document.write( "Question 1023772: X over 4 equals 7.5 over 2.5 and x is 12 \n" ); document.write( "
Algebra.Com's Answer #639208 by Edwin McCravy(20066) $\"\"$ $\"About$
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document.write( "3x+2y+z=8\r\n" );
document.write( "2x+y+3z=7\r\n" );
document.write( "5x-3y+4z=3\r\n" );
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document.write( "3x + 2y +  z = 8\r\n" );
document.write( "2x +  y + 3z = 7\r\n" );
document.write( "5x - 3y + 4z = 3\r\n" );
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document.write( "The augmented matrix is\r\n" );
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document.write( "[3   2   1  |  8]\r\n" );
document.write( "[2   1   3  |  7]\r\n" );
document.write( "[5  -3   4  |  3]\r\n" );
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document.write( "We need to get 0's in the three lower left\r\n" );
document.write( "positions, that is, in the positions below\r\n" );
document.write( "the upper left to lower-right diagonal:\r\n" );
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document.write( "To get a 0 where the 2 in row 2 column 1 is,\r\n" );
document.write( "we multiply row 1 temporarily by -2 and add it\r\n" );
document.write( "to 3 times row 2. This is easy to do mentally\r\n" );
document.write( "if you write -2 to the left of row 1 and 3 left\r\n" );
document.write( "of row 2:\r\n" );
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document.write( "-2[3   2   1  |  8]\r\n" );
document.write( " 3[2   1   3  |  7]\r\n" );
document.write( "  [5  -3   4  |  3]\r\n" );
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document.write( "  [3   2   1  |  8]\r\n" );
document.write( "  [0  -1   7  |  5]\r\n" );
document.write( "  [5  -3   4  |  3]\r\n" );
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document.write( "To get a 0 where the 5 in row 3 column 1 is,\r\n" );
document.write( "we multiply row 1 temporarily by -5 and add it\r\n" );
document.write( "to 3 times row 3. This is easy to do mentally\r\n" );
document.write( "if you write -5 to the left of row 1 and 3 left\r\n" );
document.write( "of row 3:\r\n" );
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document.write( "-5[3   2   1  |  8]\r\n" );
document.write( "  [0  -1   7  |  5]\r\n" );
document.write( " 3[5  -3   4  |  3]\r\n" );
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document.write( "  [3   2   1  |   8]\r\n" );
document.write( "  [0  -1   7  |   5]\r\n" );
document.write( "  [0 -19   7  | -31]\r\n" );
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document.write( "To get a 0 where the -19 in row 3 column 2 is,\r\n" );
document.write( "we multiply row 2 temporarily by 19 and add it\r\n" );
document.write( "to -1 times row 3. This is easy to do mentally\r\n" );
document.write( "if you write 19 to the left of row 2 and 1 left\r\n" );
document.write( "of row 3:\r\n" );
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document.write( "  [3   2    1  |   8]\r\n" );
document.write( "19[0  -1    7  |   5]\r\n" );
document.write( "-1[0 -19    7  | -31]\r\n" );
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document.write( "  [3   2    1  |   8]\r\n" );
document.write( "  [0  -1    7  |   5]\r\n" );
document.write( "  [0   0  126  | 126]\r\n" );
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document.write( "Now that we have the three zeros, \r\n" );
document.write( "the above is the abbreviation of this system:\r\n" );
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document.write( "   3x + 2y +   1z =   8\r\n" );
document.write( "   0x - 1y +   7z =   5\r\n" );
document.write( "   0x + 0y + 126z = 126\r\n" );
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document.write( "or\r\n" );
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document.write( "   3x +  2y +  z =   8\r\n" );
document.write( "         -y + 7z =   5\r\n" );
document.write( "            126z = 126\r\n" );
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document.write( "The bottom equation,\r\n" );
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document.write( "            126z = 126\r\n" );
document.write( "               z = 1\r\n" );
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document.write( "Substitute (1) for z in\r\n" );
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document.write( "          -y + 7z =  5\r\n" );
document.write( "        -y + 7(1) =  5\r\n" );
document.write( "           -y + 7 =  5\r\n" );
document.write( "               -y = -2  \r\n" );
document.write( "                y = 2\r\n" );
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document.write( "Substitute (1) for z and (2) for y in\r\n" );
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document.write( "     3x + 2y +  z =   8\r\n" );
document.write( "  3x + 2(2) + (1) = 8 \r\n" );
document.write( "       3x + 4 + 1 = 8\r\n" );
document.write( "           3x + 5 = 8\r\n" );
document.write( "               3x = 3\r\n" );
document.write( "                x = 1\r\n" );
document.write( "Solution:\r\n" );
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document.write( "(x, y, z) = (1, 2, 1)    \r\n" );
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document.write( "Edwin
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