document.write( "Question 1116629: How to find the value of k for which the planes intersects in a line?
\n" ); document.write( "x+y+z=0
\n" ); document.write( "kx+y-2z=-6
\n" ); document.write( "2y+(k+2)z=k-2
\n" ); document.write( "Thank you. \n" ); document.write( "

Algebra.Com's Answer #731548 by ikleyn(52792) $\"\"$ $\"About$

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document.write( "The fact that the three planes intersect in a line MEANS that the given system of equation has infinitely many solutions.\r\n" );
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document.write( "It may happen if and only if the determinant of the coefficient matrix of the system is equal to zero:\r\n" );
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document.write( "det  = 0.\r\n" );
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document.write( "It is your equation to find \"k\".\r\n" );
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document.write( "So, calculate the determinant of the matrix as a function of \"k\"; equate it to zero and from this equation find the value/the values of \"k\".\r\n" );
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document.write( "Another way is to reduce the matrix of coefficients to its diagonal form using elementary operations on rows of the matrix \r\n" );
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document.write( "(the Gauss elimination algorithm), and then to require that all the coefficients of the last row are equal to zero.\r\n" );
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document.write( "When you get the number of \"k\", you must check that the obtained (or the original) system of equations really has infinitely many solutions\r\n" );
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document.write( "(to exclude the existing alternative that the system has no solutions at all !)\r\n" );
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