Math Problem Statement

A new restaurant is to contain two-seat tables and four-seat tables. Fire codes limit the restaurant's maximum occupancy to 68 customers. If the owners have hired enough servers to handle 20 tables of customers, how many of each kind of table should they purchase?

Solution

The problem in the image involves finding how many two-seat and four-seat tables a restaurant should purchase using a system of linear equations. Here’s a breakdown of the steps to solve this:

Problem Summary:

  • The restaurant needs two types of tables: two-seat tables and four-seat tables.
  • The total maximum capacity (occupancy) of the restaurant is 68 customers.
  • There are 20 tables in total.
  • We are to find how many of each type of table should be purchased.

Step 1: Define Variables

Let:

  • xx = number of two-seat tables.
  • yy = number of four-seat tables.

Step 2: Set up the system of equations

We have two pieces of information that lead to the following equations:

  1. Equation for the total number of tables: x+y=20x + y = 20 (since there are 20 tables in total).

  2. Equation for the total number of seats: 2x+4y=682x + 4y = 68 (since the total seating capacity is 68, with each two-seat table contributing 2 seats and each four-seat table contributing 4 seats).

Step 3: Solve the system of equations

We now have the system:

x + y &= 20 \\ 2x + 4y &= 68 \end{align*}$$ We can solve this by substitution or elimination. Let’s solve it by substitution. 1. From the first equation $$x + y = 20$$, solve for $$x$$: $$x = 20 - y$$ 2. Substitute $$x = 20 - y$$ into the second equation $$2x + 4y = 68$$: $$2(20 - y) + 4y = 68$$ Simplifying: $$40 - 2y + 4y = 68$$ $$2y = 28$$ $$y = 14$$ 3. Now substitute $$y = 14$$ back into the equation $$x + y = 20$$: $$x + 14 = 20$$ $$x = 6$$ ### Step 4: Solution The restaurant should purchase **6 two-seat tables** and **14 four-seat tables**. --- Would you like more details on this process or have any specific questions? ### 5 Related Questions: 1. How would the solution change if the maximum occupancy was different? 2. How can this system of equations be solved using the elimination method? 3. What if there was a third type of table with six seats? How would that change the system of equations? 4. How would you graph the system of equations, and what would the graph look like? 5. How can systems of equations be used in other real-life situations, like budgeting or scheduling? ### Tip: When solving a system of equations, substitution is particularly useful when one equation is easy to solve for one variable, while elimination works well when both equations are in standard form.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations

Formulas

x + y = 20
2x + 4y = 68

Theorems

Substitution Method
Elimination Method

Suitable Grade Level

Grades 7-9

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