A logistics coordinator is analyzing the allocation of three types of medical supplies (represented by $x$, $y$, and $z$) across three regional hospitals. The distribution is governed by the following system of linear equations:

$\left\{\begin{array}{l} 3x + 8y + 2z = -5 \\ 2x + 5y - 3z = 0 \\ x + 2y - 2z = -1 \end{array}\right.$

When using the matrix method to solve for the quantity of each supply, what is the correct augmented matrix the coordinator must first write to represent this system?

An inventory analyst for a retail chain is determining the optimal distribution levels for three product lines ($x$, $y$, and $z$) by solving a system of three linear equations. The analyst chooses to use the matrix method to find the solution. Arrange the following steps in the correct order that the analyst must perform to solve the system using this method.

A logistics manager is solving a system of three linear equations to determine the optimal delivery routes for three different transport vehicles. Match each term used in the matrix method with its correct functional description.

An operations analyst is using the matrix method to solve a system of three linear equations for production scheduling. True or False: The primary goal of applying row operations is to reach row-echelon form, which is characterized by having 1 as each diagonal entry and 0 for all entries directly below those diagonal entries.

A supply chain analyst is using the matrix method to solve a system of three linear equations to balance inventory costs. After defining the augmented matrix, the analyst applies row operations to convert it into row-echelon form. To properly establish this form, the analyst must ensure that the diagonal entries are 1 and that all entries directly below them in their respective columns are ____.