1. The problem is to convert the given augmented matrix $$\begin{bmatrix}4 & -2 & -1 & 1 \\ -3 & 0 & -4 & -7\end{bmatrix}$$ into an equivalent system of linear equations using variables $x_1$, $x_2$, and $x_3$. 2. Recall that each row of the augmented matrix corresponds to one linear equation. The first three entries in each row are coefficients of $x_1$, $x_2$, and $x_3$ respectively, and the last entry is the constant term on the right side of the equation. 3. For the first row: coefficients are 4, -2, -1 and constant is 1. So the first equation is: $$4x_1 - 2x_2 - x_3 = 1$$ 4. For the second row: coefficients are -3, 0, -4 and constant is -7. So the second equation is: $$-3x_1 + 0x_2 - 4x_3 = -7$$ which simplifies to $$-3x_1 - 4x_3 = -7$$ 5. Therefore, the equivalent linear system is: $$\begin{cases} 4x_1 - 2x_2 - x_3 = 1 \\ -3x_1 - 4x_3 = -7 \end{cases}$$