document.write( "Question 463760: Find two natural numbers, the sum of whose squares is 25 times their sum and also equal to 50 times their difference \n" ); document.write( "

Algebra.Com's Answer #318406 by Edwin McCravy(20060) $\"\"$ $\"About$

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Find two natural numbers, the sum of whose squares is 25 times their sum and also equal to 50 times their difference
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document.write( "Let x be the larger and y be the smaller.\r\n" );
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document.write( "x² + y² = 25(x + y) = 50(x - y)\r\n" );
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document.write( "25(x + y) = 50(x - y) \r\n" );
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document.write( "Divide both sides by 25\r\n" );
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document.write( "x + y = 2(x - y)\r\n" );
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document.write( "x + y = 2x - 2y\r\n" );
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document.write( "3y = x\r\n" );
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document.write( "Substitute 3y for x in:\r\n" );
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document.write( "x² + y² = 25(x + y)\r\n" );
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document.write( "(3y)² + y² = 25(3y + y)\r\n" );
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document.write( "9y² + y² = 75y + 25y\r\n" );
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document.write( "10y² = 100y\r\n" );
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document.write( " y² = 10y\r\n" );
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document.write( "y² - 10y = 0\r\n" );
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document.write( "y(y - 10) = 0\r\n" );
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document.write( "y = 0,  y - 10 = 0\r\n" );
document.write( "             y = 10\r\n" );
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document.write( "0 is not a natural number, so y = 10 \r\n" );
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document.write( "and since\r\n" );
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document.write( "3y = x\r\n" );
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document.write( "3(10) = x\r\n" );
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document.write( "30 = x\r\n" );
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document.write( "So the two natural numbers are 30 and 10.\r\n" );
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document.write( "Checking:\r\n" );
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document.write( "The sum of their squares is 30²+10² = 900+100 = 1000\r\n" );
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document.write( "Their sum is 30+10 = 40\r\n" );
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document.write( "25 times 40 = 1000.  That checks.\r\n" );
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document.write( "Their difference is 30-10 = 20\r\n" );
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document.write( "50 times 20 = 1000.  That checks and \r\n" );
document.write( "so the answers are indeed 30 and 10.\r\n" );
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document.write( "Edwin

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