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 Consistency parameter (Posted on 2019-01-30)
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 No Rating

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 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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