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# 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

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- PatrickLv 67 years agoFavorite 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.

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