Answer:
The system is inconsistent.
Step-by-step explanation:
The system is
[tex]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[/tex]
The associated matrix to the system is
[tex]\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][/tex]
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
[tex]\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][/tex]
2. We multiply the second row of the preview step by -1/4. We obtain the matrix
[tex]\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][/tex]
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
[tex]\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][/tex]
4. We add to row 4 of the preview step, five times the row 3 and obtain the matrix
[tex]\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][/tex]
Using backward substitution we have that
[tex]0x_5=-140[/tex], then [tex]0=-140[/tex] and this is absurd. Then The system is inconsistent.
