Answer
Solution set: $\{ ( 98$ , $2t-43$ , $t )$ , $t\in \mathbb{R}\}$
Work Step by Step
Swap equations (the coefficient of x is 1 in Eq.2).
Augmented matrix:
$\left[\begin{array}{lllll}
1 & 2 & -4 & | & 12\\
-2 & -5 & 10 & | & 19
\end{array}\right]\left\{\begin{array}{l}
.\\
\leftarrow 2R_{1}+R_{2}
\end{array}\right.$
$\left[\begin{array}{lllll}
1 & 2 & -4 & | & 12\\
0 & -1 & 2 & | & 43
\end{array}\right]\left\{\begin{array}{l}
.\\
\leftarrow-R_{2}
\end{array}\right.$
$\left[\begin{array}{lllll}
1 & 2 & -4 & | & 12\\
0 & 1 & -2 & | & -43
\end{array}\right]$
The system is consistent and does not have a unique solution.
Let $z=t, t\in \mathbb{R}$.
Substituting into row 2,
$y-2t=-42$
$y=2t-43$
Substituting into row 1,
$x+2(2t-43)-4t=12$
$x+4t-86-4t=12$
$x=12+86$
$x=98$
Solution set: $\{ ( 98$ , $2t-43$ , $t )$ , $t\in \mathbb{R}\}$
