document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=398582>Question 398582</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all solutions of the simultaneous equations\r<BR>\n" );
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document.write( "2x^2=14+yz<BR>\n" );
document.write( "2y^2=14+zx<BR>\n" );
document.write( "2z^2=14+xy\r<BR>\n" );
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document.write( "I tried everything but i seen to go in a circle I came up to notice that there is pattern meaning that each variable changes position hence it could mean that x=z=y.. but im not sure at all.. \r<BR>\n" );
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document.write( "help please<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 #282396 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=282396\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It is symmetric, but doesn't necessarily mean that x = y = z (we could prove that it's true if it is).\r<BR>\n" );
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document.write( "Without loss of generality, suppose <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=+x+%3C=+y+%3C=+z ALIGN=MIDDLE  ALT=\"0+%3C=+x+%3C=+y+%3C=+z\"   DISABLED_style= \"max-width:90%; height:auto;\" > (we can make such an assumption because it's symmetric, but we must also consider negative solutions, if they exist). Then,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2z%5E2+%3E=+2y%5E2+%3E=+2x%5E2 ALIGN=MIDDLE  ALT=\"2z%5E2+%3E=+2y%5E2+%3E=+2x%5E2\"   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=14+%2B+xy+%3E=+14+%2B+xz+%3E=+14+%2B+yz ALIGN=MIDDLE  ALT=\"14+%2B+xy+%3E=+14+%2B+xz+%3E=+14+%2B+yz\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy+%3E=+xz+%3E=+yz ALIGN=MIDDLE  ALT=\"xy+%3E=+xz+%3E=+yz\"   DISABLED_style= \"max-width:90%; height:auto;\" > (using substitution)\r<BR>\n" );
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document.write( "Taking subsets of the inequality, we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy+%3E=+xz ALIGN=MIDDLE  ALT=\"xy+%3E=+xz\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy+%3E=+yz ALIGN=MIDDLE  ALT=\"xy+%3E=+yz\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xz+%3E=+yz ALIGN=MIDDLE  ALT=\"xz+%3E=+yz\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which implies <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+z ALIGN=MIDDLE  ALT=\"y+%3E=+z\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=+z ALIGN=MIDDLE  ALT=\"x+%3E=+z\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=+y ALIGN=MIDDLE  ALT=\"x+%3E=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since we already have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+%3E=+y ALIGN=MIDDLE  ALT=\"z+%3E=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+%3E=+x ALIGN=MIDDLE  ALT=\"z+%3E=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+x ALIGN=MIDDLE  ALT=\"y+%3E=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > from our first assumption, this implies <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+y+=+z ALIGN=MIDDLE  ALT=\"x+=+y+=+z\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Setting all variables equal to x, we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+=+14+%2B+x%5E2 ALIGN=MIDDLE  ALT=\"2x%5E2+=+14+%2B+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+sqrt%2814%29 ALIGN=MIDDLE  ALT=\"x+=+sqrt%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2814%29 ALIGN=MIDDLE  ALT=\"sqrt%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2814%29 ALIGN=MIDDLE  ALT=\"sqrt%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2814%29 ALIGN=MIDDLE  ALT=\"sqrt%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is a solution.\r<BR>\n" );
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document.write( "If we assume all x,y,z are negative, then by the same logic we get the ordered triple (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%2814%29 ALIGN=MIDDLE  ALT=\"-sqrt%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%2814%29 ALIGN=MIDDLE  ALT=\"-sqrt%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%2814%29 ALIGN=MIDDLE  ALT=\"-sqrt%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) as another solution.\r<BR>\n" );
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document.write( "However, what happens if some of the numbers are positive and others are negative? In this case we can assume <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C+abs%28x%29+%3C=+abs%28y%29+%3C=+abs%28z%29 ALIGN=MIDDLE  ALT=\"0+%3C+abs%28x%29+%3C=+abs%28y%29+%3C=+abs%28z%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since the magnitudes of the squares of the numbers are still in order, we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2z%5E2+%3E=+2y%5E2+%3E=+2x%5E2 ALIGN=MIDDLE  ALT=\"2z%5E2+%3E=+2y%5E2+%3E=+2x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the inequalities <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+z ALIGN=MIDDLE  ALT=\"y+%3E=+z\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=+z ALIGN=MIDDLE  ALT=\"x+%3E=+z\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=+y ALIGN=MIDDLE  ALT=\"x+%3E=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" > similar to above. Now, suppose we assume that z is positive. If this is the case, x and y are both positive, and their magnitudes are larger, thus contradiction, so z is negative.\r<BR>\n" );
