document.write( "Question 1013631: 2x-y+4z+t=-2
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Algebra.Com's Answer #629926 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "This comes out with very terrible answers, so I'll bet anything\r\n" );
document.write( "you copied something wrong, like a number or a sign.  So you'll\r\n" );
document.write( "probably have to do it over after correcting your error of \r\n" );
document.write( "copying, because the tiniest error in a problem like this will\r\n" );
document.write( "make the answers come out terrible.  But here it is exactly\r\n" );
document.write( "as you copied it.\r\n" );
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document.write( "Line up the terms with like letters under like letters,\r\n" );
document.write( "like this: \r\n" );
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document.write( "Put in all the understood zero terms for missing letters\r\n" );
document.write( "and understood 1 coefficients:\r\n" );
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document.write( "Move the - signs close to the terms, erase the + signs\r\n" );
document.write( "and all the letters and put a vertical line where the\r\n" );
document.write( "= signs are.  Then enclose the whole thing in parentheses.\r\n" );
document.write( "This is called an augmented matrix, augmented because of\r\n" );
document.write( "the vertical line:\r\n" );
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document.write( "It's easier if you have a 1 as the upper left element.\r\n" );
document.write( "So we switch the 1st and 3rd rows.  We write that operation as\r\n" );
document.write( "R1<->R3\r\n" );
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document.write( "To get a 0 where the 3 is in the 2nd row, 1st column, multiply\r\n" );
document.write( "R1 by -3 and add R2 to it:\r\n" );
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document.write( "            -3  -6   0  -6  |  30\r\n" );
document.write( "             3   2   0  -1  |  -3\r\n" );
document.write( "           ------------------------\r\n" );
document.write( "             0  -4   0  -7  | -33\r\n" );
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document.write( "Now replace R2 by that.  This operation is written as\r\n" );
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document.write( "-3R1+R2->R2\r\n" );
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document.write( "--\r\n" );
document.write( "To get a 0 where the 2 is in the 3rd row, 1st column, multiply\r\n" );
document.write( "R1 by -2 and add R3 to it:\r\n" );
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document.write( "            -2  -4   0  -4  | -20\r\n" );
document.write( "             2  -1   4   1  |  -2\r\n" );
document.write( "           ------------------------\r\n" );
document.write( "             0  -5   4  -3  | -22\r\n" );
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document.write( "Now replace R3 by that.  This operation is written as\r\n" );
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document.write( "-2R1+R2->R2\r\n" );
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document.write( "--\r\n" );
document.write( "To get a 0 where the 1 is in the 4th row, 1st column, multiply\r\n" );
document.write( "R1 by -1 and add R4 to it:\r\n" );
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document.write( "            -1  -2   0  -2  | -10\r\n" );
document.write( "             1   1   2   0  |  -2\r\n" );
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document.write( "             0  -1   2  -2  | -12\r\n" );
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document.write( "Now replace R4 by that.  This operation is written as\r\n" );
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document.write( "-1R1+R4->R4\r\n" );
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document.write( "Now we work on the second column.  Things will be easier\r\n" );
document.write( "if we can get a 1 where the -4 is on the 2nd row 2nd column.\r\n" );
document.write( "So we multiply R3 by -1 and swap it with R2.\r\n" );
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document.write( "That operation is written as\r\n" );
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document.write( "-R4<->R2\r\n" );
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document.write( " \r\n" );
document.write( "-----------------------------------------\r\n" );
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document.write( "To get a 0 where the -5 is in the 3rd row, 2nd column, multiply\r\n" );
document.write( "R2 by 5 and add R3 to it:\r\n" );
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document.write( "             0   5  -10  10  |  60\r\n" );
document.write( "             0  -5    4  -3  | -22\r\n" );
document.write( "           ------------------------\r\n" );
document.write( "             0   0   -6   7  |  38\r\n" );
