Answer:
a. Let's define the variables and create a system of equations or inequalities:
Let:
- S be the number of student tickets sold.
- A be the number of adult tickets sold.
We can set up the following equations and inequalities:
1. The total number of tickets sold should not exceed the theater's capacity of 250 seats:
S + A ≤ 250
2. Each club member has been given a goal of selling a minimum of $140 in student tickets:
7S ≥ 140
3. The total revenue from ticket sales should cover the minimum goal:
7S + 12.50A ≥ 140
b. To graph this system of linear inequalities, you can plot the lines on a graph. In the case of inequalities, you'll shade the areas that represent the valid solutions.
1. S + A ≤ 250: This inequality represents the capacity constraint, and it can be plotted as a line where S + A = 250. Shade the area below this line, as it represents the valid solutions.
2. 7S ≥ 140: This inequality represents the minimum goal for student ticket sales. It can be plotted as a line where 7S = 140. Shade the area above this line, as it represents the valid solutions.
3. 7S + 12.50A ≥ 140: This inequality represents the revenue constraint. It can be plotted as a line where 7S + 12.50A = 140. Shade the area above this line, as it represents the valid solutions.
c. To find a solution, we can look for a point that lies within the shaded areas of all three inequalities. Let's consider the point (20, 230). This point satisfies all the conditions:
1. S + A = 20 + 230 = 250 (within the capacity limit).
2. 7S = 7 * 20 = 140 (meets the minimum goal for student tickets).
3. 7S + 12.50A = 7 * 20 + 12.50 * 230 = 140 + 2875 = 3015 (covers the revenue constraint).
So, the point (20, 230) represents a valid solution, meaning that if 20 student tickets and 230 adult tickets are sold, the theater's capacity is not exceeded, the minimum goal for student tickets is met, and the revenue constraint is satisfied.
Explanation: