document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=73968>Question 73968</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1/(y-4)-2/(y-8)=-1/(y+6) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #53007 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=53007\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28y-4%29-2%2F%28y-8%29=-1%2F%28y%2B6%29 ALIGN=MIDDLE  ALT=\"1%2F%28y-4%29-2%2F%28y-8%29=-1%2F%28y%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Note that we can make everything have a common denominator.  The common denominator is<BR>\n" );
document.write( "formed by multiplying together the denominators of all three fractions and is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-4%29%2A%28y-8%29%2A%28y%2B6%29 ALIGN=MIDDLE  ALT=\"%28y-4%29%2A%28y-8%29%2A%28y%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "The first fraction on the left side already has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-4%29 ALIGN=MIDDLE  ALT=\"%28y-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as its denominator.  Suppose that <BR>\n" );
document.write( "we multiply the first fraction by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28y-8%29%2A%28y%2B6%29%29%2F%28%28y-8%29%2A%28y%2B6%29%29 ALIGN=MIDDLE  ALT=\"%28%28y-8%29%2A%28y%2B6%29%29%2F%28%28y-8%29%2A%28y%2B6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Notice that since this<BR>\n" );
document.write( "multiplier has the same numerator as its denominator it is the same as multiplying by 1.<BR>\n" );
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document.write( "With this multiplication the first fraction becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F%28y-4%29%29%2A%28%28y-8%29%2A%28y%2B6%29%2F%28%28y-8%29%2A%28y%2B6%29%29%29=+%28y-8%29%2A%28y%2B6%29%2F%28%28y-4%29%2A%28y-8%29%2A%28y%2B6%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now let's work on the second fraction on the left side.<BR>\n" );
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document.write( "We do a similar process on the second fraction which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2F%28y-8%29 ALIGN=MIDDLE  ALT=\"-2%2F%28y-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This fraction<BR>\n" );
document.write( "already has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-8 ALIGN=MIDDLE  ALT=\"y-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the denominator.  So this fraction we multiply by<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28y-4%29%2A%28y%2B6%29%29%2F%28%28y-4%29%2A%28y%2B6%29%29 ALIGN=MIDDLE  ALT=\"%28%28y-4%29%2A%28y%2B6%29%29%2F%28%28y-4%29%2A%28y%2B6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "because this fraction contains the two factors that are missing for the common denominator.  <BR>\n" );
document.write( "The result of this multiplication is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2%2F%28y-8%29%29%2A%28%28y-4%29%2A%28y%2B6%29%2F%28%28y-4%29%2A%28y%2B6%29%29%29=+%28-2%29%2A%28y-4%29%2A%28y%2B6%29%2F%28%28y-4%29%2A%28y-8%29%2A%28y%2B6%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Finally, let's work on the fraction on the right side of the equal sign.  This fraction is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F%28y%2B6%29 ALIGN=MIDDLE  ALT=\"-1%2F%28y%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and it is missing the two terms <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-4 ALIGN=MIDDLE  ALT=\"y-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-8 ALIGN=MIDDLE  ALT=\"y-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > if it is to have the common denominator.<BR>\n" );
document.write( "Therefore let's multiply this fraction by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-4%29%2A%28y-8%29%2F%28%28y-4%29%2A%28y-8%29%29 ALIGN=MIDDLE  ALT=\"%28y-4%29%2A%28y-8%29%2F%28%28y-4%29%2A%28y-8%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The result is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%2F%28y%2B6%29%29%2A%28%28y-4%29%2A%28y-8%29%2F%28%28y-4%29%2A%28y-8%29%29%29=+%28-1%29%2A%28y-4%29%2A%28y%2B6%29%2F%28%28y-4%29%2A%28y-8%29%2A%28y%2B6%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now we can substitute all 3 of these converted fractions into the original problem to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-8%29%2A%28y%2B6%29%2F%28%28y-4%29%2A%28y-8%29%2A%28y%2B6%29%29-%282%29%2A%28y-4%29%2A%28y%2B6%29%2F%28%28y-4%29%2A%28y-8%29%2A%28y%2B6%29%29=%28-1%29%2A%28y-4%29%2A%28y%2B6%29%2F%28%28y-4%29%2A%28y-8%29%2A%28y%2B6%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now we can multiply all terms on both sides by the common denominator and that will cancel<BR>\n" );
