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

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.