Show that there do not exist scalars c1, c2, and c3 such that c1(1, 0, 1, 0) + c2(1, 0, -2, 1) + c3(2, 0, 1, 2) = (1, -2, 2, 3)
Aloiza [94]
Write the system in augmented-matrix form:
![\iff\left[\begin{array}{ccc|c}1&1&2&1\\0&0&0&-2\\1&-2&1&2\\0&1&2&3\end{array}\right]](https://tex.z-dn.net/?f=%5Ciff%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Cc%7D1%261%262%261%5C%5C0%260%260%26-2%5C%5C1%26-2%261%262%5C%5C0%261%262%263%5Cend%7Barray%7D%5Cright%5D)
Row reduce this matrix:
![\left[\begin{array}{ccc|c}1&1&2&1\\0&0&0&-2\\0&-3&-1&1\\0&1&2&3\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Cc%7D1%261%262%261%5C%5C0%260%260%26-2%5C%5C0%26-3%26-1%261%5C%5C0%261%262%263%5Cend%7Barray%7D%5Cright%5D)
![\left[\begin{array}{ccc|c}1&1&2&1\\0&0&0&-2\\0&0&5&10\\0&1&2&3\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Cc%7D1%261%262%261%5C%5C0%260%260%26-2%5C%5C0%260%265%2610%5C%5C0%261%262%263%5Cend%7Barray%7D%5Cright%5D)
![\left[\begin{array}{ccc|c}1&1&2&1\\0&0&0&-2\\0&0&1&2\\0&1&2&3\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Cc%7D1%261%262%261%5C%5C0%260%260%26-2%5C%5C0%260%261%262%5C%5C0%261%262%263%5Cend%7Barray%7D%5Cright%5D)
![\left[\begin{array}{ccc|c}1&1&2&1\\0&0&0&-2\\0&0&1&2\\0&1&0&-1\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Cc%7D1%261%262%261%5C%5C0%260%260%26-2%5C%5C0%260%261%262%5C%5C0%261%260%26-1%5Cend%7Barray%7D%5Cright%5D)
- Add -2(row 3) and -1(row 4) to row 1:
![\left[\begin{array}{ccc|c}1&0&0&-2\\0&0&0&-2\\0&0&1&2\\0&1&0&-1\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Cc%7D1%260%260%26-2%5C%5C0%260%260%26-2%5C%5C0%260%261%262%5C%5C0%261%260%26-1%5Cend%7Barray%7D%5Cright%5D)
This matrix tells us that 
, 
, and 
, but clearly 
, so there is no solution.
A = 1/2 (12x² + 8x + 24x + 16)
A = 1/2 (12x² + 32x + 16)
A = 6x² + 16x + 8
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Answer: 6x² + 16x + 8
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Time spent with paul : 1/8 h
Total time spent chatting with friends: 2/3h+1/4h=11/12 h
The fraction of the time she spent with paul is (time spent with paul)/(total time spent chatting)
hence the fraction spent with paul is (1/8)/(11/12)=3/11
Notice the picture below
so.. whatever the parabola y= -3x²+k is, will pass over (3,-2) and (-3,-2)
so.. .let us pick say hmmm 3,-2
thus
solve for "k"
reflection transformation i believe. Correct me if im wrong im not that smart lol.
