An analyst is using Cramer's Rule to solve a system of three linear equations. Match each algebraic component used in the process to its correct functional description.

An inventory manager at a logistics company is determining the exact number of three different types of shipping containers ($x$, $y$, and $z$) to load onto a cargo plane based on weight, volume, and budget constraints. They have modeled these constraints as a system of three linear equations and will solve it using Cramer's Rule. Arrange the procedural steps the manager must follow to find the solution in the correct chronological order.

A project manager at a construction firm is using Cramer's Rule to solve a system of three linear equations to allocate budget across three different departments ($x$, $y$, and $z$). If $D$ represents the determinant of the coefficient matrix and $D_y$ represents the determinant calculated by replacing the $y$-column with the constants, which formula correctly identifies the budget for department $y$?

Auditing a Cramer's Rule Calculation

A resource manager is using Cramer's Rule to solve a system of three linear equations that determines the allocation of labor across three project phases, $x$, $y$, and $z$. To calculate the determinant $D_y$, the manager must replace the column containing the $y$-coefficients in the main coefficient matrix with the constant terms from the equations.

An inventory management specialist for a retail distributor is using a resource allocation model to determine the optimal stock levels of three eco-friendly packaging materials: $x$, $y$, and $z$.

The system of linear equations representing the daily warehouse constraints is:

$\left\{\begin{array}{l} 3x - 5y + 4z = 5 5x + 2y + z = 0 2x + 3y - 2z = 3 \end{array}\right.$

To begin solving this system using Cramer's Rule, the specialist constructs the coefficient matrix and evaluates the main determinant $D$.

Recall the value of the main determinant $D$ for this specific system.

The main determinant $D$ is ____.