document.write( "Question 1134916: Find necessary and sufficient conditions on a, b, c and d to ensure that
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Algebra.Com's Answer #752432 by ikleyn(52786) $\"\"$ $\"About$

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document.write( "From the first glance to the problem, the answer is obvious:\r\n" );
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document.write( "    The necessary and sufficient condition is proportionality of the pairs (a,b) and (c,d).\r\n" );
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document.write( "To get a formal proof, you can take the derivative of the function.\r\n" );
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document.write( "The numerator of the derivative function then is \r\n" );
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document.write( "    a*(cx + d) - (ax + b)*c = (ac - ac)*x + ad - bc = ad - bc.\r\n" );
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document.write( "The function (the original rational function / (fraction) ) is a constant if and only if the numerator of the derivative is identically zero, \r\n" );
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document.write( "which leads to the equality  ad - bc = 0.\r\n" );
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document.write( "In turn, it means that  ad = bc,  or, equivalently,   = .\r\n" );
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document.write( "It is precisely the same condition as proportionality of the pairs (a,b)  and  (c,d).\r\n" );
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