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Consistency parameter (Posted on 2019-01-30) Difficulty: 2 of 5
For what value K is the following system consistent?
(i) p+q=6
(ii) kp+q=18
(iii) p+kq=30

Source: Terry Stickel, Challenging Math Problems, 2015

No Solution Yet Submitted by Ady TZIDON    
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Solution Determinant solution Comment 3 of 3 |
For the system to be consistent it must be overdetermined, which implies the vectors representing the equations; (1,1,6), (k,1,18), (1,k,30); must be linearly dependent.  Which then implies the determinant formed from the three vectors must equal zero.
| 1 1 6  |
| k 1 18 |
| 1 k 30 |
The determinant evaluates to 6k^2-48k+42=0.  This equation has two roots k=1 and k=7.  Checking each root shows that only k=7 actually makes the system consistent.

  Posted by Brian Smith on 2019-01-30 12:36:41
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