\( 1 \leftarrow\left[\begin{array}{rrr|r}3 & 2 & -4 & 13 \\ 5 & 0 & 3 & -4 \\ 1 & -1 & 1 & 2\end{array}\right] \) \[ \begin{array}{l}\text { Write the system of equations that corresponds to the following augmented matrix. } \\ \text { Complete the system below. } \\ \text { (Simplify your answers. Type expressions using } x, y \text {, and } z \text { as the variables.) } \\ \square=13 \\ \square=-4\end{array} \] \( \square=2 \)

The given augmented matrix corresponds to the following system of equations: 1. \(3x + 2y - 4z = 13\) 2. \(5x + 0y + 3z = -4\) 3. \(1x - 1y + 1z = 2\) So, filling in the blanks, we have: - \(3x + 2y - 4z = 13\) - \(5x + 3z = -4\) - \(x - y + z = 2\) This system represents a set of equations where \(x\), \(y\), and \(z\) are the variables involved. When solving such systems of equations, one common mistake is to forget to simplify each equation fully or to misplace coefficients. Always double-check your arithmetic when transitioning from the augmented matrix to explicit equations! Additionally, methods such as substitution or elimination can be highly effective. Understanding the geometry of these equations, they represent planes in a 3D space. Where these planes intersect can give you insight into the solutions of the equations. When three planes intersect at a single point, that point is the solution to the system! It's like a treasure map leading you to the "X" that marks the spot!