document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=132604>Question 132604</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two numbers differ by 6 and their reciprocals differ by 8/15.Find the numbers </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #96944 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=96944\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First number: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Second number: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The two numbers differ by 6: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-y=6 ALIGN=MIDDLE  ALT=\"x-y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >  We'll just assume that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3Ey ALIGN=MIDDLE  ALT=\"x%3Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The reciprocal of the first number: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx ALIGN=MIDDLE  ALT=\"1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E0 ALIGN=MIDDLE  ALT=\"x%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and because of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-y=6 ALIGN=MIDDLE  ALT=\"x-y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C%3E-6 ALIGN=MIDDLE  ALT=\"y%3C%3E-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The reciprocal of the second number: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fy ALIGN=MIDDLE  ALT=\"1%2Fy\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C%3E0 ALIGN=MIDDLE  ALT=\"y%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and because of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-y=6 ALIGN=MIDDLE  ALT=\"x-y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E6 ALIGN=MIDDLE  ALT=\"x%3C%3E6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Their reciprocals differ by 8/15:  Since we have assumed that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3Ey ALIGN=MIDDLE  ALT=\"x%3Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx%3C1%2Fy ALIGN=MIDDLE  ALT=\"1%2Fx%3C1%2Fy\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fy-1%2Fx=8%2F15 ALIGN=MIDDLE  ALT=\"1%2Fy-1%2Fx=8%2F15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Re-write the first equation to solve it for x:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=y%2B6 ALIGN=MIDDLE  ALT=\"x=y%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Substitute this expression for x into the second equation in place of x.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fy-1%2F%28y%2B6%29=8%2F15 ALIGN=MIDDLE  ALT=\"1%2Fy-1%2F%28y%2B6%29=8%2F15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Multiply both sides by 15:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%2Fy-15%2F%28y%2B6%29=8 ALIGN=MIDDLE  ALT=\"15%2Fy-15%2F%28y%2B6%29=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Lowest common denominator is just the product of the two denominators since there are no common factors.  So:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2815%28y%2B6%29-15y%29%2F%28y%28y%2B6%29%29=%288y%28y%2B6%29%29%2F%28y%28y%2B6%29%29 ALIGN=MIDDLE  ALT=\"%2815%28y%2B6%29-15y%29%2F%28y%28y%2B6%29%29=%288y%28y%2B6%29%29%2F%28y%28y%2B6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Distribute and collect like terms all on the left:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-8y%5E2-48y%2B15y%2B90-15y%29%2F%28y%28y%2B6%29%29 ALIGN=MIDDLE  ALT=\"%28-8y%5E2-48y%2B15y%2B90-15y%29%2F%28y%28y%2B6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-8y%5E2-48y%2B90%29%2F%28y%28y%2B6%29%29 ALIGN=MIDDLE  ALT=\"%28-8y%5E2-48y%2B90%29%2F%28y%28y%2B6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Solve the numerator expression for 0, remembering to exclude 0 and -6 as values not in the domain of the function in terms of y.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8y%5E2-48y%2B90=0 ALIGN=MIDDLE  ALT=\"-8y%5E2-48y%2B90=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Divide by -2:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y%5E2%2B24y-45=0 ALIGN=MIDDLE  ALT=\"4y%5E2%2B24y-45=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factor:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-3%29%282x%2B15%29=0 ALIGN=MIDDLE  ALT=\"%282x-3%29%282x%2B15%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so y = 1.5 or y = -7.5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = 1.5 => x = 7.5 because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7.5-1.5=6 ALIGN=MIDDLE  ALT=\"7.5-1.5=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F1.5-1%2F7.5=5%2F7.5-1%2F7.5=4%2F7.5=8%2F15 ALIGN=MIDDLE  ALT=\"1%2F1.5-1%2F7.5=5%2F7.5-1%2F7.5=4%2F7.5=8%2F15\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the first root checks.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = -7.5 => x = -1.5 because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1.5-%28-7.5%29=6 ALIGN=MIDDLE  ALT=\"-1.5-%28-7.5%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28-7.5%29-1%2F%28-1.5%29=-1%2F7.5%2B5%2F7.5=4%2F7.5=8%2F15 ALIGN=MIDDLE  ALT=\"1%2F%28-7.5%29-1%2F%28-1.5%29=-1%2F7.5%2B5%2F7.5=4%2F7.5=8%2F15\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the second root checks. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );