document.write( "Question 1093252: You must show all of your work to receive credit. All or nothing. Solve the system using any method. 3/x+4/y=5/2. 6/x+1/y=1/2 \n" ); document.write( "
Algebra.Com's Answer #707874 by ikleyn(52790)\"\" \"About 
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\n" ); document.write( "There is a standard method to solve such non-linear systems.\r
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document.write( "Introduce new variables  u = 1/x  and  v = 1/y.\r\n" );
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document.write( "Then the original system of non-linear equations becomes (by a magical way) a linear system relative new variables \"u\" and \"v\".\r\n" );
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document.write( "3u + 4v = 5/2,   (1)\r\n" );
document.write( "6u +  v = 1/2.   (2)\r\n" );
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document.write( "Now you can apply any method you want / you know to solve (1), (2).\r\n" );
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document.write( "I will use Elimination method. \r\n" );
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document.write( "Multiply equation (1) by 4 (both sides), and equation (2) by 2 (both sides).\r\n" );
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document.write( "You will get\r\n" );
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document.write( "12u + 16v = 10,   (3)\r\n" );
document.write( "12u +  2v = 1.    (4)\r\n" );
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document.write( "Subtract eq(4) from eq(3) to get\r\n" );
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document.write( "14v = 9.\r\n" );
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document.write( "Then  v = 9/14.\r\n" );
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document.write( "Having this, you easily get (from (4))  12u = 1 - 2*(9/14) = 1 - 18/14 = -4/14 = -2/7.\r\n" );
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document.write( "Hence,  u = -2/(7*12) = -1/42.\r\n" );
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document.write( "Thus you get  u = -1/42,  v = 9/14.\r\n" );
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document.write( "Now recall how you introduced u and v:  u = 1/x = -1/42;  hence,  x = -42.\r\n" );
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document.write( "                                   and   v = 1/y = 9/14;  hence,  y = 14/9.\r\n" );
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document.write( "The original system solved.  The solutions are  x = -42  and  y = 14/9.\r\n" );
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\n" ); document.write( "\n" ); document.write( "You can substitute it into the original equations to check it on your own (I did it).\r
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\n" ); document.write( "To see more tricks of this kind,  look into the lesson\r
\n" ); document.write( "\n" ); document.write( "    - Solving systems of non-linear equations in two unknowns using the Cramer's rule \r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic
\n" ); document.write( "     \"2x2-Matrices, determinants, Cramer's rule for systems in two unknowns\" \r
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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
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