document.write( "Question 727655: How do you solve this system of non-linear equations?\r
\n" ); document.write( "\n" ); document.write( "xy(x^2 + y^2) = 300\r
\n" ); document.write( "\n" ); document.write( "x^2 + xy + y^2 = 37\r
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Algebra.Com's Answer #445165 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "   xy(x² + y²) = 300 \r\n" );
document.write( "  x² + xy + y² = 37\r\n" );
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document.write( "There is an x² + y² in both equations.\r\n" );
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document.write( "Isolate x² + y² on the left of both equations:\r\n" );
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document.write( "  x² + y² = \"300%2F%28xy%29\"\r\n" );
document.write( "  x² + y² = 37 - xy \r\n" );
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document.write( "Equate the right sides:\r\n" );
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document.write( "  \"300%2F%28xy%29\" = 37 - xy\r\n" );
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document.write( "Multiply through by xy\r\n" );
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document.write( "          300 = 37xy - (xy)²\r\n" );
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document.write( "  (xy)² - 37xy + 300 = 0\r\n" );
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document.write( "That is a quadratic in xy and it factors as\r\n" );
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document.write( "  (xy - 25)(xy - 12) = 0\r\n" );
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document.write( "  xy - 25 - 0;  xy - 12 = 0\r\n" );
document.write( "       xy = 25;      xy = 12\r\n" );
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document.write( "Solve each for y\r\n" );
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document.write( "       y = \"25%2Fx\";      y = \"12%2Fx\"\r\n" );
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document.write( "Substitute in the first equation:\r\n" );
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document.write( "   xy(x² + y²) = 300;                               xy(x² + y²) = 300\r\n" );
document.write( "   25(x² +\"25%2Fx\"²) = 300;                               12(x² +\"12%2Fx\"²) = 300\r\n" );
document.write( "       x² +\"625%2Fx%5E2\" = 12;                                     x² +\"144%2Fx%5E2\" = 25\r\n" );
document.write( "      x4 + 625 = 12x²                                   x4 + 144 = 25x²\r\n" );
document.write( "x4 - 12x + 625 = 0                               x4 - 25x² + 144 = 0\r\n" );
document.write( "This equation has imaginary \r\n" );
document.write( "solutions,                                    (x² - 16)(x² - 9) = 0 \r\n" );
document.write( "which are too much trouble                     x² - 16 = 0;  x² - 9 = 0\r\n" );
document.write( "to find, so I'll just stick         (x - 4)(x + 4) = 0;   (x - 3)(x + 3) = 0\r\n" );
document.write( "with the real solutions only.        x - 4 = 0; x + 4 = 0;  x - 3 = 0; x + 3 = 0\r\n" );
document.write( "                                        x = 4;     x = -4;     x = 3;    x = -3\r\n" );
document.write( "                                 Substituting each of those in y = \"12%2Fx\",\r\n" );
document.write( "                                        y = 3;     y = -3;     y = 4;    y = -4\r\n" );
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document.write( "So the real solutions are \r\n" );
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document.write( "(x,y) = (4,3), (x,y) = (-4,-3), (x,y) = (3,4),  (x,y) = (-3,-4)\r\n" );
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document.write( "There are four more imaginary solutions coming from the\r\n" );
document.write( "equation: x4 - 12x + 625 = 0\r\n" );
document.write( "which involve square roots or expressions that involve\r\n" );
document.write( "square roots.  It's too much trouble to find those\r\n" );
document.write( "exactly, so I'll just do the 4 real solutions, and only\r\n" );
document.write( "approximate the 4 imaginary solutions with a calculator.\r\n" );
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document.write( "x =  3.937 + 3.082i, y =  3.937 + 3.082i\r\n" );
document.write( "x =  3.937 - 3.082i, y =  3.937 + 3.082i\r\n" );
document.write( "x = -3.937 + 3.082i, y = -3.937 - 3.082i\r\n" );
document.write( "x = -3.937 - 3.082i, y = -3.937 + 3.082i\r\n" );
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document.write( "Edwin
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