document.write( "Question 1182189: Solve using Gaussian Elimination Form:
\n" ); document.write( "x + 2y - z = 2
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\n" ); document.write( "2x + 4y - z + 2w = 7 \n" ); document.write( "

Algebra.Com's Answer #812513 by greenestamps(13200) $\"\"$ $\"About$

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document.write( "Original matrix:\r\n" );
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document.write( "  [ 1  2 -1  0   2 ]\r\n" );
document.write( "  [ 1  2  0  2   5 ]\r\n" );
document.write( "  [ 2  4 -1  2   7 ]\r\n" );
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document.write( "Note we start with four unknowns and only three equations, so we are going to need at least one parameter to describe the solution set; and of course describing the solution set using parameters means there are an infinite number of solutions.\r\n" );
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document.write( "We have a 1 in row 1 column 1, so leave row 1 alone and use the 1 in row 1 column 1 to get 0's in column 1 in rows 2 and 3:\r\n" );
document.write( "Replace row 2 with row 2 minus row 1; replace row 3 with row 3 minus 2 times row 1.\r\n" );
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document.write( "  [ 1  2 -1  0   2 ]\r\n" );
document.write( "  [ 0  0  1  2   3 ]\r\n" );
document.write( "  [ 0  0  1  2   3 ]\r\n" );
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document.write( "We are stuck with the 2 in row 1 column 2; there is no way to get rid of it.\r\n" );
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document.write( "We have a 1 as the first nonzero entry in row 2, so keep row 2 and use the 1 in row 2 column 3 to get 0's in column 3 rows 1 and 3:\r\n" );
document.write( "Replace row 1 with row 1 plus row 2; replace row 3 with row 3 minus row 2.\r\n" );
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document.write( "  [ 1  2  0  2   5 ]\r\n" );
document.write( "  [ 0  0  1  2   3 ]\r\n" );
document.write( "  [ 0  0  0  0   0 ]\r\n" );
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document.write( "The last row gives us no useful information, so the system only has two independent equations.  That means we will need to describe the solution set using two parameters.\r\n" );
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document.write( "The two non-zero rows of the matrix tell us\r\n" );
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document.write( "x+2y+2w = 5\r\n" );
document.write( "z+2w = 3\r\n" );
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document.write( "Let t=w be one parameter.  Then use the second equation to get z+2w=3, z = 3-2w, and z=3-2t.\r\n" );
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document.write( "w=t; z=3-2t\r\n" );
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document.write( "Substitute those in the first equation:\r\n" );
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document.write( "x+2y+2t = 5\r\n" );
document.write( "x+2y = 5-2t\r\n" );
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document.write( "We need another parameter.  Let u=y be the second parameter.  Then\r\n" );
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document.write( "x = 5-2t-2y = 5-2t-2u.\r\n" );
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document.write( "y=u; x=5-2t-2u\r\n" );
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document.write( "The solution set is\r\n" );
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document.write( "x=5-2t-2u\r\n" );
document.write( "y=u\r\n" );
document.write( "z=3-2t\r\n" );
document.write( "w=t\r\n" );
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document.write( "Check that the three original equations are satisfied....\r\n" );
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document.write( "x + 2y - z = (5-2t-2u) + 2u - (3-2t) = 5-2t-2u+2u-3+2t = 2  yes\r\n" );
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document.write( "x + 2y + 2w = (5-2t-2u) + 2u + 2t = 5  yes\r\n" );
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document.write( "2x + 4y - z + 2w = (10-4t-4u) + 4u - (3-2t) + 2t = 10-4t-4u+4u-3+2t+2t = 7  yes\r\n" );
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