document.write( "Question 1015937: find k if
\n" ); document.write( "kx+y+z=4-k
\n" ); document.write( "x+ky+z=2+k
\n" ); document.write( "x+y+kz=3
\n" ); document.write( "for
\n" ); document.write( "1-unique solu
\n" ); document.write( "2-infinite solu
\n" ); document.write( "3-no solu \n" ); document.write( "

Algebra.Com's Answer #632361 by Fombitz(32388) $\"\"$ $\"About$

You can put this solution on YOUR website!
Look at the coefficient matrix,
\n" ); document.write( " $\"A=%28matrix%283%2C3%2C%0D%0Ak%2C1%2C1%2C%0D%0A1%2Ck%2C1%2C%0D%0A1%2C1%2Ck%29%29\"$
\n" ); document.write( "Find the determinant,
\n" ); document.write( " $\"D=k%28k%5E2-1%29%2B1%281-k%29%2B1%281-k%29\"$
\n" ); document.write( " $\"D=k%5E3-k%2B2-2k\"$
\n" ); document.write( " $\"D=k%5E3-3k%2B2\"$
\n" ); document.write( " $\"D=%28k-1%29%5E2%28k%2B2%29\"$
\n" ); document.write( "So $\"k=1\"$ and $\"k=-2\"$ are solutions.
\n" ); document.write( "When $\"k=1\"$ ,
\n" ); document.write( " $\"A=%28matrix%283%2C3%2C%0D%0A1%2C1%2C1%2C%0D%0A1%2C1%2C1%2C%0D%0A1%2C1%2C1%29%29\"$
\n" ); document.write( "and the right hand side becomes,
\n" ); document.write( " $\"b=%28matrix%283%2C1%2C3%2C3%2C3%29%29\"$
\n" ); document.write( "So this system is dependent and has infinitely many solutions.
\n" ); document.write( ".
\n" ); document.write( ".
\n" ); document.write( "When $\"k=-2\"$ ,
\n" ); document.write( " $\"A=%28matrix%283%2C3%2C%0D%0A-2%2C1%2C1%2C%0D%0A1%2C-2%2C1%2C%0D%0A1%2C1%2C-2%29%29\"$
\n" ); document.write( "and the right hand side becomes,
\n" ); document.write( " $\"b=%28matrix%283%2C1%2C6%2C0%2C3%29%29\"$ .
\n" ); document.write( "In this case, it is an inconsistent system because $\"abs%28A%5Bx%5D%29%3C%3E0\"$ ,
\n" ); document.write( " $\"abs%28A%5By%5D%29%3C%3E0\"$ , and $\"abs%28A%5Bz%5D%29%3C%3E0\"$ ,
\n" ); document.write( ".
\n" ); document.write( ".
\n" ); document.write( "For all other values of k, you would have a unique solution. \n" ); document.write( "

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