document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=45922>Question 45922</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi again,\r<BR>\n" );
document.write( "\n" );
document.write( "This type of problem is still giving me some trouble:\r<BR>\n" );
document.write( "\n" );
document.write( "The sum of the squares of two numbers is 117. The difference of the squares of the same two numbers is 45. Find the numbers. \r<BR>\n" );
document.write( "\n" );
document.write( "Thanks again,<BR>\n" );
document.write( "Louis </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #30509 by </B><A HREF=/tutors/aboutme.mpl?userid=abhijitvakil><B>abhijitvakil(7)</B></A><img src=\"/images/stars/stars-05.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=abhijitvakil><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=30509\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hi Louis,\r<BR>\n" );
document.write( "\n" );
document.write( "The solution isgiven in the problem itself. Always try to convert the statements of the problem into equations. Note -\r<BR>\n" );
document.write( "\n" );
document.write( "Lets say the two numbers are x & y. Now as per the first statement, the sum of the squares of these numbers is 117. In mathematical language this would be written as:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+y%5E2+=+117 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+y%5E2+=+117\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Lets put this as eq. no. 1\r<BR>\n" );
document.write( "\n" );
document.write( "The second condition is that the difference of the squares is 45, therefore it can be written as:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+y%5E2+=+45 ALIGN=MIDDLE  ALT=\"x%5E2+-+y%5E2+=+45\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "This gives us equation no. 2 ok ?<BR>\n" );
document.write( "Now to solve a problem having two unknown quantities, we need atleast 2 sets of relations between them. Here we have the 2 relations as eq 1 & eq 2. Thus we may now proceed to solve them. In order to solve these equations, the approach is to manipulate them in such a manner that we are left with only one variable. Here if you simply add these two equations to each other, what we get is :\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+y%5E2+%2B+x%5E2+-+y%5E2+=+117+%2B+45 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+y%5E2+%2B+x%5E2+-+y%5E2+=+117+%2B+45\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "I hope this much is clear\r<BR>\n" );
document.write( "\n" );
document.write( "Or\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+=+162 ALIGN=MIDDLE  ALT=\"2x%5E2+=+162\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "or\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+81 ALIGN=MIDDLE  ALT=\"x%5E2+=+81\"   DISABLED_style= \"max-width:90%; height:auto;\" >    (divide both sides by 2)\r<BR>\n" );
document.write( "\n" );
document.write( "Obviously x = 9.\r<BR>\n" );
document.write( "\n" );
document.write( "Now put this value of x in eq 1, we get\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%5E2+%2B+y%5E2+=+117 ALIGN=MIDDLE  ALT=\"9%5E2+%2B+y%5E2+=+117\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "or\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=81+%2B+y%5E2+=+117 ALIGN=MIDDLE  ALT=\"81+%2B+y%5E2+=+117\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "or \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2+=+117+-+81+=+36 ALIGN=MIDDLE  ALT=\"y%5E2+=+117+-+81+=+36\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Thus y = 6.\r<BR>\n" );
document.write( "\n" );
document.write( "Hence we have the solution as x= 9, y = 6. Please note that i have given detailed explanations which are not required when you actually solve the problem in an exam. Enjoy solving </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );