document.write( "Question 643660: Use matrices to help find a general solution for this system of equations.\r
\n" ); document.write( "\n" ); document.write( "2x − y + 3z = 5
\n" ); document.write( "–x + 4y + 4z = –1 \n" ); document.write( "
Algebra.Com's Answer #404611 by Edwin McCravy(20086)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "This is an under-determined system because there are fewer equations\r\n" );
document.write( "than there are variables. \r\n" );
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document.write( "2x -  y + 3z =  5\r\n" );
document.write( "-x + 4y + 4z = -1\r\n" );
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document.write( "Put in 1's\r\n" );
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document.write( " 2x - 1y + 3z =  5\r\n" );
document.write( "-1x + 4y + 4z = -1\r\n" );
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document.write( "Write the system as a matrix [by eliminating the letters, putting | for\r\n" );
document.write( "equal signs and enclosing in brackrts]:\r\n" );
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document.write( "[ 2  -1    3 |  5]\r\n" );
document.write( "[-1   4    4 | -1]\r\n" );
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document.write( "We want to use row operations\r\n" );
document.write( "to make it look like this:\r\n" );
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document.write( "[ 1  0  # |  #]\r\n" );
document.write( "[ 0  1  # |  #]\r\n" );
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document.write( "where there are numbers where the #'s are\r\n" );
document.write( "but 1's and 0's as indicated:\r\n" );
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document.write( "[ 2  -1    3 |  5]\r\n" );
document.write( "[-1   4    4 | -1]\r\n" );
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document.write( "Multiply the 2nd row by 2\r\n" );
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document.write( "[ 2  -1   3  |  5]\r\n" );
document.write( "[-2   8   8  | -2]\r\n" );
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document.write( "Add the 1st row to the 2nd row to\r\n" );
document.write( "get a 0 in the lower left corner:\r\n" );
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document.write( "[ 2  -1   3  | 5]\r\n" );
document.write( "[ 0   7  11  | 3]\r\n" );
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document.write( "Multiply the 1st row by 7\r\n" );
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document.write( "[14  -7  21  |35]\r\n" );
document.write( "[ 0   7  11  | 3]\r\n" );
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document.write( "Add the 2nd row to the 1st row to\r\n" );
document.write( "get a 0 where the -7 is:\r\n" );
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document.write( "[14   0  32 |38]\r\n" );
document.write( "[ 0   7  11 | 3]\r\n" );
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document.write( "Get a 1 where the 14 is by dividing the \r\n" );
document.write( "1st row through by 14\r\n" );
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document.write( "Get a 1 where the 7 is by dividing the\r\n" );
document.write( "2nd row through by 7\r\n" );
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document.write( "[ 1   0 \"32%2F14\" | \"38%2F14\"]\r\n" );
document.write( "[ 0   1 \"11%2F7\" | \"3%2F7\"]\r\n" );
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document.write( "Reduce the fractions:\r\n" );
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document.write( "[ 1   0 \"16%2F7\" | \"19%2F7\"]\r\n" );
document.write( "[ 0   1 \"11%2F7\" | \"3%2F7\"]\r\n" );
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document.write( "Write the matrix as a system of equations by putting\r\n" );
document.write( "the letters and the equal signs back in:\r\n" );
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document.write( " 1x + 0y + \"16%2F7\"z = \"19%2F7\"\r\n" );
document.write( " 0x + 1y + \"11%2F7\"z =  \"3%2F7\"\r\n" );
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document.write( "Simplifying:\r\n" );
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document.write( " x + \"16%2F7\"z = \"19%2F7\"\r\n" );
document.write( " y + \"11%2F7\"z =  \"3%2F7\"\r\n" );
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document.write( "Solving for x and y\r\n" );
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document.write( " x = \"19%2F7\" - \"16%2F7\"z\r\n" );
document.write( " y =  \"3%2F7\" - \"11%2F7\"z\r\n" );
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document.write( "Now you can choose any arbitrary number for z to equal,\r\n" );
document.write( "say z = a\r\n" );
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document.write( " x = \"19%2F7\" - \"16%2F7\"z\r\n" );
document.write( " y =  \"3%2F7\" - \"11%2F7\"z\r\n" );
document.write( " z = a\r\n" );
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document.write( "And write the general solution this way, letting z = a:\r\n" );
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document.write( " x = \"19%2F7\" - \"16%2F7\"a\r\n" );
document.write( " y =  \"3%2F7\" - \"11%2F7\"a\r\n" );
document.write( " z = a\r\n" );
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document.write( "Or as an ordered triple:\r\n" );
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document.write( "(x, y, z) = (\"19%2F7\" - \"16%2F7\"a, \"3%2F7\" - \"11%2F7\"a, a)\r\n" );
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document.write( "That's the general solution:\r\n" );
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document.write( "Then if you want some sample solutions, you can pick various\r\n" );
document.write( "values for a. If you pick a= 0, you have this solution:\r\n" );
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document.write( "(x, y, z) = (\"19%2F7\", \"3%2F7\", 0)\r\n" );
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document.write( "which has fractions.\r\n" );
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document.write( "But by picking the right values for a=z you can get integer solutions.\r\n" );
document.write( "Here are some particular solutions:\r\n" );
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document.write( "(x,y,z) = (5,2,-1)\r\n" );
document.write( "(x,y,z) = (21,13,-8)\r\n" );
document.write( "(x,y,z) = (37,24,-15)\r\n" );
document.write( "(x,y,z) = (53,35,-22)\r\n" );
document.write( "(x,y,z) = (69,46,-29)\r\n" );
document.write( "(x,y,z) = (85,57,-36)\r\n" );
document.write( "(x,y,z) = (101,68,-43)\r\n" );
document.write( "(x,y,z) = (117,79,-50)\r\n" );
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document.write( "Edwin
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