To write the system of equations that corresponds to a given augmented matrix, remember that each row of the matrix represents a single equation. Within each row, every numerical entry to the left of the vertical line represents a coefficient of a given variable in standard form, and the final entry on the right side of the vertical line represents the isolated constant. The vertical line itself replaces the equal sign. For example, a $3 \times 4$ augmented matrix:

$\left[\begin{array}{ccc|c} 4 & -3 & 3 & -1 \\ 1 & 2 & -1 & 2 \\ -2 & -1 & 3 & -4 \end{array}\right]$

translates directly into a system of three equations with three variables:

$\left\{\begin{array}{l} 4x - 3y + 3z = -1 \\ x + 2y - z = 2 \\ -2x - y + 3z = -4 \end{array}\right.$