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The problem asks to solve how many two-seat and four-seat tables a new restaurant should purchase, using a system of linear equations. The following details are provided:

- The restaurant has a total of 19 tables (both two-seat and four-seat combined).
- The maximum occupancy is 68 customers.
- Two types of tables: two-seat tables and four-seat tables.

### Step-by-step solution:

Let:
- \( x \) = the number of two-seat tables.
- \( y \) = the number of four-seat tables.

From the problem, we can form two equations:

1. The total number of tables is 19:
   \[
   x + y = 19
   \]
   
2. The total seating capacity is 68 (each two-seat table seats 2 people, and each four-seat table seats 4 people):
   \[
   2x + 4y = 68
   \]

### Solving the system:

**Step 1: Simplify the second equation**
   \[
   2x + 4y = 68 \implies x + 2y = 34
   \]
   
Now we have the system:
\[
\begin{align*}
x + y &= 19 \\
x + 2y &= 34
\end{align*}
\]

**Step 2: Eliminate \( x \)**

Subtract the first equation from the second:
\[
(x + 2y) - (x + y) = 34 - 19
\]
\[
x + 2y - x - y = 15 \implies y = 15
\]

**Step 3: Substitute \( y = 15 \) into the first equation**
\[
x + 15 = 19 \implies x = 19 - 15 = 4
\]

### Conclusion:
- The restaurant should purchase **4 two-seat tables** and **15 four-seat tables**.

Let me know if you want further details or have any questions.

Here are some related questions:
1. How would the answer change if the occupancy limit was different?
2. What if the restaurant wanted a different proportion of two-seat to four-seat tables?
3. How can we represent this problem graphically?
4. What other methods could be used to solve this system of equations?
5. How would this system change if there were a third type of table?

**Tip:** Always double-check your system of equations for accuracy before solving to avoid errors in calculations.

A new restaurant is to contain two-seat tables and four-seat tables. Fire codes limit the restaurant's maximum occupancy to 68 customers. The owners have hired enough servers to handle 19 tables of customers. How many of each kind of table should they purchase?

Learn how to solve a system of linear equations to determine how many two-seat and four-seat tables a restaurant should purchase. This problem involves algebraic concepts, such as systems of equations, and is suitable for students in Grades 8-10.

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