Answer
The system is inconsistent (has no solution).
Solution set =$ \emptyset.$
Work Step by Step
Method: Gauss-Jordan. Row reduce the augmented matrix:
$\left[\begin{array}{rrr|r}
{5}&{-1}&{4}&{2}\\
{-1}&{5}&{-4}&{3}\\
{7}&{13}&{-4}&{17}\end{array}\right]\qquad\left\{\begin{array}{l}
R_{1}=-r_{2}.\\
R_{2}=r_{1}.\\
.
\end{array}\right\}\rightarrow$
$\rightarrow\left[\begin{array}{rrr|r}
{1}&{-5}&{4}&{-3}\\
{5}&{-1}&{4}&{2}\\
{7}&{13}&{-4}&{17}\end{array}\right]\qquad\left\{\begin{array}{l}
.\\
R_{2}=r_{2}-5r_{1}.\\
R_{3}=r_{3}-7r_{1}.
\end{array}\right\}\rightarrow$
$\left[\begin{array}{ccc|c}
{1}&{-5}&{4}&{-3}\\
{0}&{24}&{-16}&{17}\\
{0}&{48}&{-32}&{38}\end{array}\right]\qquad\left\{\begin{array}{l}
.\\
.\\
R_{3}=r_{3}-2r_{2}.
\end{array}\right\}\rightarrow$
$\left[\begin{array}{ccc|c}
{1}&{-5}&{4}&{-3}\\
{0}&{24}&{-16}&{17}\\
{0}&{0}&{0}&{4}\end{array}\right]$
the last row represents the equation$ \quad 0=4,$
which is never satisfied.
The system is inconsistent (has no solution).
Solution set =$ \emptyset.$
