Question

A movie theater charges different prices for children and adults. On Friday, 10 adults and 25 children went to the movie, and it cost $670. On Sunday 5 adults and 10 children went and it cost $290.

a) what equation would represent the cost of going to a movie on Friday?
b) what equation would represent the cost of going to a movie on Sunday?
c) what would be the best algebraic method to solve given how the equations are set up?
d) what is the cost of an adult ticket?

Answer 1

To solve this problem, we can set up variables for the ticket prices and create equations based on the information given.

Let:
- \( x \) = price of an adult ticket
- \( y \) = price of a children's ticket

### a) Equation for the cost of going to a movie on Friday
On Friday, 10 adults and 25 children attended, costing $670. This can be represented by the equation:
\[
10x + 25y = 670
\]

### b) Equation for the cost of going to a movie on Sunday
On Sunday, 5 adults and 10 children attended, costing $290. This can be represented by the equation:
\[
5x + 10y = 290
\]

### c) Best algebraic method to solve the system of equations
The best algebraic method for this set of equations is the **substitution method** or the **elimination method**. In this case, the elimination method might be particularly effective since both equations can be manipulated to eliminate \( y \).

To use elimination, we can simplify the second equation by dividing every term by 5:
\[
x + 2y = 58
\]

Now we have a system:
1. \( 10x + 25y = 670 \)
2. \( x + 2y = 58 \)

### d) Solve for the costs
Now we can use the substitution method. From equation 2:
\[
x + 2y = 58 \implies x = 58 - 2y
\]

Substituting this value of \( x \) into equation 1:
\[
10(58 - 2y) + 25y = 670
\]
\[
580 - 20y + 25y = 670
\]
\[
580 + 5y = 670
\]
\[
5y = 670 - 580
\]
\[
5y = 90 \implies y = 18
\]

Now that we have \( y \), substitute it back to find \( x \):
\[
x = 58 - 2(18)
\]
\[
x = 58 - 36 = 22
\]

Thus, the cost of an adult ticket is:
\[
\boxed{22}
\]

The cost of a children's ticket is:
\[
\boxed{18}
\]

In conclusion, an adult ticket costs $22 and a children's ticket costs $18.