Answer
$\mathbf{x}=\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{bmatrix}=x_{3}\begin{bmatrix}5\\-2\\1\\0\end{bmatrix}+x_{4}\begin{bmatrix}7\\6\\0\\1\end{bmatrix}$
Work Step by Step
$\begin{bmatrix}1&0&-5&-7\\0&1&2&-6\end{bmatrix}$.
This is equivalent to the set of equations
$\begin{cases}x_{1}-5x_{3}-7x_{4}=0\\x_{2}+2x_{3}-6x_{4}=0\end{cases}$.
We solve for each variable in terms of the free variables $x_{3}$ and $x_{4}$
$\begin{cases}x_{1}=5x_{3}+7x_{4}\\x_{2}=-2x_{3}+6x_{4}\\x_{3}=x_{3}\\x_{4}=x_{4}\end{cases}$,
which we finally rewrite as a single vector equation.
