document.write( "Question 155400: Find the solution to the system using Gaussin Elimination Method:\r
\n" ); document.write( "\n" ); document.write( "2x + 2y + 4z = 40
\n" ); document.write( "0x - 2y + 2z = 12
\n" ); document.write( "2x + 3y + 3z = 34\r
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Algebra.Com's Answer #114451 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"system%282x%2B2y%2B4z=40%2C0x-2y%2B2z=12%2C2x%2B3y%2B3z=34%29\" \r\n" );
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document.write( "We form the augmented coefficient matrix\r\n" );
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document.write( " [ 2   2   4 | 40]\r\n" );
document.write( " [ 0  -2   2 | 12]\r\n" );
document.write( " [ 2   3   3 | 34]\r\n" );
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document.write( "We must get three 0's in the lower \r\n" );
document.write( "left hand corner, where the numbers\r\n" );
document.write( "are that I have colored red below:\r\n" );
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document.write( " [ 2   2   4 | 40]\r\n" );
document.write( " [ 0  -2   2 | 12]\r\n" );
document.write( " [ 2   3   3 | 34]\r\n" );
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document.write( "Notice that one of them is already\r\n" );
document.write( "0, so we get a zero where the red 2\r\n" );
document.write( "is.\r\n" );
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document.write( "The top row will never change. \r\n" );
document.write( "The other two will. To get a 0 \r\n" );
document.write( "where the red 2 is, We multiply -1 \r\n" );
document.write( "times the first row and add it \r\n" );
document.write( "to 1 times the 3rd row. It's a \r\n" );
document.write( "good idea to put what you're\r\n" );
document.write( "going to multiply a row by out\r\n" );
document.write( "to the left of the row, even \r\n" );
document.write( "when it is just 1, like this:\r\n" );
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document.write( "-1[ 2   2   4 | 40]\r\n" );
document.write( "  [ 0  -2   2 | 12]\r\n" );
document.write( " 1[ 2   3   3 | 34]\r\n" );
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document.write( "The next matrix is then\r\n" );
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document.write( " [ 2   2   4 | 40]\r\n" );
document.write( " [ 0  -2   2 | 12]\r\n" );
document.write( " [ 0   1  -1 | -6]\r\n" );
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document.write( "To get a 0 where the 1 is, \r\n" );
document.write( "we multiply 1 times the second \r\n" );
document.write( "row and add it to 2 times the \r\n" );
document.write( "3rd row.\r\n" );
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document.write( " [ 2   2   4 | 40]\r\n" );
document.write( "1[ 0  -2   2 | 12]\r\n" );
document.write( "2[ 0   1  -1 | -6]\r\n" );
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document.write( "We get this:\r\n" );
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document.write( " [ 2   2   4 | 40]\r\n" );
document.write( " [ 0  -2   2 | 12]\r\n" );
document.write( " [ 0   0   0 |  0]\r\n" );
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document.write( "Now that we have three 0's in the\r\n" );
document.write( "lower left corner, the red places,\r\n" );
document.write( "we go back to a system of equations:\r\n" );
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document.write( "\"system%282x%2B2y%2B4z=40%2C0x-2y%2B2z=12%2C0x%2B0y%2B0z=0+%29\"\r\n" );
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document.write( "or just:\r\n" );
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document.write( "\"system%282x%2B2y%2B4z=40%2C-2y%2B2z=12%2C0z=0+%29\"\r\n" );
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document.write( "Start at the bottom, and go back up.\r\n" );
document.write( "This is called back-substitution.\r\n" );
document.write( "Start with the bottom equation, solve for\r\n" );
document.write( "z:\r\n" );
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document.write( "\"0z=0\"\r\n" );
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document.write( "That's a strange sort of equation.  We see\r\n" );
document.write( "that any arbitrarily chosen value of z will \r\n" );
document.write( "satisfy that equation.  This means that z \r\n" );
document.write( "is any arbitrarily chosen number.  Some books\r\n" );
document.write( "use the letter \"c\" and others simply keep \"z\"\r\n" );
document.write( "and write \"z=z\".\r\n" );
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document.write( "I will write as its solution\r\n" );
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document.write( "\"z=a\" because \"a\" stands for \"arbitrary\"\r\n" );
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document.write( "Substitute \"a\" for \"z\" in the\r\n" );
document.write( "second equation:\r\n" );
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document.write( "\"-2y%2B2z=12\"\r\n" );
document.write( "\"-2y%2B2%28a%29=12\"\r\n" );
document.write( "\"-2y%2B2a=12\"\r\n" );
document.write( "\"-2y=12-2a\"\r\n" );
document.write( "Divide through by -2\r\n" );
document.write( "\"y=-6%2Ba\"\r\n" );
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document.write( "Finally substitute both \"a\" for \"z\"\r\n" );
document.write( "and \"-6%2Ba\" for \"y\" in the original\r\n" );
document.write( "first equation:\r\n" );
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document.write( "\"2x%2B2y%2B4z=40\"\r\n" );
document.write( "\"2x%2B2%28-6%2Ba%29%2B4%28a%29=40\"\r\n" );
document.write( "\"2x%2B2%28-6%2Ba%29%2B4a=40\"\r\n" );
document.write( "\"2x-12%2B2a%2B4a=40\"\r\n" );
document.write( "\"2x-12%2B6a=40\"\r\n" );
document.write( "\"2x=52-6a\"\r\n" );
document.write( "Divide through by 2\r\n" );
document.write( "\"x=26-3a\"\r\n" );
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document.write( "So the solution is\r\n" );
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document.write( "(x,y,z)=(26-3a, -6+a, a)\r\n" );
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document.write( "Edwin

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