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document.write( "Now replace R3 by that.  This operation is written as\r\n" );
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document.write( "-5R2+R3->R3\r\n" );
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document.write( "--\r\n" );
document.write( "To get a 0 where the -4 is in the 4th row, 2nd column, multiply\r\n" );
document.write( "R2 by 4 and add R4 to it:\r\n" );
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document.write( "             0   4  -8   8  |  48\r\n" );
document.write( "             0  -4   0  -7  | -33\r\n" );
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document.write( "             0   0  -8   1  |  15\r\n" );
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document.write( "Now replace R4 by that.  This operation is written as\r\n" );
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document.write( "-4R2+R4->R4\r\n" );
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document.write( "To get a 0 where the -8 is in the 4th row, 3rd column, multiply\r\n" );
document.write( "R3 by 4, and add -3 times R4 to it:\r\n" );
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document.write( "             0   0  -24  28  | 152\r\n" );
document.write( "             0   0   24  -3  | -45\r\n" );
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document.write( "             0   0    0  25  | 107\r\n" );
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document.write( "Now replace R4 by that.  This operation is written as\r\n" );
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document.write( "-4R2-3R4->R4\r\n" );
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document.write( "Convert the augmented matrix back to a system of equations:\r\n" );
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document.write( "Get rid of the understood 0 terms and 1 coefficients:\r\n" );
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document.write( "Solve the 4th equation to find t\r\n" );
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document.write( "\"25t\"\"%22%22=%22%22\"\"107\"\r\n" );
document.write( "\"t\"\"%22%22=%22%22\"\"107%2F25\"\r\n" );
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document.write( "Substitute in the 3rd equation to find z\r\n" );
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document.write( "\"-6z%2B7t\"\"%22%22=%22%22\"\"38\"\r\n" );
document.write( "\"-6z%2B7%28107%2F25%29\"\"%22%22=%22%22\"\"38\"\r\n" );
document.write( "\"-6z%2B749%2F25\"\"%22%22=%22%22\"\"38\"\r\n" );
document.write( "Clear of fractions by multiplying through by 25\r\n" );
document.write( "\"-150z%2B749\"\"%22%22=%22%22\"\"950\"\r\n" );
document.write( "\"-150z\"\"%22%22=%22%22\"\"201\"\r\n" );
document.write( "\"z\"\"%22%22=%22%22\"\"-201%2F150\"\r\n" );
document.write( "Reduce the fraction:\r\n" );
document.write( "\"z\"\"%22%22=%22%22\"\"-67%2F50\"\r\n" );
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document.write( "Substitute in the 2nd equation to find y\r\n" );
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document.write( "\"y-2z%2B2t\"\"%22%22=%22%22\"\"12\"\r\n" );
document.write( "\"y-2%28-67%2F50%29%2B2%28107%2F25%29\"\"%22%22=%22%22\"\"12\"\r\n" );
document.write( "\"y%2B134%2F50%2B214%2F25\"\"%22%22=%22%22\"\"12\"\r\n" );
document.write( "Clear of fractions by multiplying through by 50\r\n" );
document.write( "\"50y%2B134%2B428\"\"%22%22=%22%22\"\"600\"\r\n" );
document.write( "\"50y%2B562\"\"%22%22=%22%22\"\"600\"\r\n" );
document.write( "\"50y\"\"%22%22=%22%22\"\"38\"\r\n" );
document.write( "\"y\"\"%22%22=%22%22\"\"38%2F50\"\r\n" );
document.write( "Reduce the fraction:\r\n" );
document.write( "\"y\"\"%22%22=%22%22\"\"19%2F25\"\r\n" );
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document.write( "Substitute in the 1st equation to find x\r\n" );
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document.write( "\"x%2B2y%2B2t\"\"%22%22=%22%22\"\"10\"\r\n" );
document.write( "\"x%2B2%2819%2F25%29%2B2%28107%2F25%29\"\"%22%22=%22%22\"\"10\"\r\n" );
document.write( "\"x%2B38%2F25%2B214%2F25\"\"%22%22=%22%22\"\"10\"\r\n" );
document.write( "Clear of fractions by multiplying through by 25\r\n" );
document.write( "\"25x%2B38%2B214\"\"%22%22=%22%22\"\"250\"\r\n" );
document.write( "\"25x%2B252\"\"%22%22=%22%22\"\"250\"\r\n" );
document.write( "\"25x\"\"%22%22=%22%22\"\"-2\"\r\n" );
document.write( "\"x\"\"%22%22=%22%22\"\"-2%2F25\"\r\n" );
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document.write( "So the terrible solution is\r\n" );
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document.write( "\"%28matrix%281%2C7%2Cx%2C%22%2C%22%2Cy%2C%22%2C%22%2Cz%2C%22%2C%22%2Ct%29%29\"\"%22%22=%22%22\"\r\n" );
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document.write( "Now find your mistake in copying the problem and\r\n" );
document.write( "re-do it with the correction.\r\n" );
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document.write( "Edwin
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