The ticket office makes $5 profit each evening ticket sold and $3 on each matinee ticket sold. The ticket office wants to sell at least 50 evening tickets per week and at least 40 matinee tickets per week. Its goal is to earn at least $250 profit in sales per week. Find all possible combinations of evening tickets and matinee tickets that the ticket office needs to sell to meet their goal.

5e + 3m =250

To find all possible combinations of evening tickets and matinee tickets that meet the ticket office's goal, we can set up a system of inequalities based on the given information.

Let's denote the number of evening tickets as 'x' and the number of matinee tickets as 'y'.

The profit made from selling evening tickets can be calculated as 5x, and the profit made from selling matinee tickets can be calculated as 3y.

Based on the given conditions, we can establish the following inequalities:

1. Number of evening tickets sold: x ≥ 50
- The ticket office wants to sell at least 50 evening tickets per week.

2. Number of matinee tickets sold: y ≥ 40
- The ticket office wants to sell at least 40 matinee tickets per week.

3. Total profit goal: 5x + 3y ≥ 250
- The ticket office aims to earn at least $250 in profit per week.

To solve this system of inequalities, we can plot the equations on a coordinate plane and shade the feasible region that satisfies all the conditions. However, since our goal is to find all possible combinations, we can use a numerical approach by substituting values and checking which combinations meet the conditions.

Let's start by finding the smallest possible value for 'x' and 'y' that satisfy all the conditions:

- Let's assume x = 50 and y = 40. In this case, the total profit would be (5 * 50) + (3 * 40) = 250. Thus, this combination meets the goal.

Now, let's find the next possible combination by increasing the number of matinee tickets:

- Let's assume x = 50 and y = 41. The total profit would be (5 * 50) + (3 * 41) = 253. This combination also meets the goal.

We can continue this process by increasing the value of 'y' while keeping 'x' constant at 50. Here are the combinations that meet the goal:

- x = 50, y = 40 (total profit = $250)
- x = 50, y = 41 (total profit = $253)
- x = 50, y = 42 (total profit = $256)
- x = 50, y = 43 (total profit = $259)
- .
- .
- .
- x = 50, y = 79 (total profit = $445)
- x = 50, y = 80 (total profit = $448)

As you can see, there are 41 possible combinations of evening tickets and matinee tickets that the ticket office can sell to meet their goal of earning at least $250 in profit per week.