To solve the given system of equations using the echelon matrix form, we can start by expressing the system in augmented matrix form. The equations are:
1. \( 5x_1 + 20x_3 = 50 \)
2. \( 10x_2 + 20x_3 = -30 \)
3. \( 5x_1 - 10x_3 = 80 \)
The augmented matrix for this system is:
\[
\begin{bmatrix}
5 & 0 & 20 & | & 50 \\
0 & 10 & 20 & | & -30 \\
5 & 0 & -10 & | & 80
\end{bmatrix}
\]
We will perform row operations to convert this matrix into