document.write( "Question 1042309: given the homogenous system x+2y-z=0,3x-3y+2z=0,-x-11y+kz=0 . a)find the value of k for the system has a non trival solution .b)solve for the value of k \n" ); document.write( "

Algebra.Com's Answer #657354 by ikleyn(52778) $\"\"$ $\"About$

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\n" ); document.write( "given the homogeneous system x+2y-z=0,3x-3y+2z=0,-x-11y+kz=0.
\n" ); document.write( "a) find the value of k for the system has a non trival solution .
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document.write( " x +  2y -  z = 0,   (1)\r\n" );
document.write( "3x -  3y + 2z = 0,   (2)\r\n" );
document.write( "-x - 11y + kz = 0.   (3)\r\n" );
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document.write( "I know two ways to solve it.\r\n" );
document.write( "One way is to calculate the determinant of the 3x3 matrix on the left and to equalize it to zero, and then find \r\n" );
document.write( "the value of \"k\" from this condition.\r\n" );
document.write( "This way is quite boring and is not so much educational.\r\n" );
document.write( "So I choose another way.\r\n" );
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document.write( "Let us multiply the equation (1) by 4 (both sides) and write the modified equation (1) along with the equation (2). You will have\r\n" );
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document.write( "4x + 8y - 4z = 0,    (1')\r\n" );
document.write( "3x - 3y + 2z = 0.    (2')\r\n" );
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document.write( "Now distract equation (1') from (2'). You will get\r\n" );
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document.write( "-x - 11y + 6z = 0.   (4)\r\n" );
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document.write( "Now I will write the equation (3) right under the equation (4).\r\n" );
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document.write( "-x - 11y + kz = 0.   (3')\r\n" );
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document.write( "Now compare (3') and (4), and you momentarily see that the equations (4) and (3') are identical at k = 6, and are different at any other value of k.\r\n" );
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document.write( "In other words, at k = 6 the system (1), (2) and (3) has linearly dependent equations.\r\n" );
document.write( "It means that at k = 6 the system is degenerated (singular) and has non-trivial solution.\r\n" );
document.write( "And we can easily find it by taking, for example, z = 1 (actually, any arbitrary value) and then find x and y from (1) and (2)\r\n" );
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document.write( " x +  2y - 1 = 0,   (1'')    ( I substituted z=1 into (1) and (2) ! )\r\n" );
document.write( "3x -  3y + 2 = 0.   (2'')\r\n" );
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document.write( "It is the same as\r\n" );
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document.write( " x + 2y =  1,       (5)\r\n" );
document.write( "3x - 3y = -2.       (6)\r\n" );
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document.write( "Now express x = 1-2y from (5) and substitute it into (6). You will get\r\n" );
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document.write( "3(1-2y) - 3y = -2  --->  3 - 6y - 3y = -2  --->  -9y = -5  --->  y = .\r\n" );
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document.write( "Then x = 1-2y = 1 -  = .\r\n" );
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document.write( "Thus, a non-trivial solution at k = 6 is  x = , y =   and  z = 1.\r\n" );
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document.write( "Or, if you love integer numbers more than rational, take \r\n" );
document.write( "x = -1, y = 5 and z = 9.\r\n" );
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document.write( "Answer.  The singular value for k is 6 and the non-trivial solution in this case is (x,y,z) = (-1, 5, 9).\r\n" );
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