We subtract the first equation from the others, simplify expressions and receive:
⎩⎨⎧12a−8b+16c=180−18a+12b−20c=1656a−4b−4c=3
We multiply the last equation by (-2) and add to the first one. Then we get 24c=174 . This implies c=7.25. Now we multiply the last equation by 3 and add to the second. We receive −32c=174. Then c=−5.4375 . Thus, we received different values of c. Therefore,
the system has no solutions. Thus, there is no sphere passing through points (4,0,1),(10,-4,9),(-5,6,-11)and (1,2,3).
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