A corporate purchasing department bought three types of gift baskets for a client event: Premium (let $x$ be the number of units), Deluxe (let $y$), and Basic (let $z$). Recall the standard method for translating application statements into a system of three linear equations by matching each statement to its correct mathematical representation.

When managing a complex purchase involving three different types of items, you must model the transaction using a system of three linear equations. Arrange the following steps in the correct order to successfully formulate this system based on a project scenario.

A logistics company is ordering three types of specialized equipment: Heavy Duty ($x$), Standard ($y$), and Light ($z$). The unit costs are 20 dollars for Heavy Duty, 12 dollars for Standard, and 10 dollars for Light. The company ordered a total of 350 units for a total cost of 4,650 dollars. They also ensured that the number of Heavy Duty units ordered was exactly the same as the number of Light units.

Which equation in the resulting system correctly models the total cost of the order?

A corporate procurement officer is ordering three types of specialized hardware: Type A ($x$), Type B ($y$), and Type C ($z$). The unit costs are 20 dollars, 12 dollars, and 10 dollars, respectively. The officer ordered a total of 350 units for a total budget of 4,650 dollars. True or False: In the system of equations used to model this purchase, the equation $x + y + z = 4650$ correctly represents the total number of units ordered.

Modeling Variable Relationships

A corporate training coordinator is ordering three types of registration passes for an upcoming leadership conference: Executive passes ($x$), Manager passes ($y$), and Associate passes ($z$). The coordinator is setting up a system of three linear equations to model the purchase. One of the constraints is that the number of Associate passes ($z$) purchased must be exactly the same as the number of Executive passes ($x$). When written in standard form $Ax + By + Cz = D$ with $A = 1$, this constraint is modeled by the equation $x -$ ____ $= 0$. Fill in the blank with the correct variable.

Formulating a Corporate Event Budget