document.write( "Question 1187273: 8. Determine the values of k such that the system of linear equations is consistent
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Algebra.Com's Answer #818223 by ikleyn(52788) $\"\"$ $\"About$

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document.write( "Consider the second and the third equations of the system.\r\n" );
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document.write( "Their left sides are almost identical: they differ only by signs of their terms.\r\n" );
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document.write( "THEREFORE, the necessary condition for the system to be consistent is that their RIGHT sides should have the opposite signs.\r\n" );
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document.write( "So, the value of \"k\" must be -2:  k = -2.\r\n" );
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document.write( "Then the second and the third equation are, actually, IDENTICAL and represent two equivalent equations.\r\n" );
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document.write( "In other words, you can think that the second and the third equation represent the same equation \r\n" );
document.write( "and not two different equations.\r\n" );
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document.write( "Since these two equations include only two variables, one of them can take arbitrary values \r\n" );
document.write( "and the other variable can be expressed / calculated as the function of the other.\r\n" );
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document.write( "Next, having y and z with any assigned values, we always can fit the third variable x,\r\n" );
document.write( "so the system is always consistent and have (infinitely many) solutions.\r\n" );
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document.write( "Thus the condition k = -2 is not only necessary for consistency of the system - it is a SUFFICIENT condition, too.\r\n" );
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document.write( "ANSWER.  k = -2 is the necessary and sufficient condition for the given system to be consistent.\r\n" );
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