document.write( "Question 1042310: what condition a,b,c must staisfy so that the system x+y=2,y+z=2,x+z=2,ax+by+cz=0 is consistent \n" ); document.write( "

Algebra.Com's Answer #657349 by ikleyn(52778) $\"\"$ $\"About$

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\n" ); document.write( "what condition a,b,c must staisfy so that the system x+y=2,y+z=2,x+z=2,ax+by+cz=0 is consistent
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document.write( "First, let us consider this system of equations\r\n" );
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document.write( "x + y = 2,       (1)\r\n" );
document.write( "y + z = 2,       (2)\r\n" );
document.write( "x + z = 2        (3)\r\n" );
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document.write( "which is a sub-system of the given one.\r\n" );
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document.write( "This system (1)-(3) has a unique solution, and we can easily find it. \r\n" );
document.write( "Add all three equations 1, (2) and (3) (both sides). You will get\r\n" );
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document.write( "2x + 2y + 2z = 6,   or   \r\n" );
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document.write( "x + y + z = 3.   (4)\r\n" );
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document.write( "Now, distract (1) from (4). You will get z = 1.\r\n" );
document.write( "     \r\n" );
document.write( "     Distract (2) from (4). You will get x = 1.\r\n" );
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document.write( "     Distract (3) from (4). You will get y = 1.\r\n" );
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document.write( "So, (x,y,z) = (1,1,1) is the unique solution of the system (1), (2) and (3).\r\n" );
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document.write( "Now we have the requirement that these values should satisfy the equation\r\n" );
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document.write( "ax + by + cz = 0.\r\n" );
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document.write( "Substitute here x=1, y=1 and z=1, and you will get\r\n" );
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document.write( "a*1 + b*1 + c*1 = 0,   or\r\n" );
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document.write( "a + b + c = 0.\r\n" );
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document.write( "So, the condition \r\n" );
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document.write( "a + b + c = 0\r\n" );
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document.write( "is the necessary and sufficient condition for the system\r\n" );
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document.write( "x+y=2, y+z=2, x+z=2, ax+by+cz=0 \r\n" );
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document.write( "to be consistent.\r\n" );
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