\(\begin{bmatrix}3&1&-1&1&0\\2&2&4&-6&0\\2&1&3&-1&0\end{bmatrix}\)

Write the augmented matrix of the coefficients and constants

\(\begin{bmatrix}1&0&0&2&0\\0&1&0&-5&0\\0&0&1&0&0\end{bmatrix}\)

Transform the matrix in its reduced row echelon form.

\(x_1=-2x_4\)

\(x_2=5x_4\)

\(x_3=0\)

\(x_4=x_4 \text{ free}\)

Determine the general solution

\(\begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}=x_4 \begin{bmatrix}-2\\ 5 \\ 0 \\ 1 \end{bmatrix}\)

Rewrite the solution in vector form

Write the augmented matrix of the coefficients and constants

\(\begin{bmatrix}1&0&0&2&0\\0&1&0&-5&0\\0&0&1&0&0\end{bmatrix}\)

Transform the matrix in its reduced row echelon form.

\(x_1=-2x_4\)

\(x_2=5x_4\)

\(x_3=0\)

\(x_4=x_4 \text{ free}\)

Determine the general solution

\(\begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}=x_4 \begin{bmatrix}-2\\ 5 \\ 0 \\ 1 \end{bmatrix}\)

Rewrite the solution in vector form