Answer
System has no solutions. It is inconsistent.
Work Step by Step
The procedure is shown with the matrix notation for simpler understanding.
$x_{2} + 4x_{3} = -5 $
$x_{1} + 3x_{2} + 5x_{3} = -2$
$3x_{1} + 7x_{2} + 7x_{3} = 6$
This can be depicted in the augmented matrix notation as follows :
$\begin{bmatrix}
0 & 1 & 4 & -5 \\
1 & 3 & 5 & -2 \\
3 & 7 & 7 & 6
\end{bmatrix}$
To obtain an $x_{1}$ in the first equation,interchange rows 1 and 2:
$\begin{bmatrix}
1 & 3 & 5 & -2 \\
0 & 1 & 4 & -5 \\
3 & 7 & 7 & 6
\end{bmatrix}$
To eliminate the $3x_{1}$ term in the third equation, add $-3$ times row 1 to row 3:
$\begin{bmatrix}
1 & 3 & 5 & -2 \\
0 & 1 & 4 & -5 \\
0 & -2 & -8 & 12
\end{bmatrix}$
Next we use the $x_{2}$ term in the second equation to eliminate the $-2x_{2}$ term from the third equation. Add $2$ times
row 2 to row 3:
$\begin{bmatrix}
1 & 3 & 5 & -2 \\
0 & 1 & 4 & -5 \\
0 & 0 & 0 & 2
\end{bmatrix}$
Now the augmented matrix is in a triangular form. For interpreting it, we go back to the equation notation:
$x_{1} + 3x_{2} + 5x_{3} = -2$
$x_{2} + 4x_{3} = -5 $
$0 = 2$
The equation $0 = 2$ is a short form of $0x_{1} + 0x_{2} + 0x_{3} = 2$.
This is a contradiction, and no values of $ x_{1}, x_{2} $, and $x_{3} $ can satisfy this equation.
Since the final and original equations have the same solution set, we conclude that the given system of equations
$x_{2} + 4x_{3} = -5 $
$x_{1} + 3x_{2} + 5x_{3} = -2$
$3x_{1} + 7x_{2} + 7x_{3} = 6$
has no solutions. It is inconsistent.
