A warehouse manager uses an augmented matrix to track the labor hours required for two different types of shipping orders. The first column represents the hours for order type $x$, the second column represents the hours for order type $y$, and the final column represents the total hours available in two different departments. The matrix is shown below:

$\left[\begin{array}{cc|c} 6 & 1 & 120 \\ 2 & 4 & 180 \end{array}\right]$

Which system of equations correctly translates this matrix into the warehouse's labor model?

A logistics coordinator for a construction firm uses an augmented matrix to manage the delivery of Gravel ($x$), Sand ($y$), and Cement ($z$) for three different projects. The constants on the right represent the total weight in tons for each project.

$\left[\begin{array}{ccc|c} 10 & -5 & 2 & 50 \\ 1 & 3 & -1 & 20 \\ -4 & 1 & 6 & 80 \end{array}\right]$

Match each project's row to its corresponding linear equation in the system.

A production supervisor uses the following augmented matrix to model the material costs for two different product lines, $x$ and $y$:

$\left[\begin{array}{cc|c} 12 & 18 & 540 \\ 10 & 15 & 450 \end{array}\right]$

True or False: The first row of this matrix correctly translates to the linear equation ${}12x + 18y = 540$.

Translating Augmented Matrices in Business

A project manager uses an augmented matrix to model the cost of labor ($x$), materials ($y$), and overhead ($z$). In any given row of this matrix, the vertical line (or bar) that separates the final column from the others is used to represent the ________ sign in the resulting linear equation.