document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=61781>Question 61781</A>: <I><SPAN STYLE=\"word-break: break-all;\"> When solving a rational equation, why it is OK to remove the denominator by multiplying both sides by the LCD and why can you not do the same operation when simplifying a rational expression? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #42569 by </B><A HREF=/tutors/aboutme.mpl?userid=mathick><B>mathick(4)</B></A><img src=\"/images/stars/iconNewId_16x16.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathick><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=mathick><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=42569\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color = \"blue\"><BR>\n" );
document.write( "Good question.  The difference between expressions and equations is key to the answer.\r<BR>\n" );
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document.write( "A simple example of an expression is:   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  \r<BR>\n" );
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document.write( "And a simple example of an equation is:     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5 ALIGN=MIDDLE  ALT=\"x+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "If the equation were a scale, the left side and right side would balance each other perfectly.  Now if the same weight (say 3) is added to both sides of the equation:\r<BR>\n" );
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document.write( "<center><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2B+3+=+5+%2B+3 ALIGN=MIDDLE  ALT=\"x+%2B+3+=+5+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,</center>\r<BR>\n" );
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document.write( "you get an equation that is equivalent to (basically the same as) the original equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5 ALIGN=MIDDLE  ALT=\"x+=+5\"   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=x+=+5 ALIGN=MIDDLE  ALT=\"x+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > is basically the same equation as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2B+3+=+5+%2B+3 ALIGN=MIDDLE  ALT=\"x+%2B+3+=+5+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (they have the same answer).\r<BR>\n" );
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document.write( "In an equation, the left and right side are balanced.  The main idea in solving equations is:  if you start with a balanced scale and then do the same thing to both sides of the scale, the scale will still end up balanced.\r<BR>\n" );
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document.write( "With an expression, however there's no scale.  An expression is like a weight just sitting there on it's own.  So if 3 is added to an expression, it's no longer the same expression.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not the same expression as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5+%2B+3 ALIGN=MIDDLE  ALT=\"5+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Moving to your question, the reason you can multiply both sides of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F3+%2B+x%2F2+=+10 ALIGN=MIDDLE  ALT=\"x%2F3+%2B+x%2F2+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" > by 6 is that you're preserving the balance by doing the same thing (multiplying by 6) to both sides.\r<BR>\n" );
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document.write( "Multiplying the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F3 ALIGN=MIDDLE  ALT=\"x%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > by 6, however, results in a different expression that's not equivalent.  So multiplying an expression by a number typically changes the expression and so isn't allowed.\r<BR>\n" );
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document.write( "Sometimes, though you do multiply an expression by 1.  Multiplying by any other number usually changes the expression, but multiplying by 1 doesn't change it.  That's why multiplying by 1 is allowed.  Usually you multiply by 1 in a different form, such as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F3 ALIGN=MIDDLE  ALT=\"3%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2F2x ALIGN=MIDDLE  ALT=\"2x%2F2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  For example, starting with\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F3+%2B+x%2F2+=+10 ALIGN=MIDDLE  ALT=\"x%2F3+%2B+x%2F2+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" >,\r<BR>\n" );
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document.write( "you can multiply the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F3 ALIGN=MIDDLE  ALT=\"x%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F2 ALIGN=MIDDLE  ALT=\"2%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F2 ALIGN=MIDDLE  ALT=\"x%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F3 ALIGN=MIDDLE  ALT=\"3%2F3\"   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=2x%2F%282%2A3%29+%2B+3x%2F%283%2A2%29+=+10 ALIGN=MIDDLE  ALT=\"2x%2F%282%2A3%29+%2B+3x%2F%283%2A2%29+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "to get the common denominator:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2F6+%2B+3x%2F6+=+10 ALIGN=MIDDLE  ALT=\"2x%2F6+%2B+3x%2F6+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "I hope this helps - let me know if you have questions about any part of it.<BR>\n" );
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