document.write( "Question 80796: Solving systems of equations using matrices:\r
\n" ); document.write( "\n" ); document.write( "4x + y = 7
\n" ); document.write( "-5x - y + z - w = 1
\n" ); document.write( "z+w = 7\r
\n" ); document.write( "\n" ); document.write( "I have z = 7 and know that w is any real number so far but am stuck solving for x and y. Help please. \n" ); document.write( "
Algebra.Com's Answer #57961 by Edwin McCravy(20060) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "Write the equations this way\r\n" );
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document.write( " 4x +  y         = 7\r\n" );
document.write( "-5x -  y + z - w = 1\r\n" );
document.write( "           z + w = 7 \r\n" );
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document.write( "Then put 1's in front of the letters\r\n" );
document.write( "without coefficients showing, and +0\r\n" );
document.write( "times the missing letters:\r\n" );
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document.write( " 4x + 1y + 0z + 0w = 7\r\n" );
document.write( "-5x - 1y + 1z - 1w = 1\r\n" );
document.write( " 0x + 0y + 1z + 1w = 7 \r\n" );
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document.write( "Erase all the letters and\r\n" );
document.write( "+ signs, move the minus\r\n" );
document.write( "signs next to the coefficients\r\n" );
document.write( "as negative signs, draw a line\r\n" );
document.write( "where the equal signs are, and\r\n" );
document.write( "put brackets around:\r\n" );
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document.write( "[ 4  1  0  0 | 7]\r\n" );
document.write( "[-5 -1  1 -1 | 1]\r\n" );
document.write( "[ 0  0  1  1 | 7]\r\n" );
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document.write( "Get a 0 where the -5 is by\r\n" );
document.write( "adding 5 times the 1st row to\r\n" );
document.write( "4 times the second row, and\r\n" );
document.write( "restore the 1st row\r\n" );
document.write( "\r\n" );
document.write( "5[ 4  1  0  0 | 7]\r\n" );
document.write( "4[-5 -1  1 -1 | 1]\r\n" );
document.write( " [ 0  0  1  1 | 7]\r\n" );
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document.write( " [ 4  1  0  0 |  7] \r\n" );
document.write( " [ 0  1  4 -4 | 39]\r\n" );
document.write( " [ 0  0  1  1 |  7]\r\n" );
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document.write( "Get a 0 where the 1 in row \r\n" );
document.write( "1 is by adding -1 times row\r\n" );
document.write( "2 to 1 times row 1, and\r\n" );
document.write( "restore the 2nd row\r\n" );
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document.write( " 1 [ 4  1  0  0 |  7] \r\n" );
document.write( "-1 [ 0  1  4 -4 | 39]\r\n" );
document.write( "   [ 0  0  1  1 |  7] \r\n" );
document.write( "\r\n" );
document.write( "   [ 4  0 -4  4 |-32] \r\n" );
document.write( "   [ 0  1  4 -4 | 39]\r\n" );
document.write( "   [ 0  0  1  1 |  7] \r\n" );
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document.write( "Get a 0 where the -4 is in\r\n" );
document.write( "the 1st row by multiplying\r\n" );
document.write( "the 3rd row by 4 and adding\r\n" );
document.write( "it to 1 times the 1st row,\r\n" );
document.write( "and restore the 3rd row\r\n" );
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document.write( "  1[ 4  0 -4  4 |-32] \r\n" );
document.write( "   [ 0  1  4 -4 | 39]\r\n" );
document.write( "  4[ 0  0  1  1 |  7]\r\n" );
document.write( "\r\n" );
document.write( "   [ 4  0  0  8 | -4] \r\n" );
document.write( "   [ 0  1  4 -4 | 39]\r\n" );
document.write( "   [ 0  0  1  1 |  7]\r\n" );
document.write( "\r\n" );
document.write( "Get a 0 where the 4 is in \r\n" );
document.write( "the 2nd row by multiplying\r\n" );
document.write( "the 3rd row by -4 and\r\n" );
document.write( "adding it to 1 times the \r\n" );
document.write( "2nd row, and restore the \r\n" );
document.write( "3rd row:\r\n" );
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document.write( "   [ 4  0  0  8 | -4] \r\n" );
document.write( "  1[ 0  1  4 -4 | 39]\r\n" );
document.write( " -4[ 0  0  1  1 |  7]\r\n" );
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document.write( "   [ 4  0  0  8 | -4] \r\n" );
document.write( "   [ 0  1  0 -8 | 11]\r\n" );
document.write( "   [ 0  0  1  1 |  7]\r\n" );
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document.write( "Get a 1 where the first 4 is\r\n" );
document.write( "in the 1st row by multiplying\r\n" );
document.write( "the 1st row through by 1/4\r\n" );
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document.write( "   [ 1  0  0  2 | -1] \r\n" );
document.write( "   [ 0  1  0 -8 | 11]\r\n" );
document.write( "   [ 0  0  1  1 |  7]\r\n" );
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document.write( "This is in row reduced\r\n" );
document.write( "echelon form.\r\n" );
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document.write( "Put the letters, plus signs,\r\n" );
document.write( "and equal signs back in:\r\n" );
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document.write( "   1x + 0y + 0z + 2w = -1\r\n" );
document.write( "   0x + 1y + 0z - 8w = 11\r\n" );
document.write( "   0x + 0y + 1z + 1w =  7\r\n" );
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document.write( "Simplify to\r\n" );
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document.write( "    x           + 2w = -1\r\n" );
document.write( "         y      - 8w = 11\r\n" );
document.write( "              z +  w =    7 \r\n" );
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document.write( "Eliminate the spaces (close it up:\r\n" );
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document.write( "x + 2w = -1\r\n" );
document.write( "y - 8w = 11\r\n" );
document.write( "z +  w =  7\r\n" );
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document.write( "Solve for x, y and z\r\n" );
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document.write( "     x = -1 - 2w\r\n" );
document.write( "     y = 11 + 8w\r\n" );
document.write( "     z = 7 - w\r\n" );
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document.write( "w can be any number, say c, then\r\n" );
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document.write( "     x = -1 - 2c\r\n" );
document.write( "     y = 11 + 8c\r\n" );
document.write( "     z = 7 - c\r\n" );
document.write( "     w = c\r\n" );
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document.write( "So the solution is\r\n" );
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document.write( "(x, y, z, w) = (-1-c, 11+8c, 7-c, c) \r\n" );
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document.write( "It does turn out that w can be any number,\r\n" );
document.write( "but as you see, z is not necessarily 7, \r\n" );
document.write( "unless c happens to be chosen to be 0.\r\n" );
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document.write( "Edwin
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