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document.write( "Suppose we subtract the first equation from the second to obtain\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y%5E2+-+2x%5E2+=+zx+-+yz+=+z%28x-y%29 ALIGN=MIDDLE  ALT=\"2y%5E2+-+2x%5E2+=+zx+-+yz+=+z%28x-y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The left side is positive since the magnitude of y is larger than the magnitude of x. Since z is negative, it follows that x-y is also negative. This implies either:\r<BR>\n" );
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document.write( "x is positive, and since y has a magnitude larger than x, y is also positive. However this would not satisfy the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=+y ALIGN=MIDDLE  ALT=\"x+%3E=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" > (unless x = y).\r<BR>\n" );
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document.write( "x is negative, and since y has a larger magnitude and x-y is less than zero, then y is positive. However this would not satisfy <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=+y ALIGN=MIDDLE  ALT=\"x+%3E=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" > in any case (since none of the variables can be zero).\r<BR>\n" );
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document.write( "The only possibility is if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+y ALIGN=MIDDLE  ALT=\"x+=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+%3C+0 ALIGN=MIDDLE  ALT=\"z+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. If we replace y with x we get the system:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2z%5E2+=+14+%2B+x%5E2 ALIGN=MIDDLE  ALT=\"2z%5E2+=+14+%2B+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Subtracting the first equation from the second equation,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2z%5E2+-+2x%5E2+=+x%5E2+-+xz+=+x%28x+-+z%29 ALIGN=MIDDLE  ALT=\"2z%5E2+-+2x%5E2+=+x%5E2+-+xz+=+x%28x+-+z%29\"   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=2%28z-x%29%28z%2Bx%29+=+x%28x-z%29 ALIGN=MIDDLE  ALT=\"2%28z-x%29%28z%2Bx%29+=+x%28x-z%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Cancel z-x\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28z-x%29+=+-x ALIGN=MIDDLE  ALT=\"2%28z-x%29+=+-x\"   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=2z+-+2x+=+-x ALIGN=MIDDLE  ALT=\"2z+-+2x+=+-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2z+=+x ALIGN=MIDDLE  ALT=\"2z+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+x%2F2 ALIGN=MIDDLE  ALT=\"z+=+x%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Therefore we can replace y with x, z with x/2 in the first and third equations to obtain:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+=+14+%2B+x%28x%2F2%29 ALIGN=MIDDLE  ALT=\"2x%5E2+=+14+%2B+x%28x%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F4%29x%5E2+=+14 ALIGN=MIDDLE  ALT=\"%283%2F4%29x%5E2+=+14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28x%2F2%29%5E2+=+14+%2B+x%5E2 ALIGN=MIDDLE  ALT=\"2%28x%2F2%29%5E2+=+14+%2B+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%2F2%29x%5E2+=+14 ALIGN=MIDDLE  ALT=\"%28-1%2F2%29x%5E2+=+14\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "These two equations cannot be satisfied, since it implies <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F4+=+-1%2F2 ALIGN=MIDDLE  ALT=\"3%2F4+=+-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (note that x cannot be zero, it can be proven using contradiction that none of x,y,z can equal zero). Therefore there are no solutions in this case.\r<BR>\n" );
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document.write( "Thus, the only solutions are when x = y = z, or (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2814%29 ALIGN=MIDDLE  ALT=\"sqrt%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2814%29 ALIGN=MIDDLE  ALT=\"sqrt%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2814%29 ALIGN=MIDDLE  ALT=\"sqrt%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%2814%29 ALIGN=MIDDLE  ALT=\"-sqrt%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%2814%29 ALIGN=MIDDLE  ALT=\"-sqrt%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%2814%29 ALIGN=MIDDLE  ALT=\"-sqrt%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >). </SPAN>\n" );
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