Answer
Infinitely many solutions; dependent equations.
Work Step by Step
The given system of equations is
$x-2y+z=4$
$5x-10y+5z=20$
$-2x+4y-2z=-8$
The augmented matrix is
$\Rightarrow \left[\begin{array}{ccc|c}
1 & -2 & 1& 4\\
5 & -10 & 5& 20 \\
-2&4&-2&-8
\end{array}\right]$
Perform $R_2\rightarrow R_2-5\times R_1$ and $R_3\rightarrow R_3+ 2R_1$.
$\Rightarrow \left[\begin{array}{ccc|c}
1 & -2 & 1& 4\\
5-5(1) & -10-5(-2) & 5-5(1) & 20-5(4) \\
-2+2(1)&4+2(-2)&-2+2(1)&-8+2(4)
\end{array}\right]$
Simplify.
$\Rightarrow \left[\begin{array}{ccc|c}
1 & -2 & 1& 4\\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]$
Use back substitution in the second and third row.
$\Rightarrow x(0)+y(0)+z(0)=0$
$\Rightarrow 0=0$.
Hence, the system of linear equation contains dependent equations and has infinitely many solutions.
