As an inventory analyst, your logistics software outputs daily shipping data as an augmented matrix. The first three columns represent the quantities of three box types: standard ($x$), deluxe ($y$), and premium ($z$). The vertical line separates these coefficients from the total weight in pounds (the constants).

Given the following augmented matrix:

$\left[\begin{array}{ccc|c} 3 & 1 & 2 & 50 \\ 1 & 0 & 4 & 35 \\ 0 & 2 & -1 & 10 \end{array}\right]$

Which of the following systems of linear equations correctly represents this matrix?

In a logistics application, your shipping software represents resource constraints using an augmented matrix. If the first row of the matrix is ${}5 \quad 0 \quad -2 \quad | \quad 12$ and the variables represent quantities of standard boxes ($x$), deluxe boxes ($y$), and premium boxes ($z$), write the resulting linear equation for that row.

Equation: ____

A logistics coordinator is verifying the output of a warehouse management system that tracks inventory levels across three different storage zones ($x$, $y$, and $z$). The system outputs these levels as an augmented matrix. Match each row of the matrix to its corresponding linear equation to ensure the reporting module is functioning correctly.

A logistics analyst is translating a system-generated augmented matrix into linear equations to verify warehouse capacity. If a row in the matrix is $\left[\begin{array}{ccc|c} 4 & 1 & -2 & 100 \end{array}\right]$ and the variables are $x$, $y$, and $z$, the analyst correctly derives the equation ${}4x + y - 2z = 0$.

You are developing a training manual for new data analysts at a logistics firm. The manual must explain how to verify 'Constraint Matrices'—augmented matrices where each row represents a resource limitation. To ensure the analysts understand the conversion logic, you need them to document the standard procedure for translating a matrix row into a linear equation using variables $x$, $y$, and $z$.

Arrange the following steps in the correct order to translate a row of an augmented matrix into its corresponding linear equation.