document.write( "Question 1091579: The difference between two positive numbers is 18 and 4 times their GM is equal to 5 times their HM find the numbers \n" ); document.write( "

Algebra.Com's Answer #706017 by ikleyn(52790) $\"\"$ $\"About$

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document.write( "1)  x - y = 18, with x, y > 0.    (1)\r\n" );
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document.write( "    GM is geometric mean, GM for x, y is defined as .\r\n" );
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document.write( "    HM is harmonic mean, HM for x, y is defined as . \r\n" );
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document.write( "2)  =  ====>\r\n" );
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document.write( "    =   ====>\r\n" );
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document.write( "    =   ====>  square both sides  ====>\r\n" );
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document.write( "    =   ====>  cancel xy in both sides ====>\r\n" );
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document.write( "   16 =   ====>  = 100xy  ====>   = 25xy      (2)\r\n" );
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document.write( "   Now from (1) express  x = y + 18 and substitute it into (2). You will get\r\n" );
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document.write( "    = 25*(y+18)*y,   or\r\n" );
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document.write( "    = 25*y*(y+18)  ====>   =  ====>   = 0  ====>  cancel 9 in both sides  ====>\r\n" );
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document.write( "    = 0\r\n" );
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document.write( "   (y+24)*(y-6) = 0.\r\n" );
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document.write( "   Since y must be positive, we have only ONE solution for y: y = 6.\r\n" );
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document.write( "   Then  x = y + 18 = 24.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Answer. x = 24, y = 6 is the only solution.\r
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\n" ); document.write( "\n" ); document.write( " The lesson to learn from this solution:\r
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\n" ); document.write( "\n" ); document.write( " If you choose the correct way, you reduce the problem to the quadratic equation, and there is no need to solve the equation of degree 4.\r
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