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. (4 points)
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?

Note: The variables are a for adults, and c for children.

Answer 1

Let \( a \) represent the cost of an adult ticket and \( c \) represent the cost of a children's ticket.

### a) The equation for the cost of going to a movie on Friday:
On Friday, 10 adults and 25 children went to the movie, and the total cost was $670. The equation can be set up as follows:

\[
10a + 25c = 670
\]

### b) The equation for the cost of going to a movie on Sunday:
On Sunday, 5 adults and 10 children went to the movie, and the total cost was $290. The equation can be established as:

\[
5a + 10c = 290
\]

### c) The best algebraic method to solve the equations:
Since we have a system of two equations, we can use the **substitution method** or the **elimination method** to solve for the variables \( a \) and \( c \). For this case, elimination might be more straightforward since both equations can be adjusted to eliminate one of the variables easily.

### d) Solving the equations:

We have the following two equations:

1. \( 10a + 25c = 670 \) (1)
2. \( 5a + 10c = 290 \) (2)

First, let's simplify equation (2) by dividing everything by 5:

\[
a + 2c = 58 \quad (3)
\]

Now we can solve equation (3) for \( a \):

\[
a = 58 - 2c
\]

Next, substitute \( a \) from equation (3) into equation (1):

\[
10(58 - 2c) + 25c = 670
\]

Distributing the 10:

\[
580 - 20c + 25c = 670
\]

Combine like terms:

\[
580 + 5c = 670
\]

To isolate \( c \), subtract 580 from both sides:

\[
5c = 90
\]

Now divide by 5:

\[
c = 18
\]

Now, substitute \( c = 18 \) back into equation (3) to find \( a \):

\[
a = 58 - 2(18) = 58 - 36 = 22
\]

### Summary:
- The cost of an adult ticket (\( a \)) is **$22**.
- The cost of a children's ticket (\( c \)) is **$18**.

Thus, the final answers are:
- **Cost of adult ticket**: \( \boxed{22} \)
- **Cost of children's ticket**: \( \boxed{18} \)