document.write( "Question 167734: i have some problems with this system ? can you tell me how to solution sytem\r
\n" ); document.write( "\n" ); document.write( "sytem is x^3 + y^3 = 1
\n" ); document.write( " x^2y + 2xy^2 + Y^3 = 2\r
\n" ); document.write( "\n" ); document.write( "please contact with me as soon as you can \n" ); document.write( "
Algebra.Com's Answer #123725 by Edwin McCravy(20054) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "Let \r\n" );
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document.write( "Now we can solve each for  as long as we require that\r\n" );
document.write( " and \r\n" );
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document.write( "So we will take Case 1 as when they are both true:\r\n" );
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document.write( "Case 1:   and \r\n" );
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document.write( "Then we can solve both for  without\r\n" );
document.write( "dividing by 0:\r\n" );
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document.write( "Since the left sides are equal, so are the right \r\n" );
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document.write( "Cross multiply:\r\n" );
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document.write( "Get 0 on the right and left side in descending\r\n" );
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document.write( "Factor k^2 out of the first two terms on the left,\r\n" );
document.write( "and factor -1 out of the last two terms on the left:\r\n" );
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document.write( "Factor  out of both terms on the left:\r\n" );
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document.write( "Factor the second parentheses on the left:\r\n" );
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document.write( "Using the zero-factor principle:\r\n" );
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document.write( "Remember that the requirements for Case 1 were:\r\n" );
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document.write( " and \r\n" );
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document.write( "We investigate to see whether  satisfies\r\n" );
document.write( "both of those requirements. Substituting,\r\n" );
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document.write( " and \r\n" );
document.write( " and \r\n" );
document.write( " and \r\n" );
document.write( " and \r\n" );
document.write( " and \r\n" );
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document.write( "So there are both satisfied, so we can use \r\n" );
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document.write( "We now investigate to see whether  satisfies\r\n" );
document.write( "both of those requirements. Substituting,\r\n" );
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document.write( " and \r\n" );
document.write( " and \r\n" );
document.write( " and \r\n" );
document.write( " and \r\n" );
document.write( " and \r\n" );
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document.write( "So these are both satisfied, so we can also use \r\n" );
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document.write( "We now investigate to see whether  satisfies\r\n" );
document.write( "both of those requirements. Substituting,\r\n" );
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document.write( " and \r\n" );
document.write( " and \r\n" );
document.write( " and \r\n" );
document.write( " and \r\n" );
document.write( " and \r\n" );
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document.write( "Since the first is not satisfied we cannot use \r\n" );
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document.write( "Now we use and since ,  in the original:\r\n" );
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document.write( "solving both for \r\n" );
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document.write( "So these are the same, so we can\r\n" );
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document.write( "get the real solution by taking cube roots\r\n" );
document.write( "of both sides:\r\n" );
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document.write( "rationalizing:\r\n" );
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document.write( "And since in this case \r\n" );
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document.write( "we have solution\r\n" );
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document.write( "But that's not necessarily the only solution.\r\n" );
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document.write( "Now we use and since ,  in the original:\r\n" );
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document.write( "solving both for \r\n" );
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document.write( "So these are the same, so we can\r\n" );
document.write( "get a real solution by taking cube roots\r\n" );
document.write( "of both sides:\r\n" );
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document.write( "rationalizing:\r\n" );
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document.write( "And since in this case \r\n" );
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document.write( "we have solution\r\n" );
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document.write( "So we have found two solutions.  But this\r\n" );
document.write( "was only for Case 1, when \r\n" );
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document.write( " and \r\n" );
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document.write( "We must investigate the possibility of\r\n" );
document.write( "solutions when one or both of these are\r\n" );
document.write( "violated:\r\n" );
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document.write( "Case 2: \r\n" );
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document.write( "This means that\r\n" );
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document.write( " or \r\n" );
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document.write( "Now we use and since ,  in the original:\r\n" );
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document.write( "This is certainly not true, so Case 2\r\n" );
document.write( " is not possible since it produces\r\n" );
document.write( "no solutions.\r\n" );
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document.write( "Case 3: \r\n" );
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document.write( "Factoring out k on the left:\r\n" );
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document.write( "Factoring the trinomial in parenheses:\r\n" );
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document.write( "Using the zero-factor principle:\r\n" );
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document.write( "We have already seen that  is not\r\n" );
document.write( "possible, so we only need investigate \r\n" );
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document.write( "Now we use and since ,  in the original:\r\n" );
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document.write( "This is certainly not true, so Case 3\r\n" );
document.write( " is not possible since it produces\r\n" );
document.write( "no solutions.\r\n" );
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document.write( "So the only real solutions are:\r\n" );
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document.write( "and\r\n" );
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document.write( "However, there are also some imaginary solutions.  I did not\r\n" );
document.write( "find those. They would require finding the 2 imaginary\r\n" );
document.write( "cube roots each of 4 and 3.  I only considered the real\r\n" );
document.write( "cube roots of 4 and 3.  Post again if you want all the\r\n" );
document.write( "imaginary solutions as well.\r\n" );
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document.write( "Edwin
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