document.write( "Question 163433: Using the Gauss-Jordan elimination method, solve the following linear system. \r
\n" ); document.write( "\n" ); document.write( "7x +5y -3z = 16
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Algebra.Com's Answer #120403 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "In the Gauss-Jordan elimination, you start with a system\r\n" );
document.write( "of three equations and three unknowns:\r\n" );
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document.write( "  \r\n" );
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document.write( "Then you convert it to this matrix:\r\n" );
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document.write( "Then you use row operations and end up with a matrix like this:\r\n" );
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document.write( "That is, you get 0's in the lower left three elements.\r\n" );
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document.write( "Then you convert back to a system of equations:\r\n" );
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document.write( "   \r\n" );
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document.write( "Then you do what is called \"back-substitution\":\r\n" );
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document.write( "1. Solve the bottom equation for z.\r\n" );
document.write( "2. Substitute that value of z in the middle equation, \r\n" );
document.write( "   and solve for y.\r\n" );
document.write( "3. Substitute the values of y and z in the top equation\r\n" );
document.write( "   and solve for x.  \r\n" );
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document.write( "----------\r\n" );
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document.write( "To get a 0 where the 3 is on the middle row:\r\n" );
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document.write( "Multiply the top row by -3 and the middle row by 7, and\r\n" );
document.write( "add them together:\r\n" );
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document.write( "\"%28matrix%282%2C5%2C-21%2C-15%2C9%2C%22%7C%22%2C-48%2C21%2C-35%2C14%2C%22%7C%22%2C-56%29%29\"\r\n" );
document.write( "--------------\r\n" );
document.write( "\"%28matrix%281%2C5%2C0%2C-50%2C23%2C%22%7C%22%2C-104%29%29\"\r\n" );
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document.write( "Replace the second row by that, leaving the rest as is\r\n" );
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document.write( "To get a 0 where the 5 is on the bottom row:\r\n" );
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document.write( "Multiply the top row by -5 and the bottom row by 7, and\r\n" );
document.write( "add them together:\r\n" );
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document.write( "\"%28matrix%282%2C5%2C-35%2C-25%2C15%2C%22%7C%22%2C-80%2C35%2C21%2C-49%2C%22%7C%22%2C0%29%29\"\r\n" );
document.write( "--------------\r\n" );
document.write( "\"%28matrix%281%2C5%2C0%2C-4%2C-34%2C%22%7C%22%2C-80%29%29\"\r\n" );
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document.write( "Notice that as it turns out, we can divide that through\r\n" );
document.write( "by -2, so we might as well do that too, and get:\r\n" );
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document.write( "\"%28matrix%281%2C5%2C0%2C2%2C17%2C%22%7C%22%2C40%29%29\"\r\n" );
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document.write( "We replace the bottom row by that, leaving the rest as is\r\n" );
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document.write( "To get a 0 where the 2 is on the bottom row:\r\n" );
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document.write( "Take the middle row as it is. Multiply the bottom row by 25, and\r\n" );
document.write( "add them together:\r\n" );
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document.write( "\"%28matrix%282%2C5%2C0%2C-50%2C23%2C%22%7C%22%2C-104%2C0%2C50%2C425%2C%22%7C%22%2C1000%29%29\"\r\n" );
document.write( "--------------\r\n" );
document.write( "\"%28matrix%281%2C5%2C0%2C0%2C448%2C%22%7C%22%2C896%29%29\"\r\n" );
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document.write( "Notice that as it turns out, we can divide that through\r\n" );
document.write( "by 448, so we might as well do that too, and get:\r\n" );
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document.write( "\"%28matrix%281%2C5%2C0%2C0%2C1%2C%22%7C%22%2C2%29%29\"\r\n" );
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document.write( "We replace the bottom row by that, leaving the rest as is\r\n" );
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document.write( "Then we convert that back to this system of equations:\r\n" );
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document.write( "\"system%287x%2B5y-3z=16%2C0x-50y%2B23z=-104%2C0x%2B0y%2Bz=2%29\"\r\n" );
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document.write( "or rather,\r\n" );
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document.write( "\"system%287x%2B5y-3z=16%2C-50y%2B23z=-104%2Cz=2%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( "The bottom equation is already solved for z.\r\n" );
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document.write( "Substitute \"z=2\" in the middle equation:\r\n" );
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document.write( "\"-50y%2B23z=-104\"\r\n" );
document.write( "\"-50y%2B23%282%29=-104\"\r\n" );
document.write( "\"-50y%2B46=-104\"\r\n" );
document.write( "\"-50y=-150\"\r\n" );
document.write( "\"y=3\"\r\n" );
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document.write( "Substitute \"z=2\" and \"y=3\" in the top\r\n" );
document.write( "equation:\r\n" );
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document.write( "\"7x%2B5y-3z=16\"\r\n" );
document.write( "\"7x%2B5%283%29-3%282%29=16\"\r\n" );
document.write( "\"7x%2B15-6=16\"\r\n" );
document.write( "\"7x%2B9=16\"\r\n" );
document.write( "\"7x=7\"\r\n" );
document.write( "\"x=1\"\r\n" );
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document.write( "So the solution is  \r\n" );
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document.write( "Edwin

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