document.write( "Question 1209170: Find one ordered pair (x,y) of real numbers such that x + y = 8 and x^3 + y^3 = 200 - 4x^2 - 4y^2. \n" ); document.write( "

Algebra.Com's Answer #847752 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Find one ordered pair (x,y) of real numbers such that x + y = 8 and x^3 + y^3 = 200 - 4x^2 - 4y^2.
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document.write( "Left side  x^3 + y^3  can be composed  as  (x+y)*(x^2 - xy + y^2).\r\n" );
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document.write( "The factor  (x+y)  can be replaced by 8,  based on the first equation.\r\n" );
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document.write( "After that, second equation can be written in this form\r\n" );
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document.write( "    8(x^2 - xy + y^2) = 200 - 4(x^2 + y^2),\r\n" );
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document.write( "or\r\n" );
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document.write( "    12(x^2 + y^2) - 8xy = 200,\r\n" );
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document.write( "    12(x+y)^2 - 24xy - 8xy = 200,\r\n" );
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document.write( "    12(x+y)^2 - 32xy = 200.\r\n" );
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document.write( "Again, we can replace here  (x+y)  by 8, and we get then\r\n" );
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document.write( "    12*8^2 - 32xy = 200,\r\n" );
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document.write( "    32xy = 12*64 - 200 = 568,\r\n" );
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document.write( "      xy = 568/32 = 17.75.\r\n" );
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document.write( "But under the condition x+y = 8,  well known minimax property says \r\n" );
document.write( "that the product xy may have the maximum value of  4*4 = 16  at  x = y = 8/2 = 4.\r\n" );
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document.write( "In other words, under given conditions, the original system has no real solutions.\r\n" );
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document.write( "So, there is no any pair of real numbers (x,y) satisfying the imposed conditions.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved completely in Math terms.\r
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