For what values of `k ,` the following system of equations possesses a nontrival solution over the set of rationals:`x+k y+3z=0,3x+k y-2z=0,2x+3y-4z=0.` Also find the solution for this value of `kdot`

Explanation

The system
x+ky+ 3z=0
3x+ky-2z=0
2x+3y-4z=0
as nontrivial solution (i.e., nonzero solution ) if the determinant if coefficients of x,y and z is zero , Here,
`Delta=|{:(1,,k,,3),(3,,k,,-2),(2,,3,,-4):}|=0`
`" or " 2k-33=0 " ok " k= 33//2`
then the equations become
`2x+33y+6z=0`
`6x+33y-4z=0`
`2x+3y-4z=0`
Eliminating x we get from (2) and (4)
30y+ 10z=0 i.e., 3y+z=0
`" Let "y=lambda in R . " then " z=- 3lambda " and so "`
`2x=-33lambda + 18lambda =- 15lambda`
`:. x=-(15)/(2)lambda , y=lambda z=-3lambda , lambda in R`