# For what value of k does the system have a non-trivial solution? kx + 3y - 4z = 0 x + y = 0 3x + ky + 2z = 0

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sciencesolve | Certified Educator

Notice that the right column of terms consists of zeroes, hence, the system is called homogenous.

If the system has an unique solution, this solution is trivial solution, meaning that `x=0,y=0,z=0` .

The system has an unique solution if the determinant of matrix, `Delta` , of system has a value different form zero.

Since the problem provides the information that the system has non-trivial solution, hence determinant of matrix of system is zero such that:

`Delta = [[k,3,-4],[1,1,0],[3,k,2]]`

`Delta = 2k - 4k + 0 + 12 - 0 - 6`

But `Delta = 0 => 2k - 4k + 0 + 12 - 0 - 6 = 0`

-`2k + 6 = 0 => -2k = -6 => k = 3`

**Hence, evaluating the value of k for the system to have non-trivial solution yields k = 3.**