document.write( "Question 173732: Solve the system of equations using matrices (row operations).
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Algebra.Com's Answer #128623 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Put 1's in front of the single letters\r\n" );
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document.write( "\"system%28-4x-6y-1z=-25%2C+1x-4y%2B3z=9%2C+-7x%2B1y%2B1z=-7%29\" \r\n" );
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document.write( "Separate into terms, erasing the plus signs, keeping\r\n" );
document.write( "the minus signs as negative signs:\r\n" );
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document.write( "Erase all the letters and replace the equal signs\r\n" );
document.write( "with \"|\"'s:\r\n" );
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document.write( "Erase the brace and put parentheses around it:\r\n" );
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document.write( "Now we want to end up with a matrix like this,\r\n" );
document.write( "with three zeros on the the bottom left, and\r\n" );
document.write( "numbers everywhere else:\r\n" );
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document.write( "Start with this:\r\n" );
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document.write( "Swap the rows so that the smallest number in absolute\r\n" );
document.write( "value in the first column is on the far left of the \r\n" );
document.write( "top row.  Since 1 is the smallest number in absolute\r\n" );
document.write( "value in row 1, I will swap rows 1 and 2:\r\n" );
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document.write( "Now we will add 4 times the top row to the 2nd row,\r\n" );
document.write( "to get a zero where the -4 is. It's easier if you\r\n" );
document.write( "write 4 to the left of the top row and 1 to the left\r\n" );
document.write( "of the second row,and write that equal to a new matrix\r\n" );
document.write( "with the same 1st and 3rd rows, with a blank middle row:\r\n" );
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document.write( "\"matrix%283%2C1%2C4%2C1%2C%22%22%29\"\r\n" );
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document.write( "Then you can easily fill in the blank row term by term as:\r\n" );
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document.write( "Since all the numbers in the middle row are divisible by\r\n" );
document.write( "11, we can multiply it through by \"1%2F11\":\r\n" );
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document.write( "Now we will add 7 times the top row to the 3rd row,\r\n" );
document.write( "to get a zero where the -7 is. It's easier if you\r\n" );
document.write( "write 7 to the left of the top row and 1 to the left\r\n" );
document.write( "of the bottom row,and write that equal to a new matrix\r\n" );
document.write( "with the same 1st and 2nd rows, with a blank bottom row:\r\n" );
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document.write( "\"matrix%283%2C1%2C-7%2C%22%22%2C1%29\"\r\n" );
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document.write( "Then you can easily fill in the blank row term by term as:\r\n" );
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document.write( "---\r\n" );
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document.write( "Now we will add -27 times the middle row to 2 times\r\n" );
document.write( "the 3rd row, to get a zero where the -27 is. It's \r\n" );
document.write( "easier if you write -27 to the left of the middle row \r\n" );
document.write( "and 2 to the left of the bottom row,and write that \r\n" );
document.write( "equal to a new matrix with the same 1st and 2nd rows,\r\n" );
document.write( "with a blank bottom row:\r\n" );
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document.write( "\"matrix%283%2C1%2C%22%22%2C-27%2C2%29\"\r\n" );
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document.write( "Then you can easily fill in the blank row term by term as:\r\n" );
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document.write( "The bottom row can be multiplied through by \"1%2F17\"\r\n" );
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document.write( "\"matrix%283%2C1%2C%22%22%2C%22%22%2C1%2F17%29\"\r\n" );
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document.write( "Now we put the letters back as we took them out, and\r\n" );
document.write( "put equal signs where the \"|\"'s are:\r\n" );
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document.write( "So we have this system:\r\n" );
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document.write( "\"system%28x-4y%2B3z=9%2C-2y%2B1z=1%2Cz=5%29\"\r\n" );
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document.write( "Now we do what is called \"back-substitution\":\r\n" );
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document.write( "Substitute \"z=5\" into the middle equation:\r\n" );
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document.write( "\"matrix%284%2C1%2C%0D%0A-2y%2B1%285%29=1%2C%0D%0A-2y%2B5=1%2C%0D%0A-2y=-4%2C%0D%0Ay=2%29\"\r\n" );
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document.write( "Finally substitute both \"z=5\" and \"y=2\" in\r\n" );
document.write( "the top equation:\r\n" );
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document.write( "So \"x=2\", \"y=2\", \"z=5\".\r\n" );
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document.write( "Edwin

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