SOLUTION: How to find the value of k for which the planes intersects in a line? x+y+z=0 kx+y-2z=-6 2y+(k+2)z=k-2 Thank you.

Algebra ->  Matrices-and-determiminant -> SOLUTION: How to find the value of k for which the planes intersects in a line? x+y+z=0 kx+y-2z=-6 2y+(k+2)z=k-2 Thank you.       Log On


   

Question 1116629: How to find the value of k for which the planes intersects in a line?
x+y+z=0
kx+y-2z=-6
2y+(k+2)z=k-2
Thank you.
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
How to find the value of k for which the planes intersects in a line?
x+y+z=0
kx+y-2z=-6
2y+(k+2)z=k-2
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There are 3 planes.
Each pair intersects in a line --> 3 lines. 3 planes always generate 3 lines, unless 2 or more of the planes are parallel.
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Clarify what you're asking.
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PS 3-space is a LOT more difficult than 1.5 times 2-space. But it's where we live.

Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.
The fact that the three planes intersect in a line MEANS that the given system of equation has infinitely many solutions.


It may happen if and only if the determinant of the coefficient matrix of the system is equal to zero:


det %28matrix%283%2C3%2C+1%2C1%2C1%2C++k%2C1%2C-2%2C++0%2C2%2C%28k%2B2%29%29%29 = 0.


It is your equation to find "k".


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".



Another way is to reduce the matrix of coefficients to its diagonal form using elementary operations on rows of the matrix 

(the Gauss elimination algorithm), and then to require that all the coefficients of the last row are equal to zero.



When you get the number of "k", you must check that the obtained (or the original) system of equations really has infinitely many solutions

(to exclude the existing alternative that the system has no solutions at all !)