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Question

For what value of k do the following system of equations possess a non-trivial solution over the set of rationals.
x+ky+3z=0,3x+ky2z=0,2x+3y4z=0
Also, find all the solutions of the system.

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Solution

For non-trivial solution D=0
or ∣ ∣1k33k2234∣ ∣=0
Apply R23R1,R32R1
=∣ ∣1k302k11032k10∣ ∣=0
or 20k+11(32k)=0
or 332k=0
k=33/2
Putting the value of k, the equations are
x+332y+3z=0 .......(1)
3x+332y2z=0 .......(2)
2x+3y4z=0 ........(3)
Mutliply (1) by (3) and subtract from (2) and similarly multiply (1) by 2 and subtract from (3). Thus we get the equivalent system of equations as
x+332y+3z=0
33y11z=0
30y10z=0
From any of the last two, we get 3y=z
or y1=z3=λ, say.
y=λ,z=3λ.
From (1), we get
x+332y+3z=0
x+332λ9λ=0x=152λ
x:y:z=152:1:3

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