document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=130266>Question 130266</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of the digits of a certain two-digit number is 7.  Reversing its digits increases the number by 9.  What is the number?\r<BR>\n" );
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document.write( "I know that the answer is 34, and I know that one of the equations is x+y=7, but I do not know the other part to the system of equations.  Thanks for helping me!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #95188 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=95188\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are actually two ways to solve this problem.  Either way, your first equation is correct.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=7 ALIGN=MIDDLE  ALT=\"x%2By=7\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The first way is to derive two additional equations.\r<BR>\n" );
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document.write( "If x is the 10s digit and y is the ones digit then 10x plus y equals the number.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x%2By=n ALIGN=MIDDLE  ALT=\"10x%2By=n\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now, if we reverse the digits, y becomes the 10s digit and x becomes the ones digit and the number is increased by 9, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2B+10y=n%2B9 ALIGN=MIDDLE  ALT=\"x+%2B+10y=n%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Add -9 to both sides of this last equation to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B10y-9=n ALIGN=MIDDLE  ALT=\"x%2B10y-9=n\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Now we have two things that equal n so we can set these two expressions equal to each other:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x%2By=x%2B10y-9 ALIGN=MIDDLE  ALT=\"10x%2By=x%2B10y-9\"   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=9x-9y=-9 ALIGN=MIDDLE  ALT=\"9x-9y=-9\"   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-y=-1 ALIGN=MIDDLE  ALT=\"x-y=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Add this last equation to your very first equation (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=7 ALIGN=MIDDLE  ALT=\"x%2By=7\"   DISABLED_style= \"max-width:90%; height:auto;\" >) term by term:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B0y=6 ALIGN=MIDDLE  ALT=\"2x%2B0y=6\"   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=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2By=7 ALIGN=MIDDLE  ALT=\"3%2By=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get <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;\" >, therefore the number is 34.\r<BR>\n" );
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document.write( "The second way to solve the problem is to realize that the difference between any two-digit number and the result of reversing the digits of that two-digit number is a multiple of nine, and that the multiplier of 9 is the difference between the two digits (for example 25 and 52 differ by 27, 2 and 5 differ by 3 and 27 is 3 times 9).  Since reversing the digits in this problem resulted in a number that was 9 larger, the 10s digit had to be 1 smaller than the ones digit.  This way you could have written <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-y=-1 ALIGN=MIDDLE  ALT=\"x-y=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > directly and solved from there.\r<BR>\n" );
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