand, you can sometimes get a result such as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=1 ALIGN=MIDDLE  ALT=\"0=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is a false statement. This implies that there is no solution to an equation or a system of equations.\r<BR>\n" );
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document.write( "Example: Consider this system of equations: <BR>\n" );
document.write( "            <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=1 ALIGN=MIDDLE  ALT=\"x%2By=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "            <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=-1 ALIGN=MIDDLE  ALT=\"x%2By=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solve for y in the first equation: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-x%2B1 ALIGN=MIDDLE  ALT=\"y=-x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "substitute this into the second equation: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B%28-x%2B1%29=-1 ALIGN=MIDDLE  ALT=\"x%2B%28-x%2B1%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. As you can see, this will result in the statement that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1=-1 ALIGN=MIDDLE  ALT=\"1=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is false, there are no values of x or y that make that system true.<BR>\n" );
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