document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=564213>Question 564213</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what are all ordered pairs of numbers (x, y) which satisfy  x^2 – xy + y^2 = 7  and  x- xy + y = -1 ?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #365449 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=365449\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Add 3xy to both sides of the 1st equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^2 + 2xy + y^2 = 7 + 3xy \Rightarrow (x+y)^2 = 7 + 3xy\">\r<BR>\n" );
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document.write( "The second equation is equivalent to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE xy = x+y+1\">. Substituting this into the first equation,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (x+y)^2 = 7 + 3(x+y+1) = 10 + 3(x+y)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (x+y)^2 - 3(x+y) - 10 = 0\">\r<BR>\n" );
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document.write( "This is a quadratic in x+y, and the solutions are x+y = 5 and x+y = -2. If x+y = 5, substituting into the second equation we obtain\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 5 - xy = -1 \Rightarrow xy = 6\">\r<BR>\n" );
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document.write( "And if x+y = -2, substituting into the second equation yields\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -2 - xy = -1 \Rightarrow xy = -1\">\r<BR>\n" );
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document.write( "Therefore we have two cases: x+y = 5 & xy = 6, and x+y = -2 & xy = -1. The first case yields {3,2}; the second case we can let x = -2-y, in which\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (-2-y)(y) = -1 \Rightarrow y^2 + 2y - 1 = 0\">\r<BR>\n" );
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document.write( "By the quadratic formula, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y = -1 \pm \sqrt{2}\">, the solution obtained is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (-1+\sqrt{2}, -1-\sqrt{2})\"> </SPAN>\n" );
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