document.write( "Question 805934: Consider the equation (x+ 1)(y+ 1) = 2xy. Find two pairs of positive integers that satisfy this equation and explain how you found them. \n" ); document.write( "

Algebra.Com's Answer #485593 by AnlytcPhil(1806) $\"\"$ $\"About$

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document.write( "(x+1)(y+1) = 2xy\r\n" );
document.write( "Multiply the left side out:\r\n" );
document.write( "xy + x + y + 1 = 2xy\r\n" );
document.write( "Subtract xy from both sides:\r\n" );
document.write( "x + y + 1 = xy\r\n" );
document.write( "Isolate the terms in y\r\n" );
document.write( "x + 1 = xy - y\r\n" );
document.write( "Factor y out on the right\r\n" );
document.write( "x + 1 = y(x - 1)\r\n" );
document.write( "Divide both sides by (x-1)\r\n" );
document.write( " = y\r\n" );
document.write( "I like y on the left\r\n" );
document.write( "y = \r\n" );
document.write( "Divide that fraction out by long division\r\n" );
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document.write( "      1\r\n" );
document.write( "x-1)x+1\r\n" );
document.write( "    x-1\r\n" );
document.write( "      2\r\n" );
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document.write( "y = QUOTIENT + \r\n" );
document.write( "y = 1 + \r\n" );
document.write( "In order for y to be an integer, the fractional expression\r\n" );
document.write( " must equal to an integer.  That means the \r\n" );
document.write( "denominator x-1 must be a factor of the numerator, 2.\r\n" );
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document.write( "The only factors of 2 are 1 and 2. So we set the\r\n" );
document.write( "denominator x-1 equal to each of those:\r\n" );
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document.write( "If x-1 = 1\r\n" );
document.write( "     x = 2\r\n" );
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document.write( "y = \r\n" );
document.write( "y =  \r\n" );
document.write( "y = \r\n" );
document.write( "y = 3\r\n" );
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document.write( "So one possible pair of integers is x=2 and y=3.\r\n" );
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document.write( "If x-1 = 2\r\n" );
document.write( "     x = 3\r\n" );
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document.write( "y = \r\n" );
document.write( "y =  \r\n" );
document.write( "y = \r\n" );
document.write( "y = 2\r\n" );
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document.write( "So the other possible pair of integers is x=3 and y=2.\r\n" );
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document.write( "Edwin

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