In a professional setting, mathematical models are often used to solve complex resource allocation problems. Match each component of an augmented matrix with the specific element of the linear system it represents.

In a professional resource management scenario, a project lead uses a system of linear equations to track departmental spending. Before these equations can be correctly organized into an augmented matrix for analysis, what specific form must each equation be written in?

A logistics coordinator is using an augmented matrix to represent a system of linear equations for route optimization. In this matrix, the variable letters (such as $x$ and $y$) are explicitly written as entries within the columns along with their numerical coefficients.

An operations manager is preparing a budget model using a system of linear equations to allocate resources between two departments. To use an automated solving tool, the manager must first organize the system into an augmented matrix. Arrange the following steps in the correct order to construct the matrix as described in the training manual.

Matrix Structure in Financial Modeling