document.write( "Question 1170660: (1)/(x^(3))+(1)/(y^(3))=35, (1)/(x^(2))-(1)/(xy)+(1)/(y^(2))=7
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Algebra.Com's Answer #795545 by ikleyn(52805) $\"\"$ $\"About$

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document.write( "Introduce new variables  a = ,  b = .\r\n" );
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document.write( "Then the given system can be re-written in this EQUIVALENT form\r\n" );
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document.write( "    a^3 + b^3 = 35        (1)     (from the first equation)\r\n" );
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document.write( "and\r\n" );
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document.write( "    a^2 - ab + b^2 = 7    (2)     (from the second equation).\r\n" );
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document.write( "From Algebra, we know this identity\r\n" );
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document.write( "    a^3 + b^3 = (a+b)*(a^2 - ab + b^2).\r\n" );
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document.write( "Therefore, equation (1) is\r\n" );
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document.write( "    (a+b)*(a^2 - ab + b^2) = 35.    (3)\r\n" );
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document.write( "In equation (3), replace a^2 - ab + b^2 by 7, based on (2).   You will get then\r\n" );
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document.write( "    7*(a+b) = 35                    \r\n" );
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document.write( "from (3),  which implies\r\n" );
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document.write( "    a + b = 35/7 = 5.\r\n" );
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document.write( "So, instead of two equations (1) and (2), one of which is of the degree 3 and another is of the degree 2,\r\n" );
document.write( "we get an EQUIVALEN system of equations\r\n" );
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document.write( "    a^3 + b^3 = 35      (4)    (the same as equation (1) )\r\n" );
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document.write( "    a   + b   =  5      (5)    (deduced and has the degree 1)\r\n" );
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document.write( "Now, from equation (5)  express  b = 5-a  and substitute it into equation (4).  You will get then\r\n" );
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document.write( "    a^3 + (5-a)^3 = 35\r\n" );
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document.write( "    a^3 + 5^3 - 3*5^2*a + 3*5*a^2 - a^3 = 35\r\n" );
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document.write( "    15a^2 - 75a + 125 = 35\r\n" );
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document.write( "    15a^2 - 75a + 90 = 0\r\n" );
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document.write( "      a^2 - 5a  +  6 = 0      (after canceling factor 15 in previous equation)\r\n" );
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document.write( "      (a - 2)*(a - 3) = 0      (after factoring the previous equation).\r\n" );
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document.write( "So the system (1), (2)  has two solutions\r\n" );
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document.write( "    (a)  (a,b) = (2,3)   \r\n" );
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document.write( "and\r\n" );
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document.write( "    (b)  (a,b) = (3,2).\r\n" );
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document.write( "It means that the original system has two solutions\r\n" );
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document.write( "    (a)  x = ,  y = \r\n" );
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document.write( "and\r\n" );
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document.write( "    (b) x = ,  y = .\r\n" );
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document.write( "ANSWER.  The given system has two solutions  (a)  x = ,  y =   and  (b) x = ,  y = .\r\n" );
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