Answered

Use a system of linear equations to solve the following problem.



A new restaurant is to contain two-seat tables and four-seat tables. Fire codes limit the restaurant's maximum occupancy to 58 customers. If the owners have hired enough servers to handle 17 tables of customers, how many of each kind of table should they purchase?



Write a system of linear equations using the given information. Choose the correct answer below.


A$\left\{\begin{array}{l}2x + 4y = 58 \\ x + y = 17\end{array}\right.$
B$\left\{\begin{array}{l}x - y = 58 \\ 2x - 4y = 17\end{array}\right.$
C$\left\{\begin{array}{l}x + y = 58 \\ 2x + 4y = 17\end{array}\right.$
D$\left\{\begin{array}{l}2x - 4y = 58 \\ \dots\end{array}\right.$

Verified Answer :

  • Define variables \(x\) (two-seat tables) and \(y\) (four-seat tables).
  • Create a system of equations based on capacity (\(2x + 4y = 58\)) and table count (\(x + y = 17\)).
  • The system is \(\begin{cases} 2x + 4y = 58 \ x + y = 17 \end{cases}\), which is option A.
  • Final answer: \(\boxed{\text{A}}\).

Explanation

  1. Defining Variables and Translating Conditions

To determine the configuration of tables in the restaurant, we start by assigning variables to the quantities we need to find. Let \(x\) represent the number of two-seat tables and \(y\) represent the number of four-seat tables. We are given two conditions related to these variables: the total occupancy and the total number of tables. These conditions will allow us to form a system of two linear equations.

  1. Constructing the System of Equations

The first condition is that the restaurant's maximum occupancy is 58 customers. A two-seat table accommodates 2 customers, and a four-seat table accommodates 4 customers. Therefore, the total number of customers is given by the expression \(2x + 4y\). Setting this equal to the occupancy limit, we obtain the equation:
\[2x + 4y = 58\]
The second condition is that there are 17 tables in total. This gives us the simple sum of the number of tables:
\[x + y = 17\]
Combining these two equations gives the system:
\[\begin{cases} 2x + 4y = 58 \ x + y = 17 \end{cases}\]
This system corresponds to option A.

  1. Solving and Verifying the System

To verify our result, we can solve the system of equations. From the second equation, we express \(x\) in terms of \(y\):
\[x = 17 - y\]
Substituting this into the first equation:
\[2(17 - y) + 4y = 58\]
\[34 - 2y + 4y = 58\]
\[34 + 2y = 58\]
Subtracting 34 from both sides:
\[2y = 24\]
Dividing by 2:
\[y = 12\]
Now, finding \(x\):
\[x = 17 - 12 = 5\]
The restaurant should purchase 5 two-seat tables and 12 four-seat tables. This confirms that our system of equations is correct, and the solution leads to option A.

Examples

This type of problem is essential in fields like hospitality management and logistics. For instance, a manager organizing an event or a restaurant layout must balance limited resources (like total number of tables) with capacity constraints to optimize space and serve all customers effectively. Understanding how to model these real-world constraints as a system of linear equations is a foundational skill in business analytics and operational research.

Related Questions