document.write( "out the common denominator from all terms just leaving us with the numerators as follows:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-8%29%2A%28y%2B6%29-2%2A%28y-4%29%2A%28y%2B6%29+=+-1%2A%28y-4%29%2A%28y-8%29 ALIGN=MIDDLE  ALT=\"%28y-8%29%2A%28y%2B6%29-2%2A%28y-4%29%2A%28y%2B6%29+=+-1%2A%28y-4%29%2A%28y-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "On the right side drop the 1 and just keeping <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28y-4%29%2A%28y-8%29 ALIGN=MIDDLE  ALT=\"-%28y-4%29%2A%28y-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-8%29%2A%28y%2B6%29-2%2A%28y-4%29%2A%28y%2B6%29+=+-%28y-4%29%2A%28y-8%29 ALIGN=MIDDLE  ALT=\"%28y-8%29%2A%28y%2B6%29-2%2A%28y-4%29%2A%28y%2B6%29+=+-%28y-4%29%2A%28y-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Next do all the multiplications.  First, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-8%29%2A%28y%2B6%29+=+y%5E2+-+2y+-+48 ALIGN=MIDDLE  ALT=\"%28y-8%29%2A%28y%2B6%29+=+y%5E2+-+2y+-+48\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2%29%2A%28y-4%29%2A%28y%2B6%29+=+%28-2%29%2A%28y%5E2+%2B2y+-24%29=+-2%2Ay%5E2+-4y+%2B+48 ALIGN=MIDDLE  ALT=\"%28-2%29%2A%28y-4%29%2A%28y%2B6%29+=+%28-2%29%2A%28y%5E2+%2B2y+-24%29=+-2%2Ay%5E2+-4y+%2B+48\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Finally do the multiplication 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=-%28y%5E2+-12y+%2B+32%29+=+-y%5E2+%2B+12y+-+32 ALIGN=MIDDLE  ALT=\"-%28y%5E2+-12y+%2B+32%29+=+-y%5E2+%2B+12y+-+32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Put all these multiplications into the equation to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2+-+2y+-+48+-2%2Ay%5E2+-4y+%2B+48+=+-y%5E2+%2B+12y+-+32 ALIGN=MIDDLE  ALT=\"y%5E2+-+2y+-+48+-2%2Ay%5E2+-4y+%2B+48+=+-y%5E2+%2B+12y+-+32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Combine the like terms on the left side.  First you have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2+-2y%5E2+=+-y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2+-2y%5E2+=+-y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Next you have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2y+-4y+=+-6y ALIGN=MIDDLE  ALT=\"-2y+-4y+=+-6y\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Finally you have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-48+%2B48=+0 ALIGN=MIDDLE  ALT=\"-48+%2B48=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. When you substitute these results<BR>\n" );
document.write( "on the left side the entire equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-y%5E2+-6y+=+-y%5E2+%2B12y+-32 ALIGN=MIDDLE  ALT=\"-y%5E2+-6y+=+-y%5E2+%2B12y+-32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This is further reduced by adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides and this eliminates the squared<BR>\n" );
document.write( "terms entirely.  What is left is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6y+=+12y+-32 ALIGN=MIDDLE  ALT=\"-6y+=+12y+-32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Subtract 12y from both sides to eliminate the 12y on the right side and the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18y+=+-32 ALIGN=MIDDLE  ALT=\"-18y+=+-32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get rid of the minus signs by multiplying both sides by -1 and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18y+=+32 ALIGN=MIDDLE  ALT=\"18y+=+32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and finally divide both sides by 18 to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+32%2F18 ALIGN=MIDDLE  ALT=\"y+=+32%2F18\"   DISABLED_style= \"max-width:90%; height:auto;\" > which reduces to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+16%2F9 ALIGN=MIDDLE  ALT=\"y+=+16%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "That's the answer. You can convert it to a proper fraction (1 and 7/9) or a decimal if you<BR>\n" );
document.write( "like.<BR>\n" );
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document.write( "It's been a long trip to get to the answer, but there is a lot of good practice involved in<BR>\n" );
document.write( "getting there.  Hope this helps you to gain some insight into algebraic manipulations. </SPAN>\n" );
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