document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=124341>Question 124341</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you know when an equation has infinitely<BR>\n" );
document.write( "many solutions? <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #91132 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=91132\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If both sides of the equation can be reduced such that they are identical,<BR>\n" );
document.write( "then there are <BR>\n" );
document.write( "an infinite number of solutions.<BR>\n" );
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document.write( "For example, take the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+=+%2812%282x+-+2%29%29%2F6+%2B+4 ALIGN=MIDDLE  ALT=\"4x+=+%2812%282x+-+2%29%29%2F6+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Looks pretty normal, but let's solve it. Begin by doing the distributed multiplication <BR>\n" );
document.write( "in the numerator on the right side to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+=+%2824x+-+24%29%2F6+%2B+4 ALIGN=MIDDLE  ALT=\"4x+=+%2824x+-+24%29%2F6+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Divide the two terms in the numerator by the denominator 6 to reduce the equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+=+4x+-+4+%2B+4 ALIGN=MIDDLE  ALT=\"4x+=+4x+-+4+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The -4 and +4 on the right side cancel out, so the equation is just:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+=+4x ALIGN=MIDDLE  ALT=\"4x+=+4x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and if you divide both sides by 4 you get the identity:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+x ALIGN=MIDDLE  ALT=\"x+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This tells you that no matter what value of x you use, the right side of the equation will <BR>\n" );
document.write( "equal the left side. Try it for some values of x ... <BR>\n" );
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document.write( "Start with the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+=+%2812%282x+-+2%29%29%2F6+%2B+4 ALIGN=MIDDLE  ALT=\"4x+=+%2812%282x+-+2%29%29%2F6+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and let x = 0<BR>\n" );
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document.write( "The equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+%2812%280+-+2%29%29%2F6+%2B+4+=+-24%2F6+%2B+4+=+-4+%2B+4+=+0 ALIGN=MIDDLE  ALT=\"0+=+%2812%280+-+2%29%29%2F6+%2B+4+=+-24%2F6+%2B+4+=+-4+%2B+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The result is <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;\" > so that works.<BR>\n" );
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document.write( "Suppose you now let x = 10. The equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+=+%2812%282x+-+2%29%29%2F6+%2B+4 ALIGN=MIDDLE  ALT=\"4x+=+%2812%282x+-+2%29%29%2F6+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "then becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40+=+%2812%2820+-+2%29%29%2F6+%2B+4 ALIGN=MIDDLE  ALT=\"40+=+%2812%2820+-+2%29%29%2F6+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "the numerator on the right side reduces to 12(18) = 216 and the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40+=+216%2F6+%2B+4 ALIGN=MIDDLE  ALT=\"40+=+216%2F6+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Do the division on the right side and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40+=+36+%2B+4+ ALIGN=MIDDLE  ALT=\"40+=+36+%2B+4+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "That is true so x = 10 works. You can try other values for x, and no matter what value you<BR>\n" );
document.write( "substitute for x the result will always be the same ... the left side will equal the right <BR>\n" );
document.write( "side. There are an infinite number of solutions because the equation can be reduced to <BR>\n" );
document.write( "the identity x &#8801; x.<BR>\n" );
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document.write( "Hope this clarifies things a little bit.<BR>\n" );
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