If the following system of equations possess a $$non-trivial$$ solution over the set of rationals $$x+y$$−$$2z=0$$, $$2x$$−$$3y+z=0andx$$−$$5y+4z=k$$ $$thenk+11000=$$

Question

If the following system of equations possess a $$non-trivial$$ solution over the set of rationals $$x+y$$−$$2z=0$$, $$2x$$−$$3y+z=0andx$$−$$5y+4z=k$$ $$thenk+11000=$$

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Augmented matrix $$AD=11-202-3101-54k$$                                                apply $$R2$$→$$R2-2R1$$, $$R3$$→$$R3-R1$$                                             ~$$11-200-5500-66k$$                                                 apply  $$R2$$→$$R25$$,  $$R3$$→$$R36$$                                              ~$$11-200-1100-11k6$$                                                    apply $$R3$$→$$R3-R2$$                                              ~$$11-200-110000k6$$ Since the system possesses a $$non-trivial$$ solution, the system is consistent. ⇒$$rankA=rankAD$$ ⇒$$k6=0$$ ⇒$$k=0$$ ∴$$k+11000=11000=0.001$$

If the following system of equations possess a non-trivial solution over the set of rationals x+y−2z=0, 2x−3y+z=0andx−5y+4z=k thenk+11000=

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