document.write( "Question 1132097: How many ordered pairs of positive integers (x,y) satisfy the equation x=(4-x)/(y^2-x)? \r
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\n" ); document.write( "\n" ); document.write( "My work: I am pretty sure that the only possible pair is (1,2), but I don’t know how to mathematically explain it... I just need help on this process.
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Algebra.Com's Answer #748989 by ikleyn(52787) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " The tutor @rothauserc was in one step from the final answer.\r
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\n" ); document.write( "\n" ); document.write( " I came to make this last step and to complete the solution.\r
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document.write( "x = (4-x)/(y^2-x)    (1)\r\n" );
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document.write( "x*(y^2-x) = 4 - x\r\n" );
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document.write( "xy^2 -x^2 = 4 - x\r\n" );
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document.write( "xy^2 = x^2 - x + 4\r\n" );
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document.write( "y^2 = (x^2 -x +4)/x\r\n" );
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document.write( "y =  = \r\n" );
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document.write( "Now look into the last formula.\r\n" );
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document.write( "The number    under the square root operator is integer only when\r\n" );
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document.write( "    x=1  and  x= 2  and  x= 4.\r\n" );
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document.write( "For x larger than 4 we have  the fraction under the square root, which CAN NOT give an integer number after taking the square root.\r\n" );
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document.write( "x= 1  gives  y= +/-2.\r\n" );
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document.write( "x= 2  gives  y= +/-, which is not integer.\r\n" );
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document.write( "x= 4 gives  y= +/-2.  But it makes the denominator of the equation (1) equal to zero, so it is ERRONEOUS solution.\r\n" );
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\n" ); document.write( "\n" ); document.write( "ANSWER. The only solutions in integer numbers to equation (1) are these two pairs (x,y) = (1,2) and (x,y) = (1,-2).\r
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\n" ); document.write( "\n" ); document.write( " Of them, only the pair (x,y) = (1,2) is the solution in positive integer numbers.\r
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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