Question

The price of admission at a movie theater is $6 for an adult and $4 for a child. In one day, the movie theater sold 80 tickets and made $420. How many adults and how many children bought tickets to the movie theater that day?

x + y = 80,

6x + 4y = 420

Which is a solution of the system of equations, and what does it represent?
(20, 60); 20 adult tickets and 60 child tickets
(30, 50); 30 adult tickets and 50 child tickets
(40, 40); 40 adult tickets and 40 child tickets
(50, 30); 50 adult tickets and 30 child tickets

Answer 1

To solve the system of equations, we have:

1. \( x + y = 80 \) (Equation 1)
2. \( 6x + 4y = 420 \) (Equation 2)

Where:
- \( x \) is the number of adult tickets sold.
- \( y \) is the number of child tickets sold.

Let's analyze the options given for their validity against these equations:

1. **Testing (20, 60)**:
- \( x = 20 \) and \( y = 60 \)
- From Equation 1: \( 20 + 60 = 80 \) (correct)
- From Equation 2: \( 6(20) + 4(60) = 120 + 240 = 360 \) (not correct)

2. **Testing (30, 50)**:
- \( x = 30 \) and \( y = 50 \)
- From Equation 1: \( 30 + 50 = 80 \) (correct)
- From Equation 2: \( 6(30) + 4(50) = 180 + 200 = 380 \) (not correct)

3. **Testing (40, 40)**:
- \( x = 40 \) and \( y = 40 \)
- From Equation 1: \( 40 + 40 = 80 \) (correct)
- From Equation 2: \( 6(40) + 4(40) = 240 + 160 = 400 \) (not correct)

4. **Testing (50, 30)**:
- \( x = 50 \) and \( y = 30 \)
- From Equation 1: \( 50 + 30 = 80 \) (correct)
- From Equation 2: \( 6(50) + 4(30) = 300 + 120 = 420 \) (correct)

The only solution that satisfies both equations is **(50, 30)**.

This means that **50 adult tickets and 30 child tickets** were sold that day at the movie theater.