document.write( "Question 1168642: If for three distinct positive numbers x, y, and z,
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Algebra.Com's Answer #793283 by ikleyn(52782) $\"\"$ $\"About$

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\n" ); document.write( "If for three distinct positive numbers x, y, and z,
\n" ); document.write( " $\"y%2F%28x-z%29\"$ = $\"%28x%2By%29%2Fz\"$ = $\"x%2Fy\"$
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document.write( "You are given two equations\r\n" );
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document.write( "     =          (1)\r\n" );
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document.write( "     =          (2)\r\n" );
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document.write( "In equation (1), divide the numerator and denominator of the left side by \"y\".\r\n" );
document.write( "    You can safely do it, because \"y\" is assumed to be positive.\r\n" );
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document.write( "In equation (2), divide the numerator and denominator of the left side by \"y\".\r\n" );
document.write( "    You can safely do it, due to the same reason.\r\n" );
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document.write( "Both left side sides in equations (1) and (2) will not change their values, so you will get an EQUIVALENT equations\r\n" );
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document.write( "     =         (1')\r\n" );
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document.write( "     =         (2')\r\n" );
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document.write( "Introduce new variables  u =   and  v = .\r\n" );
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document.write( "Our goal is to find the numerical value of  \"u\".\r\n" );
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document.write( "You can write equation (1') and (2') in the form\r\n" );
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document.write( "     = u              (3)\r\n" );
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document.write( "     = u              (4)\r\n" );
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document.write( "From equation (3), you have\r\n" );
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document.write( "    u*(u-v) = 1.             (5)\r\n" );
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document.write( "From equation (4) express  v =   and substitute it into equation (5), replacing \"v\" there.  You will get\r\n" );
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document.write( "     = 1,       or\r\n" );
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document.write( "     = 1,    or\r\n" );
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document.write( "     = 1,           or\r\n" );
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document.write( "     = 0.          (6)\r\n" );
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document.write( "Equation (6) is a quadratic equation.  Solve it by factoring\r\n" );
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document.write( "    (u-2)*(u+1) = 0.\r\n" );
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document.write( "Thus it has two roots  u= 2  and  u= -1.\r\n" );
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document.write( "According to the condition, only positive value  u= 2  is the solution to the problem.\r\n" );
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document.write( "So, the ANSWER  to the problem's question is   = u = 2.\r\n" );
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