Answer:
The system is inconsistent.
Step-by-step explanation:
The system is
![x_1+5x_2+4x_3+3x_4+9x_5=18\\2x_1+6x_2+4x_3+6x_4+6x_5=28\\3x_1+7x_2+4x_3+10x_4+x_5=41\\4x_1+6x_2+2x_3+7x_4+4x_5=29](https://tex.z-dn.net/?f=x_1%2B5x_2%2B4x_3%2B3x_4%2B9x_5%3D18%5C%5C2x_1%2B6x_2%2B4x_3%2B6x_4%2B6x_5%3D28%5C%5C3x_1%2B7x_2%2B4x_3%2B10x_4%2Bx_5%3D41%5C%5C4x_1%2B6x_2%2B2x_3%2B7x_4%2B4x_5%3D29)
The associated matrix to the system is
![\left[\begin{array}{cccccc}1&5&4&3&9&18\\2&6&4&6&6&28\\3&7&4&10&1&41\\4&6&2&7&4&29\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccccc%7D1%265%264%263%269%2618%5C%5C2%266%264%266%266%2628%5C%5C3%267%264%2610%261%2641%5C%5C4%266%262%267%264%2629%5Cend%7Barray%7D%5Cright%5D)
Now we use row operations to find the echelon form of the matrix:
1. We substract to row 2, two times the row 1.
We substract to row 3, three times the row 1.
We substract to row 4, four times the row 1 and obtain the matrix
![\left[\begin{array}{cccccc}1&5&4&3&9&18\\0&-4&-4&0&-12&-8\\0&-8&-8&1&-26&-13\\0&-14&-14&-5&-32&-43\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccccc%7D1%265%264%263%269%2618%5C%5C0%26-4%26-4%260%26-12%26-8%5C%5C0%26-8%26-8%261%26-26%26-13%5C%5C0%26-14%26-14%26-5%26-32%26-43%5Cend%7Barray%7D%5Cright%5D)
2. We multiply the second row of the preview step by -1/4. We obtain the matrix
![\left[\begin{array}{cccccc}1&5&4&3&9&18\\0&1&1&0&3&2\\0&-8&-8&1&-26&-13\\0&-14&-14&-5&-32&-43\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccccc%7D1%265%264%263%269%2618%5C%5C0%261%261%260%263%262%5C%5C0%26-8%26-8%261%26-26%26-13%5C%5C0%26-14%26-14%26-5%26-32%26-43%5Cend%7Barray%7D%5Cright%5D)
3.
We add to row 3, eight times the row 2.
We add to row 4, fourtheen times the row 2 and obtain the matrix
![\left[\begin{array}{cccccc}1&5&4&3&9&18\\0&1&1&0&3&2\\0&0&0&1&-2&3\\0&0&0&-5&10&-155\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccccc%7D1%265%264%263%269%2618%5C%5C0%261%261%260%263%262%5C%5C0%260%260%261%26-2%263%5C%5C0%260%260%26-5%2610%26-155%5Cend%7Barray%7D%5Cright%5D)
4. We add to row 4 of the preview step, five times the row 3 and obtain the matrix
![\left[\begin{array}{cccccc}1&5&4&3&9&18\\0&1&1&0&3&2\\0&0&0&1&-2&3\\0&0&0&0&0&-140\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccccc%7D1%265%264%263%269%2618%5C%5C0%261%261%260%263%262%5C%5C0%260%260%261%26-2%263%5C%5C0%260%260%260%260%26-140%5Cend%7Barray%7D%5Cright%5D)
Using backward substitution we have that
, then
and this is absurd. Then The system is inconsistent.