\r\n" );
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document.write( "This will likely have infinitely many solutions,\r\n" );
document.write( "since there are more letters, variables, than equations.\r\n" );
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document.write( "2w - 3x + 4y + z = 7\r\n" );
document.write( " w - x + 3y - 5z = 10\r\n" );
document.write( "3w + x - 2y - 2z = 6\r\n" );
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document.write( "
\r\n" );
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document.write( "Swap rows 1 and 2, to get a 1 in the\r\n" );
document.write( "upper left corner:\r\n" );
document.write( "\r\n" );
document.write( "abbreviated R1<->R2\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "To get a 0 where the 2 is, multiply row 1 \r\n" );
document.write( "by -2 and add it to row 2:\r\n" );
document.write( "\r\n" );
document.write( "abbreviated -2R1+R2->R2\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "To get a 0 where the bottom left 3 is, \r\n" );
document.write( "multiply row 1 by -3 and add it to row 3:\r\n" );
document.write( "\r\n" );
document.write( "abbreviated -2R1+R2->R2\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "To get a 1 where the -1 is on Row 2, \r\n" );
document.write( "multiply row 2 by -1\r\n" );
document.write( "\r\n" );
document.write( "abbreviated -R2->R2\r\n" );
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document.write( "
\r\n" );
document.write( " \r\n" );
document.write( "To get a 0 where the 4 is, \r\n" );
document.write( "multiply row 2 by -4 and add it to row 3:\r\n" );
document.write( "\r\n" );
document.write( "abbreviated -4R2+R3->R3\r\n" );
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document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "To get a 1 where the -19 is on Row 3, \r\n" );
document.write( "multiply row 3 by -1/19\r\n" );
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document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Then we change the matrix back to equations:\r\n" );
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document.write( " w - x + 3y - 5z = 10\r\n" );
document.write( " x + 2y - 11z = 13\r\n" );
document.write( " y - 3z = 4\r\n" );
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document.write( "Solve each equation for the first letter\r\n" );
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document.write( " w = 10 + x - 3y + 5z\r\n" );
document.write( " x = 13 - 2y + 11z \r\n" );
document.write( " y = 4 + 3z\r\n" );
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document.write( "Now we do back substitution:\r\n" );
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document.write( "Sunstitute the expression for y in the middle equation:\r\n" );
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document.write( " x = 13 - 2(4 + 3z) + 11z\r\n" );
document.write( " x = 13 - 8 - 6z + 11z\r\n" );
document.write( " x = 5 + 5z\r\n" );
document.write( "\r\n" );
document.write( "Substitute the expressions for x and y in the top equation:\r\n" );
document.write( "\r\n" );
document.write( " w = 10 + x - 3y + 5z\r\n" );
document.write( " w = 10 + (5 + 5z) - 3(4 + 3z) + 5z\r\n" );
document.write( " w = 10 + 5 + 5z - 12 - 9z + 5z\r\n" );
document.write( " w = 3 + z \r\n" );
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document.write( "So the solution is\r\n" );
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document.write( " (w,x,y,z) = (3+z, 5+5z, 4+3z, z)\r\n" );
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document.write( "Some teachers will tell you to use a different letter than z for z, \r\n" );
document.write( "such as \"a\", or \"k\". If they use \"a\" the solution would be:\r\n" );
document.write( "\r\n" );
document.write( " (w,x,y,z) = (3+a, 5+5a, 4+3a, a)\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" );
document.write( "![\"\"](\"/images/stars/stars-12.gif\")
