document.write( "Question 480456: The following equations have infinitely many solutions
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\n" ); document.write( " x + 3y =3
\n" ); document.write( " -x + 7y + 5z =18
\n" ); document.write( " -2x + 6y + 6z =24\r
\n" ); document.write( "\n" ); document.write( "Give the right hand side of the vector form of the general solution, using a parameter such as s or t. (Any lowercase letter will do as a parameter, so long as it is not x, y or z.)\r
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\n" ); document.write( "\n" ); document.write( "FOR EXAMPLE, for the equations
\n" ); document.write( "x = y + 1, y = z + 1, z = x − 2
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\n" ); document.write( "[x, y, z] = [0, −1, −2] + t [1, 1, 1]
\n" ); document.write( "and you would enter the right hand side of this equation in the space provided:
\n" ); document.write( "[0, −1, −2] + t * [1, 1, 1]
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Algebra.Com's Answer #329062 by Edwin McCravy(20060) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"system%28x+%2B+3y+=3%2C%0D%0A-x+%2B+7y+%2B+5z+=18%2C%0D%0A-2x+%2B+6y+%2B+6z+=24%29\"$
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document.write( "Put in +0z in the first equation and write -x as -1x\r\n" );
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document.write( "Write as an augmented matrix by erasing the letters and \r\n" );
document.write( "replacing the equal signs by \"|\", and put the whole thing\r\n" );
document.write( "in parentheses (some people use brackets):\r\n" );
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document.write( "Now we get it in echelon form:\r\n" );
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document.write( "Get a 0 where the -1 is by adding the top row to the\r\n" );
document.write( "middle row:\r\n" );
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document.write( "Get a 0 where the -2 is by multiplying the top row by 2 and\r\n" );
document.write( "adding it to the bottom row: \r\n" );
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document.write( "And since that can be divided through by 6, we do so and get  \r\n" );
document.write( "and replace the bottom row by that:\r\n" );
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document.write( "Get a 1 where the 10 is by dividing the 2nd row through by 10\r\n" );
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document.write( "---\r\n" );
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document.write( "Get a 0 where the 2 is by multiplying the middle row by -2 and\r\n" );
document.write( "adding it to the bottom row: \r\n" );
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document.write( "And since that can be divided through by .8, we do so and get  \r\n" );
document.write( "and replace the bottom row by that:\r\n" );
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document.write( "Now it is in echelon form.  Write it as a system again:\r\n" );
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document.write( "or\r\n" );
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document.write( "Oh! Oh!  Look at that bottom equation:\r\n" );
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document.write( "0z = 1\r\n" );
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document.write( "There is no value of z that would make that true since \r\n" );
document.write( "zero times any number gives 0, and cannot give 1, so there\r\n" );
document.write( "are no solutions.  So what is stated at the top, that it\r\n" );
document.write( "has infinitely many solutions, is incorrect.  It may have \r\n" );
document.write( "been copied wrong, or there was a typo, but as it is \r\n" );
document.write( "given above, there are no solutions to this system. \r\n" );
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document.write( "Edwin

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