document.write( "Question 1127259: HI!\r
\n" ); document.write( "\n" ); document.write( "How do I solve these linear equations using a matrix?\r
\n" ); document.write( "\n" ); document.write( "2/x+5/y+3/z=7
\n" ); document.write( "5/x-9/y-7/z=5
\n" ); document.write( "3/x+2/y-4/z=-2\r
\n" ); document.write( "\n" ); document.write( "Thank you! \n" ); document.write( "

Algebra.Com's Answer #743629 by ikleyn(52800) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " I'd like to start noticing that the given system IS NOT a linear: it is the system of NON-LINEAR equations.\r
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document.write( "The standard method to solve such a system is to introduce new variables\r\n" );
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document.write( "u = ,  v = ,  w = .    (*)\r\n" );
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document.write( "Then the system takes the form\r\n" );
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document.write( "2u + 5v + 3w = 7     (1)\r\n" );
document.write( "5u - 9v - 7w = 5     (2)\r\n" );
document.write( "3u - 2c - 4z = 2     (3)\r\n" );
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document.write( "This system is a standard 3x3-system of linear equations. It is MUCH SIMPLER than the original non-linear system.\r\n" );
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document.write( "This modified system you can solve by the Elimination method or by using the Cramer rule (determinants) - on your choice.\r\n" );
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document.write( "Or, if this standard solution procedure is boring to you, you can use standard solvers from the Internet.\r\n" );
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document.write( "If you decide to go this way, I recommend you the online free of charge solvers at the site  www.reshish.com.\r\n" );
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document.write( "In any case, when you complete the solution to the system (1), (2), (3), you must to return to original unknowns \r\n" );
document.write( "x, y and z  using the transformation formulas (*).\r\n" );
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\n" ); document.write( "\n" ); document.write( "To see many problems solved in this way, look into the lesson\r
\n" ); document.write( "\n" ); document.write( "    - Solving systems of non-linear equations in three unknowns using Cramer's rule\r
\n" ); document.write( "\n" ); document.write( "in this site.\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 lesson is the part of this online textbook under the topic
\n" ); document.write( "     \"3x3-Matrices, determinants, Cramer's rule for systems in three 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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\n" ); document.write( "\n" ); document.write( "Again, the lesson to learn from the solution is THIS :\r
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\n" ); document.write( "\n" ); document.write( "            The standard method to solve such a system is to introduce new variables.\r
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\n" ); document.write( "\n" ); document.write( "            In this way, you reduce the original system of non-linear equations to the standard system of linear equations,\r
\n" ); document.write( "\n" ); document.write( "            for which you can use many different and well known standard methods.\r
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