document.write( "Question 1007154: Solve the homogeneous system of equations. (If the system is dependent, enter a general solution in terms of c.)
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Algebra.Com's Answer #623152 by Edwin McCravy(20059)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Get a 0 where the 3 is by multiplying row 1 by -3,\r\n" );
document.write( "and row 2 by 5, adding them and replacing row 2 by\r\n" );
document.write( "the result.  That's written -3R1+5R2->R2\r\n" );
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document.write( "        -15   6    9  |  0\r\n" );
document.write( "         15  -5  -20  |  0\r\n" );
document.write( "      ---------------------\r\n" );
document.write( "          0   1  -11  |  0\r\n" );
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document.write( "  \r\n" );
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document.write( "Get a 0 where the 4 is by multiplying row 1 by -4,\r\n" );
document.write( "and row 3 by 5, adding them and replacing row 3 by\r\n" );
document.write( "the result.  That's written -4R1+5R3->R3\r\n" );
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document.write( "        -20   8   12  |  0\r\n" );
document.write( "         20  -5  -20  |  0\r\n" );
document.write( "      ---------------------\r\n" );
document.write( "          0   1  -11  |  0\r\n" );
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document.write( "Get a 0 where the 1 on the bottom row is by \r\n" );
document.write( "multiplying row 2 by -1, and row 3 by 1, \r\n" );
document.write( "adding them and replacing row 3 by\r\n" );
document.write( "the result.  That's written -R2+R3->R3\r\n" );
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document.write( "          0   1  -11  |  0\r\n" );
document.write( "          0  -1   11  |  0\r\n" );
document.write( "      ---------------------\r\n" );
document.write( "          0   0    0  |  0\r\n" );
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document.write( "Convert this back to a system of equations:\r\n" );
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document.write( "Which when simplified is:\r\n" );
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document.write( "z can be assigned any value, some people set z equal\r\n" );
document.write( "to some other letter, and other just leave it as z.\r\n" );
document.write( "I'll just leave it as z.\r\n" );
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document.write( "Solve the second equation for y\r\n" );
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document.write( "-y - 11z = 0\r\n" );
document.write( "      -y = -11z\r\n" );
document.write( "       y = 11z\r\n" );
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document.write( "Substitute 11z for y in the first equation:\r\n" );
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document.write( "    5x - 2y - 3z = 0\r\n" );
document.write( "5x - 2(11z) - 3z = 0\r\n" );
document.write( "   5x - 22z - 3z = 0\r\n" );
document.write( "        5x - 25z = 0\r\n" );
document.write( "              5x = 25z\r\n" );
document.write( "               x = 5z\r\n" );
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document.write( "We write the solution as\r\n" );
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document.write( "(x, y, z) = (5z, 11z, z)\r\n" );
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document.write( "Some books and teachers require students to use a different letter\r\n" );
document.write( "for z, such as\r\n" );
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document.write( "(x, y, z) = (5a, 11a, a) or (5k, 11k, k), etc.\r\n" );
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document.write( "This system is dependent and has infinitely many solutions. Substitute\r\n" );
document.write( "different numbers for z (or for a or k) and get different solutions.  \r\n" );
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document.write( "Edwin
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