Find all values of K for the system of linear equation?

kx-ky=k

y+ (1-k)z=0

kx-4y+(k-1)z=3

Find all k for which the system has infinite many solutions

And all k for system has no solution

1 Answer

Relevance
  • 7 years ago
    Favorite Answer

    k = 0 is clearly a value for which the system has infinitely many solutions.

    Suppose k ≠ 0.

    --

    k -k 0 k

    0 1 1-k 0

    k -4 k-1 3

    --

    1 -1 0 1

    0 1 1-k 0

    k -4 k-1 3

    --

    1 -1 0 1

    0 1 1-k 0

    0 k-4 k-1 3-k

    --

    1 0 1-k 1

    0 1 1-k 0

    0 0 k^2-4k+3 3-k

    --

    1 0 1-k 1

    0 1 1-k 0

    0 0 (k-1)(k-3) 3-k

    --

    1 0 1-k 1

    0 1 1-k 0

    0 0 (1-k)(3-k) 3-k

    If k = 3, there are infinitely many solutions.

    If k = 1, there is no solution.

    Suppose k is neither.

    --

    1 0 1-k 1

    0 1 1-k 0

    0 0 1-k 1

    --

    1 0 0 0

    0 1 0 -1

    0 0 1-k 1

    There is no way for there to be infinite solutions.

    We still get no solutions if k = 1.

    Hence, we get infinite solutions if k ∈ {0,3} and we get no solution if k = 1.

Still have questions? Get your answers by asking now.