document.write( "Question 1000276: Find x and y given that e^x + 3e^y=3 and e^2x - 9e^2y=6, expressing your answers as a logarithm to base e. \n" ); document.write( "

Algebra.Com's Answer #617810 by ikleyn(52797) $\"\"$ $\"About$

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\n" ); document.write( "Find x and y given that e^x + 3e^y=3 and e^2x - 9e^2y=6, expressing your answers as a logarithm to base e.
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document.write( "Introduce two new variables u =  and v = .\r\n" );
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document.write( "Then your system will be reduced to the system\r\n" );
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document.write( "u   +  3v  = 3,     (1)\r\n" );
document.write( "u^2 - 9v^2 = 6.     (2)\r\n" );
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document.write( "Factor the second equation\r\n" );
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document.write( "(u-3v)*(u+3v) = 6\r\n" );
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document.write( "and substitute 3 instead of u+3x in the last equation. You will get the system\r\n" );
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document.write( "u   +  3v  = 3,         (1')\r\n" );
document.write( "u   -  3v  =  = 2.    (2')\r\n" );
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document.write( "Add the two last equations. You will get\r\n" );
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document.write( "2u = 5. Hence, u = .\r\n" );
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document.write( "Now, distract (2) from (1'). You will get \r\n" );
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document.write( "6v = 1. Hence, v = .\r\n" );
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document.write( "Thus we get \r\n" );
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document.write( "u =  =  -----> x =  = .\r\n" );
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document.write( "v =  =  -----> y =  = .\r\n" );
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